Path-dependent Hamilton-Jacobi equations in infinite dimensions

TL;DR

This paper develops a new theoretical framework for solving fully nonlinear path-dependent PDEs in infinite-dimensional Hilbert spaces, establishing well-posedness and enabling applications in optimal control and differential games.

Contribution

It introduces minimax and viscosity solution concepts for infinite-dimensional path-dependent PDEs and proves well-posedness, combining techniques to handle the lack of compactness.

Findings

Established existence and uniqueness of solutions.

Extended dynamic programming to infinite-dimensional delay equations.

Enabled analysis of differential games in Hilbert spaces.

Abstract

We propose notions of minimax and viscosity solutions for a class of fully nonlinear path-dependent PDEs with nonlinear, monotone, and coercive operators on Hilbert space. Our main result is well-posedness (existence, uniqueness, and stability) for minimax solutions. A particular novelty is a suitable combination of minimax and viscosity solution techniques in the proof of the comparison principle. One of the main difficulties, the lack of compactness in infinite-dimensional Hilbert spaces, is circumvented by working with suitable compact subsets of our path space. As an application, our theory makes it possible to employ the dynamic programming approach to study optimal control problems for a fairly general class of (delay) evolution equations in the variational framework. Furthermore, differential games associated to such evolution equations can be investigated following the Krasovskii-Subbotin approach similarly as in finite dimensions.

Equations

$$\begin{array}{rl}& {\partial }_{t}u(t,x)-⟨A(t,x(t)),{\partial }_{x}u(t,x)⟩+F(t,x,{\partial }_{x}u(t,x))=0,\\ & (t,x)\in [0,T)\times C([0,T],H).\end{array}$$

$${\int }_{t_0}^T\ell (t,\{x(s,\cdot ){\}}_{s\le t},a(t,\cdot ))dt+h(\{x(t,\cdot ){\}}_{t\le T})$$

$$\frac{\partial x}{\partial t}(t,\xi )$$

$$=f(t,\xi ,\{x(s,\xi ){\}}_{s\le t},a(t,\xi )),$$

$$D^{\beta }x(t,\xi )$$

$$x(t,\xi )$$

$$F(t,x,z)=\tilde{F}(t,x(t),z),h(x)=\tilde{h}(x(T)),(t,x,z)\in [0,T]\times C([0,T],H)\times H,$$

$$v_t(t,x)-⟨Ax,v_x(t,x)⟩+G(t,x,v_x(t,x))=0,(t,x)\in (0,T)\times H,$$

$$\underset{\delta \downarrow 0}{\underline{\lim }}\text{ (or limδ↓0) }\frac{1}{\delta }{\int }_t^{t+\delta }⟨A(s,x(s)),{\partial }_{x}u(s,x)⟩ds,$$

$$\left|F(t,x,z)-F(t,x,\tilde{z})\right|\le L\Vert z-\tilde{z}{\Vert }_H,(t,x)\in [0,T]\times C([0,T],H).$$

$${\tilde{\mathcal{X}}}^L(t_0,x_0)$$

$$x^{\prime }(t)+A(t,x(t))=f^x(t),\left|f^x(t)\right|\le L\text{ a.e. on (t0,T)}\},$$

$$\begin{array}{rl}u(t,x)-u(t_0,x_0)& ={\int }_{t_0}^t{\partial }_{t}u(s,x)+⟨x^{\prime }(s),{\partial }_{x}u(s,x)⟩ds\\ & ={\int }_{t_0}^t{\partial }_{t}u(s,x)+⟨-A(s,x(s))+f^x(s),{\partial }_{x}u(s,x)⟩ds\\ & ={\int }_{t_0}^t-F(s,x,{\partial }_{x}u(s,x))+⟨f^x(s),{\partial }_{x}u(s,x)⟩ds.\end{array}$$

$$u(t,x)-u(t_0,x_0)={\int }_{t_0}^t-F(s,x,z)+(f^x(s),z)ds,t\in [t_0,T].$$

$$\left|F(t,x,z)-F(t,\tilde{x},z)\right|\le L(1+\Vert z{\Vert }_H)\underset{s\le t}{\sup }\Vert x(s)-y(s){\Vert }_H.$$

$$\begin{array}{rl}& {\partial }_{t}w(t,x,y)-⟨A(t,x(t)),{\partial }_{x}w(t,x,y)⟩-⟨A(t,y(t)),{\partial }_{y}w(t,x,y)⟩\\ & +F(t,x,{\partial }_{x}w(t,x,y))-F(t,y,-{\partial }_{y}w(t,x,y))=0,\\ & (t,x,y)\in [0,T)\times C([0,T],H)\times C([0,T],H).\end{array}$$

$$u_0(t,x):=\inf \{u(t,x):\text{ u “improper" minimax supersolution}\}$$

$$\frac{\partial x}{\partial t}(t,\xi )-{\Delta }_{\xi }x(t,\xi )$$

$$x(t,\xi )$$

$$x(0,\xi )$$

$$\frac{d}{dt}{\int }_{G}x(t,\xi )\phi (\xi )d\xi$$

$$={\int }_{G}a(t,\xi )\phi (\xi )d\xi$$

$$x(t,\xi )$$

$$x(0,\xi )$$

$$x^{\prime }(t)+Ax(t)$$

$$x(t)$$

$$x\in C([0,T],H)\cap L^2(t_0,T;V)\text{ and }{x}^{\prime }\in L^2(t_0,T;V^{\ast }).$$

$$⟨a(t),v⟩$$

$$⟨Ax,v⟩$$

$$\Vert x{\Vert }_{W_{pq}(t_0,T)}:=\Vert x{\Vert }_{L^p(t_0,T;V)}+\Vert x^{\prime }{\Vert }_{L^q(t_0,T;V^{\ast })},$$

$$(x(s),y(s))-(x(t),y(t))={\int }_t^s⟨x^{\prime }(r),y(r)⟩+⟨y^{\prime }(r),x(r)⟩dr.$$

$$\Vert x{\Vert }_{\infty }:=\underset{t\in [0,T]}{\sup }\left|x(t)\right|.$$

$${\mathbf{d}}_{\infty }((t,x),(s,y))$$

$$C(S)$$

$$\mathrm{USC}(S)$$

$$\mathrm{LSC}(S)$$

$$⟨A(t,x)-A(t,y),x-y⟩\ge 0.$$

Full text

The research of the first author was supported in part by the National Science Foundation under grant DMS-1613170

1. 1 Introduction

2. 1.1 Related research

3. 1.2 Our approach and the main difficulties

4. 1.3 Notions, main results, and methodology in a simplified setting

5. 1.4 Organization of the rest of the paper

6. 2 Setting and preliminary results

7. 2.1 Path space and related topologies

8. 2.2 The operator $A$, related trajectory spaces, and evolution equations

9. 2.3 Chain rule and standard derivatives

10. 2.4 Functional chain rule and path derivatives

11. 3 Path-dependent Hamilton-Jacobi equations

12. 3.1 Minimax solutions: Global version of the notion

13. 3.2 Minimax solutions: Infinitesimal version of the notion

14. 4 Comparison

15. 5 Existence and uniqueness

16. 6 Stability

17. 7 Applications to optimal control

18. 8 Applications to differential games

19. A Properties of solution sets of evolution equations

20. B Other notions of solutions

21. B.1 Classical solutions

22. B.2 Viscosity solutions

1. Introduction

Let $V\subseteq H\subseteq V^{\ast}$ be a Gelfand triple, i.e., $V$ is a separable reflexive Banach space with a continuous and dense embedding into a Hilbert space $H$. Moreover, this embedding is assumed to be compact. We study fully nonlinear so-called path-dependent PDEs (PPDEs) of the form

[TABLE]

Here, $A(t,\cdot):V\to V^{\ast}$, $t\in[0,T]$, are nonlinear, monotone, coercive operators. The derivatives $\partial_{t}u$ and $\partial_{x}u$ are certain path derivatives on the path space $C([0,T],H)$. A definition will be provided later. Note that they are not Fréchet derivatives. Let us also mention that any kind of solution $u$ of (1.1) should at least be non-anticipating, i.e., for every $x$, $y\in C([0,T],H)$, whenever $x=y$ on $[0,t]$, we have $u(t,x)=u(t,y)$.

A particular example of (1.1) is the Bellman equation associated to distributed control problems for divergence-form quasi-linear parabolic PDEs of order $2m$ on some bounded domain $G\subset{\mathbb{R}}^{n}$ with smooth boundary $\partial G$, e.g., the problem of minimizing a cost functional

[TABLE]

over some class of admissible controls $a=a(t,\xi)$ subject to $x=x^{t_{0},x_{0},a}$ solving

[TABLE]

where $\alpha$ and $\beta$ are multi-indices, $Dx=(D^{\beta}x)_{\left|{\beta}\right|\leq m}$, $D^{\alpha}$ and $D^{\beta}$ are partial derivatives with respect to $\xi$, and $\ell$, $h$, $f$, and $A_{\alpha}$ are appropriate functions (see [ZeidlerIIB, Section 30.4] for details regarding $A_{\alpha}$). Note that we allow delays in the data of our problem with the exception of the differential operator.

The objective of this paper is to establish wellposedness for the terminal-value problem related to (1.1) under an appropriate notion of solution. Our problem is set up in path-dependent framework, i.e., a solution $u$ depends at any fixed time $t$ on a“path segment” $\{x(s)\}_{s\leq t}$, which can be understood as history until time $t$. Formally, the usual state space $[0,T]\times H$ is replaced by $[0,T]\times C([0,T],H)$ (or a subset) and a solution is required to be non-anticipating. The original motivation to work in such a framework comes from problems involving delays in areas such as optimal control, differential games, and mathematical finance. In this work however, the path-dependent approach provides a new methodology for infinite-dimensional PDEs, which is of relevance even if our data are not path-dependent, i.e., when the function $F$ and a terminal datum $h$ are of the form

[TABLE]

for some functions $\tilde{F}:[0,T]\times H\times H\to{\mathbb{R}}$ and $\tilde{h}:H\to{\mathbb{R}}$. The reason is that, thanks to our additional structural assumptions for $A$ in contrast to the related works [Tataru92JMAA, Tataru94JDE_simplified] and [CrandallLionsVI], we can identify suitable compact subspaces of $C([0,T],H)$ and this allows us to avoid the use of perturbed optimization based on Ekeland’s variational principle. We want to remark that our approach crucially depends on the coercivity of $A$ and thus for different or more general situations perturbed optimization is an indispensable tool. Nevertheless, we want to advocate our approach when it is applicable as it often simplifies the analysis and many methods from the finite-dimensional theory can be easily carried over to the infinite-dimensional case.

1.1. Related research

Pioneering work on Hamilton-Jacobi equations on Hilbert space has been done by Barbu and Da Prato (see, e.g., their monograph [BarbuDaPrato83book]).

Viscosity solutions for PDEs in infinite dimensions of the form

[TABLE]

(and their stationary counterparts), where $A$ is an unbounded operator, have been investigated by Crandall and Lions in [CrandallLionsI, CrandallLionsII, CrandallLionsIII] for the case $A=0$, in [CrandallLionsIV, CrandallLionsV, CrandallLionsVII] for linear $A$, and in [CrandallLionsVI] for nonlinear $A$. The research on such equations with nonlinear $A$ has been started earlier by Tataru [Tataru92JMAA, Tataru92AMO] (see also [Tataru94JDE_simplified] for subsequent work). For the case of $A$ being the subdifferential of a convex function, we refer to Ishii [Ishii92JFA_HJ_Hilbert] and Shimano [Shimano02AMO].

Cannarsa, Gozzi, and Soner [CannarsaGozziSoner91AMO] studied Hamilton-Jacobi equations on Hilbert space (here with $A=0$) related to optimal control problems with exit time and state constraints. Motivated by boundary control problems for parabolic PDEs, Cannarsa, Gozzi, and Soner [CannarsaGozziSoner93JFA] and Cannarsa and Tessitore [CannarsaTessitore96SICON] studied Hamilton-Jacobi equations, where the Hamiltonian $G$ involves fractional powers of $A$.

We also want to mention the works of Lions [Lions89_Zakai] and Gozzi and Świ\polhkech [GSJFA00] as well as the works of Gozzi, Sritharan, and Świ\polhkech [GSS_2D_NS, GSS_2D_SNS], respectively, where Bellman equations associated to control problems for variational solutions of the Zakai equation and the Navier-Stokes equation, respectively, are studied. Furthermore, we want to single out Soner’s paper [Soner88], which influenced the literature on viscosity solutions for fully nonlinear 1st and 2nd order PPDEs via [Lukoyanov07_viscosity].

For the viscosity theory of second-order PDEs in infinite dimensions, we refer to the works of Lions [Lions88Acta, Lions89JFA], of Ishii [Ishii93CommPDE], of Świ\polhkech, [Swiech94CommPDE] and of Gozzi, Rouy, and Świ\polhkech [GRS00SICON] besides the already mentioned papers [Lions89_Zakai], [GSJFA00], [GSS_2D_SNS].

For further works about viscosity solutions for PDEs in infinite dimensions, one can consult the bibliographies of [FabbriThesis] and [FabbriGozziSwiech].

Results on minimax solutions in infinite dimensions can be found in the work [ClarkeLedyaev94TAMS] by Clarke and Ledyaev on mean value inequalities. We refer also to Carja [Carja00JOTA, Carja12SICON] regarding the related notion of contingent solution.

Fully nonlinear 1st order PPDEs have been studied by Lukoyanov in the minimax [Lukoyanov01minimax, Lukoyanov03a, Lukoyanov03b] as well as in the viscosity solution framework [Lukoyanov07_viscosity]. In those works, so-called coinvariant derivatives by Kim [KimBook] were used. The investigation of 2nd order PPDEs has been motivated much by Peng’s ICM plenary talk [PengICM] and by Dupire’s functional Itô calculus [dupirefunctional] (see also subsequent work by Cont and Fournié [ContFournie10_JFA]). A viscosity solution approach has been successfully initiated by Ekren, Keller, Touzi, and Zhang [EKTZ]. In a similar spirit, Isaacs equations are studied by Pham and Zhang [PhamZhang14], fully nonlinear equations by Ekren, Touzi, and Zhang [ETZ_I, ETZ_II], non-local equations by Keller [Keller16], elliptic equations by Ren [Ren16], obstacle problems by Ekren [Ekren2017], and degenerate second-order equations by Ren, Touzi, and Zhang [RTZ17]. In contrast to the papers mentioned first, the latter work covers also the first-order case.

A class of semilinear 2nd order PPDEs with a linear unbounded operator on Hilbert space has been studied by Cosso, Federico, Gozzi, Rosestolato, and Touzi [cosso2015path]. Their work is closely related to the semigroup framework, in particular to mild solutions of stochastic evolution equations involving linear operators, whereas the present work is closely related to variational solutions of (deterministic) evolution equations involving nonlinear operators.

For optimal control problems in the variational framework, we refer to Hu and Papageorgiou [HPHandbookII, Chapter IV] and the references therein.

Differential games in infinite dimensions have been investigated by Kocan, Soravia, and Świ\polhkech [KocanSoraviaSwiech97JMAA], by Kocan and Soravia [KocanSoravia00SICON], by Shaiju [Shaiju03], by Ghosh and Shaiju [GhoshShaiju04], by Ramaswamy and Shaiju [RamaswamyShaiju09], by Nowakowska and Nowakowski [NowakowskaNowakowski11], by Świ\polhkech [Swiech11games], by Banks and Shuhua [BanksShuhua12], and by Vlasenko, Rutkas, and Chikriĭ [VlasenkoRutkasChikrii15]. Stochastic differential games in infinite dimensions and the related Isaacs equations have been studied by Fleming and Nisio [FlemingNisio93], by Nisio [Nisio93, Nisio94, Nisio96, Nisio98, Nisio99a, Nisio99b], and by Świ\polhkech [Swiech02NA].

Let us remark that the literature on optimal control problems for PDEs is immense (see, e.g, the recent survey article [ControlPDE_survey] or the monographs [AhmedTeo], [BensoussanEtAl07], [FabbriGozziSwiech], [Fattorini], [Fursikov], [LasieckaTriggiani_I], [LasieckaTriggiani_II] [LiYong], [LionsJL_control], [Tiba], and [Troltzsch]; note that this list is not exhaustive).

Finally, let us conclude this literature review with mentioning two approaches how to treat optimal control problems with delays via the dynamic programming approach. A very common approach in the literature is to rewrite a controlled delay differential equation as an ordinary differential equation without delays but with values in a Hilbert space $H$ such as, e.g., $H=L^{2}(-\tau,0;{\mathbb{R}}^{d})$. This would then naturally lead to an investigation of Bellman equations on domains of the form $[0,T]\times H$ or $H$; see, e.g, the works of Federico [Federico11FinStoch], of Federico, Goldys, and Gozzi [FedericoGoldysGozzi10SICON, FedericoGoldysGozzi11SICON], of Federico and Tacconi [FedericoTacconi14SICON], of Fuhrman, Masiero, and Tessitore [FuhrmanFedericaTessitore10SICON], of Zhou and Liu [ZhouLiu12JDE], and of Zhou and Zhang [ZhouZhang11IJC]. In the PPDE approach however, the controlled delay differential equations is not rewritten as an (infinite-dimensional) evolution equation but is considered as a “path-dependent” differential equation, i.e., a differential equation of the form $x^{\prime}(t)=f(t,\{x(s)\}_{s\leq t}))$ in the deterministic case. This view would lead to a path-dependent Bellman equation. Furthermore, solutions of those PPDE should be non-anticipating. A particular feature of this work is that it also allows the study of optimal control problems associated to PDEs with delays.

1.2. Our approach and the main difficulties

We propose an appropriate adaptation of the notion of minimax solutions in the sense of Subbotin [Subbotin_book] and Lukoyanov [Lukoyanov03a] as generalized solution for (1.1). Under this notion, we establish well-posedness. Moreover, minimax solutions admit similarly as viscosity solutions an infinitesimal characterization, which is of importance in control theory, in particular, to establish verification theorems (cf. [FabbriGozziSwiech10JConvex]) and for the synthesis of optimal feedback controls (see, e.g., [Frankowska89AMO], [BardiCapuzzoDolcetta, Chapter III], or [Subbotina2006, Chapter II]).

The main difficulties are the lack of compactness in infinite-dimensional spaces and the interpretation of the duality pairing $\langle A(t,x(t)),\partial_{x}u(t,x)\rangle$ in (1.1) due to incompatibilities regarding the involved spaces: We have $A(t,x(t))\in V^{\ast}$ and thus $\partial_{x}u(t,x)$ should belong to $V$ but, in general, we only have $\partial_{x}u(t,x)\in H$ and $H$ is in most relevant situations a strict superset of $V$. The first difficulty is overcome by the path-dependent approach itself and the use of minimax solutions as we are able to identify suitable compact subsets of the path space $C([0,T],H)$. The second difficulty actually disappears in our formulation of minimax solutions. In the formulation of viscosity solutions the problem is resolved by replacing $\langle A(t,x(t)),\partial_{x}u(t,x)\rangle$ with a term of the form

[TABLE]

in which case the space incompatibilities disappear as, for a.e. $s\in(t,T)$, we have $x(s)\in V$ and thus $\partial_{x}u(s,x)\in V$ (cf. Definition 2.14 (i) and Definition 2.16 (i)) for suitable trajectories $x$.

Without rendering our problem irrelevant or trivial, we also consider smaller domains than $[0,T]\times C([0,T],H)$ such as (compact!) sets of the form $[0,T]\times\Omega^{L}$, $L\geq 0$, and the (locally compact!) set $[0,T]\times\Omega$, where $\Omega:=\cup_{L\geq 0}\Omega^{L}$. The sets $\Omega^{L}$ can be thought of as spaces of realizable trajectories (or possible histories) corresponding to a suitable family of control problems. From this point of view, it strikes us at least as debatable why a larger class of trajectories should be considered.

In the proof of the comparison principle for viscosity solutions of 1st order PPDEs in finite dimensions by Lukoyanov [Lukoyanov07_viscosity] (see also [ETZ_I, Theorem 8.4]) time and space variables are doubled, i.e., functions of the form $u(t,x)-v(s,y)+\text{ ``penalty term"}$ are used. However, this approach seems to be problematic in our setting as it is not clear how one can then exploit the monotonicity of the operator $A$ (even if $A$ is time-independent!). To avoid this issue, we double only the space variable following Ishii [IshiiRemark] and Crandall and Lions [CrandallLionsVI]. To obtain a comparison principle for viscosity solutions we need then to establish a so-called doubling theorem, i.e., a statement of the following form:

If $u$ is a viscosity subsolution and $v$ is a viscosity supersolution, then $w$ defined by $w(t,x,y):=u(t,x)-v(t,y)$ is a viscosity subsolution of some “doubled equation.”

Due to the particular nature of our path derivatives we have been unable to prove such a statement. (The problem is that in the path-dependent case the time derivative does not depend on time alone, which can also be seen in alternative, explicit definitions of path derivatives such as in [dupirefunctional].) To get around this obstacle, we utilize the stronger notion of minimax solutions. By doing so, we can prove a doubling theorem but now with $u$ being a minimax subsolution and $v$ being a minimax supersolution. However, $w$ will still be a viscosity subsolution. Our doubling theorem will then be used to prove a comparison principle for minimax solutions.

1.3. Notions, main results, and methodology in a simplified setting

Suppose that there exists a constant $L\geq 0$ such that, for every $z$, $\tilde{z}\in H$,

[TABLE]

Also note that hypothesis H($A$) from Section 2.2 is tacitly assumed to be satisfied.

In the finite-dimensional minimax solution theory, spaces of Lipschitz continuous functions with common Lipschitz constant play an important role. Their counterpart in our work are the trajectory spaces

[TABLE]

where $(t_{0},x_{0})\in[0,T)\times C([0,T],H)$. (For a precise definition, see Remark A.4; cf. also Definition 2.6.) These spaces are compact, connected, and satisfy a certain continuous-dependence type property (cf. Proposition 2.12). The last property is needed to prove existence for solutions of (1.1).

Fix $x_{\ast}\in H$. Here, our state space is $[0,T]\times\tilde{\Omega}$, where $\tilde{\Omega}=\tilde{\mathcal{X}}^{L}(0,x_{\ast})$. In the main part of this paper, we use a larger set as state space. In particular, this set does not depend on the Lipschitz constant $L$. However, our notions will then become slightly more complicated.

Assume for a moment that $u$ is smooth in some sense and satisfies (1.1). Then, for every $x\in\tilde{\mathcal{X}}^{L}(t_{0},x_{0})$, we should have

[TABLE]

(A precise definition of the path derivatives can be found in Section 2.4.) Because of (1.2), one can show that (1.3) also holds if the terms $\partial_{x}u(s,x)$, $s\in[t_{0},t]$, are replaced by an arbitrary $z\in H$. This motivates the following definition.

Definition 1.1**.**

A function $u\in C([0,T]\times\tilde{\Omega})$ is a minimax solution of (1.1) if, for every $(t_{0},x_{0},z)\in[0,T)\times\tilde{\Omega}\times H$, there exists an $x\in\tilde{\mathcal{X}}^{L}(t_{0},x_{0})$ such that

[TABLE]

Similarly, minimax semisolutions (i.e., sub- and supersolutions) can be defined.

The next statement is a weaker version of our main result.

Theorem 1.2**.**

Let $h:C([0,T],H)\to{\mathbb{R}}$ be continuous. Let $F$ be continuous and suppose that besides (1.2), for every $t\in[0,T]$, $x$, $y\in\tilde{\Omega}$, and $z\in H$,

[TABLE]

Then there exists a unique minimax solution $u$ of (1.1) that satisfies $u(T,\cdot)=h$.

To prove this result, we employ viscosity subsolutions (see Definition 4.1) of the so-called “doubled equation”

[TABLE]

Our methodology can then be summarized in the following five-step scheme:

(i) (“Doubled comparison”). Let $w$ be a viscosity subsolution of (1.4). Then $w(T,x,x)\leq 0$ for all $x\in\tilde{\Omega}$ implies $w(t,x,x)\leq 0$ for all $(t,x)\in[0,T]\times\tilde{\Omega}$.

(ii) (Doubling theorem). If $u$ is a minimax sub-, and $v$ a minimax supersolution, then $w$ defined by $w(t,x,y):=u(t,x)-v(t,y)$ is a viscosity subsolution of (1.4).

(iii) (Comparison). “(i)+(ii)” yields comparison for minimax semisolutions.

(iv) (“Perron”). Define a function $u_{0}$ by

\displaystyle u_{0}(t,x):=\inf\{u(t,x):\text{ u ``improper" minimax supersolution}\}

and consider its semi-continuous envelopes $u_{0}^{-}$ and $u_{0}^{+}$. By definition $u_{0}^{-}\leq u_{0}\leq u_{0}^{+}$. Moreover, $u_{0}^{-}$ is an l.s.c. minimax super- and $u_{0}^{+}$ is a u.s.c. minimax subsolution.

(v) (Existence). “Comparison + Perron” yields $u_{0}^{+}\leq u_{0}^{-}$, i.e., $u_{0}^{+}=u_{0}=u_{0}^{-}$.

Next, we present a special case of the distributed control problem mentioned at the beginning of this introduction to illustrate how those problems fit into our abstract setting.

Example 1.3**.**

This example has been borrowed from [ZeidlerIIA, Chapter 23]. For the sake of simplicity, we consider the problem of distributed control for the heat equation, i.e., $x=x^{t_{0},x_{0},a}$ is supposed to be a weak solution of

[TABLE]

In other words, for every $\varphi\in C_{0}^{\infty}(G)$ (the space of smooth functions with bounded support),

[TABLE]

An abstract formulation of this Cauchy-Dirichlet problem is

[TABLE]

and we require that a solution $x$ together with its generalized derivative $x^{\prime}$ satisfies

[TABLE]

Here, $H=L^{2}(G)$, $V=W_{0}^{1,2}(G)$ is the closure of $C_{0}^{\infty}(G)$ under $\lVert{\cdot}\rVert_{W^{1,2}(G)}$, $V^{\ast}$ is the dual of $V$, and the right-hand side $a:[t_{0},T]\to V^{\ast}$ and the operator $A:V\to V^{\ast}$, $x\mapsto Ax$, are defined by

[TABLE]

1.4. Organization of the rest of the paper

In Section 2, we introduce the setting for this work including most of the notation and the standing assumptions. Furthermore, some of the preliminary results concerning evolution equations will be proven. Moreover, a definition for the already mentioned path derivatives will be provided. In Section 3, we state our main object of study, namely the terminal-value problem related to (1.1), together with the standing hypotheses for the data. Then we present our notion of minimax solutions for this problem and prove basic results regarding this notion. In Section 4, we prove a comparison principle for minimax solutions using viscosity solution techniques. In Section 5, Perron’s method is used to establish existence for minimax solutions. In Section 6, a stability result for minimax solutions is proven. In Section 7, we apply the dynamic programming approach to optimal control problems associated with nonlinear evolution equations, establish (local) Lipschitz continuity for the corresponding value function in space as well as in time, and show that the value function is a minimax solution of the corresponding Bellman equation. The proof of the last statement also uses a combination of minimax and viscosity solution techniques, which is new to our best knowledge. In Section 8, we adapt the Krasovskiĭ-Subbotin approach for differential games to our setting. Under the Isaacs condition, it is shown that our differential games have value and the value function is a minimax solution of the corresponding Isaacs equation. Appendix A contains the proofs for some of the crucial results about solution sets of evolution equations. In Appendix B, we present first a notion of classical solution and discuss their relation to minimax solutions. Then we present a notion of viscosity solutions for which we have existence but cannot prove uniqueness.

2. Setting and preliminary results

Let $V\subseteq H\subseteq V^{\ast}$ be a Gelfand triple, i.e., $V$ is a separable reflexive Banach space with a continuous and dense embedding into a Hilbert space $H$. Denote by $\lVert{\cdot}\rVert$, $\left|{\cdot}\right|$, and $\lVert{\cdot}\rVert_{\ast}$ the norms on $V$, $H$, and $V^{\ast}$. (With slight abuse of notation, we also denote by $\left|{\cdot}\right|$ the Euclidean norm on ${\mathbb{R}}^{n}$.) Denote by $\langle\cdot,\cdot\rangle$ the duality pairing between $V$ and its dual $V^{\ast}$, i.e., $\langle x^{\ast},x\rangle=x^{\ast}(x)$, $x^{\ast}\in V^{\ast}$, $x\in V$. Denote by $(\cdot,\cdot)$ the inner product on $H$. Moreover, using appropriate identifications, we have $\langle x,y\rangle=(x,y)$ for every $x\in H$ and $y\in V$.

Assumption 2.1**.**

The embedding from $V$ into $H$ is compact.

Fix $p\geq 2$ and $q>1$ with $p^{-1}+q^{-1}=1$. Given $t_{0}\in[0,T)$, let $W_{pq}(t_{0},T)$ be the space of all $x\in L^{p}(t_{0},T;V)$ possessing a generalized derivative $x^{\prime}\in L^{q}(t_{0},T;V^{\ast})$. Equipped with the norm $\lVert{\cdot}\rVert_{W_{pq}(t_{0},T)}$ defined by

[TABLE]

the space $W_{pq}(t_{0},T)$ becomes a Banach space.

Remark 2.2**.**

The space $W_{pq}(t_{0},T)$ is continuously embedded into $C([t_{0},T],H)$ (see, e.g., [ZeidlerIIA, Proposition 23.23 (ii)]).

Remark 2.3**.**

As a consequence of Assumption 2.1, the embedding from $W_{pq}(t_{0},T)$ into $L^{p}(t_{0},T;H)$ is compact (see [JLLions1969, Théorème I.12.1]). This is crucial for the proof of Proposition 2.10 below. However, also note that, according to [Migorski95PAMS], it is, in general, not true that the embedding from $W_{pq}(0,T)$ into $C([0,T],H)$ is compact.

We shall frequently use the following result (see, e.g., [ZeidlerIIA, Proposition 23.23 (iv)]).

Proposition 2.4** (Integration-by-parts).**

Let $t_{0}\in[0,T)$. If $x$, $y\in W_{pq}(t_{0},T)$ and $t_{0}\leq t\leq s\leq T$, then

[TABLE]

On the left-hand side, $x$ and $y$ are to be understood as elements of $C([t_{0},T],H)$.

2.1. Path space and related topologies

The domain for solutions to our path-dependent PDE will of the form $[0,T]\times\tilde{\Omega}$, where $T>0$ and $\tilde{\Omega}$ is a subset of the path space $C([0,T],H)$, which we equip with the uniform norm $\lVert{\cdot}\rVert_{\infty}$ defined by

[TABLE]

If not mentioned otherwise, any subset of $C([0,T],H)$ is to be understood as a metric space with respect to $\lVert{\cdot}\rVert_{\infty}$.

Definition 2.5**.**

A function $u$ defined on some subset $S$ of $[0,T]\times C([0,T],H)$ is non-anticipating if $u(t,x)=u(t,y)$ whenever $x=y$ on $[0,t]$, $(t,x)$, $(t,y)\in S$.

To ensure that (semi-) continuous functions are non-anticipating, we equip $[0,T]\times\Omega$ with the pseudo-metric $\mathbf{d}_{\infty}$ defined by

[TABLE]

Here, $a\wedge b:=\min\{a,b\}$. Given a non-empty subset $S$ of $[0,T]\times C([0,T],H)$, we shall employ the following function spaces:

[TABLE]

Clearly, the elements of those function spaces are non-anticipating.

2.2. The operator $A$, related trajectory spaces, and evolution equations

Throughout this work, we fix an operator $A:[0,T]\times V\to V^{\ast}$ and assume that the following hypotheses hold.

H($A$): (i) The mappings $t\mapsto\langle A(t,x),v\rangle$, $v\in V$, are measurable.

(ii) Monotonicity: For a.e. $t\in(0,T)$ and every $x$, $y\in V$,

[TABLE]

(iii) Hemicontinuity: For a.e. $t\in(0,T)$ and every $x$, $y$, $v\in V$, the mappings $s\mapsto\langle A(t,x+sy),v\rangle$, $[0,1]\to{\mathbb{R}}$, are continuous.

(iv) Boundedness: There exist a function $a_{1}\in L^{q}(0,T)$ and a number $c_{1}\geq 0$ such that, for every $x\in V$ and a.e. $t\in(0,T)$,

[TABLE]

(v) Coercivity: There exists a number $c_{2}>0$ such that, for every $x\in V$ and a.e. $t\in(0,T)$,

[TABLE]

The following trajectory spaces are ubiquitous in the present work. They are the natural analogues to the spaces of Lipschitz continuous functions with common Lipschitz constant in the finite-dimensional theory.

Definition 2.6**.**

Given $L\geq 0$ and $(t_{0},x_{0})\in[0,T)\times C([0,T],H)$, put

[TABLE]

Remark 2.7**.**

Given $x\in\mathcal{X}^{L}(t_{0},x_{0})$, the function ${\mathbf{x}}$ is actually the unique element of $W_{pq}(t_{0},T)$ that coincides with $x$ a.e. on $(t_{0},T)$ (in particular, uniqueness of ${\mathbf{x}}^{\prime}$ in this context follows from [ZeidlerIIA, Proposition 23.20 (a)]) and also $f^{x}$ is uniquely determined in $L^{2}(t_{0},T;H)$ (actually, even in $L^{q}(t_{0},T;V^{\ast})$), which can be seen by setting $f^{x}(t)={\mathbf{x}}^{\prime}(t)+A(t,{\mathbf{x}}(t))$.

Fix $x_{\ast}\in H$. Considering $x_{\ast}$ as a constant function on $[0,T]$, we set

[TABLE]

Those path spaces will serve as our primary domains for solutions of our path-dependent PDE. In particular, $\Omega$ can be used as universal path space for all Hamiltonians $F$ satisfying the hypothesis H($F$) below, whereas the spaces $\Omega^{L}$ can only be used for Hamiltonians that are $L$-Lipschitz in the gradient (in the sense of condition H($F$)(ii) with $L_{0}$ replaced by $L$).

Remark 2.8**.**

The sets $\mathcal{X}^{L}(t_{0},x_{0})$ are not stable under stopping, i.e., from $x\in\mathcal{X}^{L}(t_{0},x_{0})$ one cannot deduce that $x(\cdot\wedge t)\in\mathcal{X}^{L}(t_{0},x_{0})$ if $t<T$. (Recall that $s\wedge t=\min\{s,t\}$.)

Remark 2.9**.**

On $\mathcal{X}^{L}(t_{0},x_{0})$, the pseudo-metrics

[TABLE]

are topologically (but not pseudo-metrically) equivalent. The non-trivial direction follows from (A.9).

The next proposition is crucial. Note that compactness of the sets $\mathcal{X}^{L}(t_{0},x_{0})$ (see, e.g., [TT1999NoDEA] for a more general result in the context of evolution inclusions) ultimately relies on $V$ being compactly embedded in $H$ (Assumption 2.1) and its ramifications (see, in particular, Remark 2.3). Due to its importance and for the sake of completeness, a proof will be provided in the appendix. Let us also mention that compactness of solution sets of monotone evolution inclusions is addressed in [Migorski96Edinburgh_compactness], [PS97AMO], and [HPHandbookII].

Proposition 2.10**.**

The sets $\mathcal{X}^{L}(t_{0},x_{0})$ are non-empty, compact, and path-connected in $C([0,T],H)$.

Proof.

Non-emptiness follows from [ZeidlerIIB, Theorem 30.A, p. 771], compactness from Lemma A.2 and Lemma A.3, and path-connectedness from Lemma A.5. ∎

As the next example shows, coercivity of $A$ is necessary for $\mathcal{X}^{L}(t_{0},x_{0})$ to be compact in $C([0,T],H)$.

Example 2.11** ($A=0$).**

Let $H=\{v\in L^{2}(0,2\pi):v(0)=v(2\pi)=0\}$ and $V=H^{1}_{0}(0,2\pi)$, which is the completion of $C^{\infty}_{0}(0,2\pi)$ under $\lVert{\cdot}\rVert_{H^{1}(0,2\pi)}$. Consider the sequence $(x_{n})_{n}$ in $W_{2,2}(0,1)$ defined by

[TABLE]

Then $x^{\prime}_{n}(t)=f_{n}(t)$, $x_{n}(0)=0$. Clearly, $x_{n}\in\mathcal{X}^{1}(0,0)$ for every $n\in{\mathbb{N}}$. Since $x_{n}(1)\xrightarrow{w}0$ in $H$ but $\left|{x_{n}(1)}\right|=1$ for every $n\in{\mathbb{N}}$, the sequence $(x_{n})_{n}$ cannot have a subsequence that converges in $C([0,T],H)$. Also note that $\sup_{n}\lVert{x_{n}}\rVert_{C([0,1],H)}\leq 1$ but $\sup_{n}\lVert{x_{n}}\rVert_{L^{2}(0,1;V)}=\infty$, i.e., $(x_{n})_{n}$ is not bounded in $W_{2,2}(0,1)$.

The following continuous dependence type result will play a central role in carrying out Perron’s method to prove our main existence result (see the proof of Lemma 5.6). It is also needed to prove stability.

Proposition 2.12**.**

*Let $L\geq\tilde{L}\geq 0$. Let $(t_{n},x_{n})\to(t_{0},x_{0})$ in $[0,T)\times\Omega^{L}$. Consider a sequence $(\tilde{x}_{n})_{n}$ in $\Omega^{L}$ with $\tilde{x}_{n}\in\mathcal{X}^{\tilde{L}}(t_{n},x_{n})$, $n\in{\mathbb{N}}$. Then $(\tilde{x}_{n})_{n}$ has a converging subsequence with a limit in $\mathcal{X}^{\tilde{L}}(t_{0},x_{0})$. *

Proof.

By Proposition 2.10, $(\tilde{x}_{n})_{n}$ has a subsequence, which, with slight abuse of notation is still denoted by $(\tilde{x}_{n})_{n}$, that converges to some $\tilde{x}_{0}\in\Omega^{L}$. To show that $\tilde{x}_{0}\in\mathcal{X}^{\tilde{L}}(t_{0},x_{0})$ we need to verify that

[TABLE]

First, we show (2.1). To this end, let $t\in[0,t_{0}]$. Since, for every $n\in{\mathbb{N}}$,

[TABLE]

and $\left|{\tilde{x}_{0}(t)-\tilde{x}_{n}(t)}\right|+\left|{x_{n}(t)-x(t)}\right|\to 0$ as $n\to\infty$, we only need to deal with the term $\left|{\tilde{x}_{n}(t)-x_{n}(t)}\right|$. Note that in the case $t_{n}\geq t$, this term equals [math] because $\tilde{x}_{n}\in\mathcal{X}^{\tilde{L}}(t_{n},x_{n})$. Thus, we shall assume from now on that $t_{n}<t$. By the a-priori estimates from Lemma A.1 applied to $\Omega^{L}=\mathcal{X}^{L}(0,x_{\ast})$, there exists a $C\geq 0$ independent from $n$ such that

[TABLE]

Letting $n\to\infty$ yields (2.1).

Next, we show (2.2). Keep in mind that $f^{\tilde{x}_{n}}\xrightarrow{w}f^{\tilde{x}}$ in $L^{2}(0,T;H)$ and that

[TABLE]

Without loss of generality, we distinguish between the following two cases.

Case 1: Let $t_{n}\uparrow t_{0}$. In this case, one can proceed as in the proof of Lemma A.3.

Case 2: Let $t_{n}\downarrow t_{0}$. Following the lines of the proof of of Lemma A.3, we can infer that

[TABLE]

Let $t\in(f^{\tilde{x}})^{-1}(\tilde{E}_{1})\cap(t_{0},T)=:E_{1}$. Then there exists an $n_{0}\in{\mathbb{N}}$ such that, for all $n\geq n_{0}$, we have $t_{0}\leq t_{n}<t$. Moreover, suppose that, in addition, we have

[TABLE]

Next, let h\in\text{ w$-$\varlimsup}\{f^{\tilde{x}_{n}}(t)\}_{n\in{\mathbb{N}}}. Then there exists a subsequence $(f^{\tilde{x}_{n_{k}}}(t))_{k}$ of $(f^{\tilde{x}_{n}}(t))_{n}$ that converges weakly to $h$ in $H$. Thus, by [ZeidlerIIA, Proposition 21.23 (c), p. 258] (Banach-Steinhaus), by (2.3), and since $\tilde{x}_{n}\to\tilde{x}$ in $C([0,T],H)$,

[TABLE]

Since $(E_{1}\cap E_{2})^{c}$ is a null set, we can deduce that (2.2) holds. ∎

We shall also need the following result concerning evolution equations with more general right-hand side than in [ZeidlerIIB].

Proposition 2.13**.**

Let $(t_{0},x_{0})\in[0,T)\times C([0,T],H)$ and $L\geq 0$. Consider the initial-value problem

[TABLE]

where $f:[t_{0},T]\times C([0,T],H)\to H$ is non-anticipating and measurable.

(i) If, for a.e. $t\in(t_{0},T)$, the mapping $x\mapsto f(t,x)$, $C([0,T],H)\to H$, is continuous, and if, for a.e $t\in(0,T)$ and every $x\in C([0,T],H)$,

[TABLE]

then (2.4) has a solution in $\mathcal{X}^{L}(t_{0},x_{0})$.

(ii) If, in addition to the assumptions in (i), there exists an $l\geq 0$ such that, for a.e. $t\in(t_{0},T)$ and every $x$, $y\in\mathcal{X}^{L}(t_{0},x_{0})$,

[TABLE]

then (2.4) has a unique solution in $\mathcal{X}^{L}(t_{0},x_{0})$.

Proof.

(i) First, let $M\in{\mathbb{N}}$ and define $f^{M}:[t_{0},T]\times C([0,T],H)\to H$ by

[TABLE]

Next, consider the set

[TABLE]

and define a mapping

[TABLE]

by

[TABLE]

This is possible according to [ZeidlerIIB, Theorem 30.A, p. 771]. Note that $\tilde{\mathcal{X}}^{L(1+M)}(t_{0},x_{0})$ is compact (the proof is similar and actually slightly easier than the proof of Proposition 2.10). Moreover, $\Gamma^{M}$ is continuous as $x_{n}\to x$ in $C([0,T],H)$ implies, for $y_{n}:=\Gamma x_{n}$, $y:=\Gamma x$, and a.e. $t\in(t_{0},T)$,

[TABLE]

thanks to the monotonicity of $A$ and thus (cf. part (ii) of this proof)

[TABLE]

by Proposition 2.4 and the dominated convergence theorem. Here, $C$ is the constant from standard a-priori estimates similarly to the ones of Lemma A.1. Thus, by Schauder’s fixed point theorem, there exists a fixed point $x^{M}$ of $\Gamma^{M}$. Moreover, note that $x^{M}\in\mathcal{X}^{L}(t_{0},x_{0})$ because, for a.e. $t\in(t_{0},T)$,

[TABLE]

Hence, we can apply Proposition 2.10 to deduce that $(x^{M})_{M}$ has a subsequence, which we still denote by $(x^{M})_{M}$, that converges to some $x\in\mathcal{X}^{L}(t_{0},x_{0})$. Furthermore, by Lemma A.2, $f^{M}(\cdot,x^{M})\xrightarrow{w}f^{x}$ in $L^{2}(t_{0},T;H)$. Since, for sufficiently large $M$, $\lVert{x^{M}}\rVert_{\infty}\leq\lVert{x}\rVert_{\infty}+1\leq M$ and $f^{M}(\cdot,x^{M})=f(\cdot,x^{M})$, it follows that $f^{M}(t,x^{M})\to f(t,x)$ for a.e. $t\in(t_{0},T)$. Therefore, we can conclude that $f^{x}=f(\cdot,x)$, i.e., we have proven existence.

(ii) Let $x$ and $y$ be solutions of (2.4) in $\mathcal{X}^{L}(t_{0},x_{0})$. Put $z:=x-y$. Then, for a.e. $t\in(t_{0},T)$, by monotonicity of $A$,

[TABLE]

where $m(t):=\sup_{s\leq t}\left|{z(s)}\right|$. Using integration-by-parts (Proposition 2.4) yields $\frac{1}{2}\frac{d}{dt}\left|{z(t)}\right|^{2}\leq l\,m(t)^{2}$, which we integrate to obtain {\color[rgb]{0,0,0}\left|{z(t)}\right|^{2}\leq\int_{t_{0}}^{t}2l\,m(s)^{2}\,ds}. By Gronwall’s inequality, $z=0$. This concludes the proof. ∎

2.3. Chain rule and standard derivatives

This section serves as motivation for the path derivatives in the following section. The results here will also be used in the treatment of differential games.

A corresponding result in the second-order case in a setting close to ours can be found in [Nisio_book, Section 5.4.3]. In particular, slight variants of the conditions ($F_{0}$) and ($F_{1}$) from [Nisio_book, pp. 186 and 190] found their way into the following definition.

Definition 2.14**.**

Denote by $\mathcal{C}_{V}^{1,1,1}([0,T]\times{\mathbb{R}}\times H)$ the set of all functions $\psi=\psi(t,\xi,x)\in C^{1,1,1}([0,T]\times{\mathbb{R}}\times H)$ such that the following holds:

(i) For a.e. $t\in(0,T)$ and every $\xi\in{\mathbb{R}}$,

[TABLE]

(ii) The functions $\left|{\psi_{t}}\right|$ and $\left|{\psi_{\xi}}\right|$ are bounded on bounded subsets of $[0,T]\times{\mathbb{R}}\times H$.

(iii) There exists a constant $L_{\psi}\geq 0$ such that, for every $x_{1}$, $x_{2}\in V$,

[TABLE]

Proposition 2.15**.**

If $\psi\in\mathcal{C}_{V}^{1,1,1}([0,T]\times{\mathbb{R}}\times H)$, then

[TABLE]

for every $\xi\in C^{1}([0,T])$, $\mathbf{x}\in W_{pq}(0,T)$, $x\in C([0,T],H)$ with ${\mathbf{x}}=x$ a.e. on $(0,T)$, and $t_{1}$, $t_{2}\in[0,T]$ with $t_{1}\leq t_{2}$.

Proof.

Fix $\xi\in C^{1}([0,T])$, $\mathbf{x}\in W_{pq}(0,T)$, and $x\in C([0,T],H)$ with ${\mathbf{x}}=x$ a.e. on $(0,T)$. We use a standard approximation argument (cf. [HPHandbookII, Proof of Theorem 1.12 (d)]). Since $C^{1}([0,T],V)$ is dense in $W_{pq}(0,T)$, we can choose a sequence $(x_{n})_{n}$ in $C^{1}([0,T],V)$ that converges to $\mathbf{x}$ in $W_{pq}(0,T)$ and thus also to $x$ in $C([0,T],H)$. Let $0\leq t_{1}\leq t_{2}\leq T$. Note that, for every $n\in{\mathbb{N}}$,

[TABLE]

Without loss of generality, $\lVert{x_{n}}\rVert_{\infty}\leq\lVert{x}\rVert_{\infty}+1$ for every $n\in{\mathbb{N}}$. Thus, by condition (ii) of Definition 2.14,

[TABLE]

Also, without loss of generality, $\lVert{x_{n}}\rVert_{L^{p}(0,T;V)}\leq\lVert{x}\rVert_{L^{p}(0,T;V)}+1$ for every $n\in{\mathbb{N}}$. Thus, by condition (ii) of Definition 2.14, there exists a constant $C>0$ such that,

[TABLE]

It is now straight-forward to complete the proof. ∎

2.4. Functional chain rule and path derivatives

Fix $n\in{\mathbb{N}}$. We shall use the spaces

[TABLE]

Definition 2.16**.**

Let $t_{\ast}\in[0,T)$. Denote by $\mathcal{C}_{V}^{1,1}([t_{\ast},T]\times C([0,T],H)^{n})$ the set of all functions $\varphi\in C([t_{\ast},T]\times C([0,T],H)^{n})$ for which there exist functions, called path derivatives,

[TABLE]

such that the following holds:

1. (i)

For every $x=(x^{j})_{j=1}^{n}\in C([0,T],H)^{n}$, and $i\in\{1,\ldots,n\}$,

[TABLE] 2. (ii)

For every $t_{0}\in[t_{\ast},T)$, every $x_{0}\in C([0,T],H)^{n}$, every $\mathbf{x}=(\mathbf{x}^{j})_{j=1}^{n}\in W_{pq}(t_{0},T)^{n}$, every $x=(x^{j})_{j=1}^{n}\in C([0,T],H)^{n}$ with $x=x_{0}$ on $[0,t_{0}]$ and $x={\mathbf{x}}$ a.e. on $(t_{0},T]$, and every $t\in[t_{0},T]$, the functional chain rule holds, i.e.,

[TABLE]

Remark 2.17**.**

The path derivatives $\partial_{t}\varphi$ and $\partial_{x^{j}}\varphi$ are uniquely determined on $[t_{\ast},T)$. Indeed, for every $t_{0}\in[t_{\ast},T)$, $x_{0}\in C([0,T],H)^{n}$ and $j\in\{1,\ldots,n\}$,

[TABLE]

and, if additionally $x_{0}^{m}(t_{0})\in V$ for every $m\in\{1,\ldots,n\}$, then

[TABLE]

Here, $\{e_{i}\}_{i}$ an orthonormal basis of $H$ with elements in $V$ and the paths $x_{i}=(x_{i}^{m})_{m=1}^{n}\in C([0,T],H)^{n}$, $i\in{\mathbb{N}}$, are defined by

[TABLE]

Note that the restrictions $x_{i}^{m}|_{[t_{0},T]}$ are $V$-valued and belong to $W_{pq}(t_{0},T)$. Since $V$ is dense in $H$ and $\partial_{x^{j}}\varphi$ is continuous, we can deduce uniqueness for $\partial_{x^{j}}\varphi(t_{0},x_{0})$ also for the general case (i.e., without requiring $x_{0}^{m}(t_{0})\in V$ for all $m$).

Example 2.18**.**

Let $n=1$ and $\varphi(t,x)=\left|{x(t)}\right|^{2}+\int_{0}^{t}\psi(s,x)\,ds$, $t\in[0,T]$, $x\in C([0,T],H)$. If $\psi\in C([0,T]\times C([0,T],H))$, then, by the integration-by-parts formula (Proposition 2.4), $\varphi\in\mathcal{C}_{V}^{1,1}([0,T]\times C([0,T],H))$ with $\partial_{t}\varphi(t,x)=\psi(t,x)$ and $\partial_{x}\varphi(t,x)=2x(t)$.

3. Path-dependent Hamilton-Jacobi equations

We consider the terminal-value problem

[TABLE]

For the functions $h$ and $F$, we impose the following standing hypotheses.

H($h$): The function $h:C([0,T],H)\to{\mathbb{R}}$ is continuous.

H($F$): The function $F:[0,T]\times C([0,T],H)\times H\to{\mathbb{R}}$ satisfies the following:

(i) $F$ is continuous.

(ii) There exists an $L_{0}\geq 0$ such that, for every $t\in[0,T]$, every $x\in C([0,T],H)$, and every $z$, $\tilde{z}\in H$,

[TABLE]

(iii) For every $L\geq 0$ and $(t_{0},x_{0})\in[0,T)\times C([0,T],H)$, there exists a modulus of continuity $m_{L,t_{0},x_{0}}$ such that, for every ${\varepsilon}>0$, every $t\in[t_{0},T]$, and every $x$, $y\in\mathcal{X}^{L}(t_{0},x_{0})$,

[TABLE]

Remark 3.1**.**

(i) It is more common to impose the stronger condition

[TABLE]

where the modulus $\tilde{m}_{L,x_{0},t_{0}}$ is independent from $z\in H$. For further discussion regarding corresponding conditions in the non-path-dependent case, we refer to [CrandallLions86NA, Remark 1].

(ii) In contrast to the conditions in [Lukoyanov03a] or [RTZ17], which only allow distributed delays, we can immediately cover concentrated delays, i.e., $F$ can be of the form $F(t,x,z)=\tilde{F}(t,x(t/2),z)$ or $F(t,x,z)=\tilde{F}(t,{\mathbf{1}}_{[1,T]}(t)\,x(t-1),z)$ for some function $\tilde{F}$. (See also Remark 2.9.) However, note that [Lukoyanov03a] has been extended in [Lukoyanov10a] and [Lukoyanov10b] to cover besides distributed delays also a finite number of concentrated delays. Let us also remark that our treatment of differential games in Section 8 will still require distributed delays.

3.1. Minimax solutions: Global version of the notion

The approach in this and the next subsection is strongly motivated by [Lukoyanov03a], where the finite-dimensional case is treated.

Definition 3.2**.**

Let $u:[0,T]\times\tilde{\Omega}\to{\mathbb{R}}$ for some subset $\tilde{\Omega}$ of $C([0,T],H)$.

(i) Let $L\geq 0$. We call $u$ a minimax $L$-subsolution (resp. $L$-supersolution, resp. $L$-solution) of (TVP) on $[0,T]\times\tilde{\Omega}$ if

• •

$u\in\mathrm{USC}([0,T]\times\tilde{\Omega})$ (resp. $\mathrm{LSC}([0,T]\times\tilde{\Omega})$, resp. $C([0,T]\times\tilde{\Omega})$),

• •

u(T,\cdot)\leq\text{ (resp.~{}\geq$, resp.~{}$=) }h on $\tilde{\Omega}$,

• •

for every $(t_{0},x_{0},z)\in[0,T)\times\tilde{\Omega}\times H$, there exists an $x\in\mathcal{X}^{L}(t_{0},x_{0})\cap\tilde{\Omega}$ such that, for every $t\in[t_{0},T]$,

[TABLE]

(ii) We call $u$ a minimax subsolution (resp. supersolution, resp. solution) of (TVP) on $[0,T]\times\tilde{\Omega}$ if $u$ is a minimax $L$-subsolution (resp. $L$-supersolution, resp. $L$-solution) of (TVP) on $[0,T]\times\tilde{\Omega}$ for some $L\geq 0$.

Remark 3.3**.**

Let $\tilde{\Omega}\subseteq C([0,T],H)$. If $u$ is a minimax $L_{1}$-supersolution of (TVP) on $[0,T]\times\tilde{\Omega}$ for some $L_{1}\geq 0$, then $u$ is a minimax $L$-supersolution of (TVP) on $[0,T]\times\tilde{\Omega}$ for all $L\geq L_{1}$.

Remark 3.4**.**

Fix $L\geq 0$. Let $\Omega^{L}\subseteq\Omega_{1}\subseteq\Omega_{2}\subseteq C([0,T],H)$. If $u$ is a minimax $L$-solution of (TVP) on $[0,T]\times\Omega_{2}$, then its restriction $u|_{[0,T]\times\Omega_{1}}$ is a minimax $L$-solution of (TVP) on $[0,T]\times\Omega_{1}$. However, a minimax solution of (TVP) on $[0,T]\times\Omega_{2}$ is not necessarily a minimax solution of (TVP) on $[0,T]\times\Omega_{1}$.

Remark 3.5**.**

Let $L\geq 0$. A minimax solution of (TVP) on $[0,T]\times\Omega^{L}$ is a minimax $L$-solution of (TVP) on $[0,T]\times\Omega^{L}$.

Lemma 3.6**.**

Fix $L\geq 0$ and a set $\tilde{\Omega}$ with $\Omega^{L}\subseteq\tilde{\Omega}\subseteq C([0,T],H)$. Consider a function $u\in\mathrm{USC}([0,T]\times\tilde{\Omega})$ (resp. $\mathrm{LSC}([0,T]\times\tilde{\Omega})$, resp. $C([0,T]\times\tilde{\Omega})$). Let $(t_{0},x_{0},z)\in[0,T)\times\tilde{\Omega}\times H$ Then the following statements are equivalent:

(i) There exists an $x\in\mathcal{X}^{L}(t_{0},x_{0})$ such that, for every $t\in[t_{0},T]$,

[TABLE]

(ii) For every $t\in[t_{0},T]$, there exists an $x\in\mathcal{X}^{L}(t_{0},x_{0})$ for which (3.3) holds.

Proof of Lemma 3.6.

We show the statement only in the case $u\in\mathrm{USC}([0,T]\times\tilde{\Omega})$.

Clearly, (i) implies (ii). Next, suppose that (ii) holds. To show (i), let $(\pi^{m})_{m\in{\mathbb{N}}}=((t_{i}^{m})_{i=1,\ldots,n(m)})_{m\in{\mathbb{N}}}$ be dyadic partitions of $[t_{0},T]$ with $t_{0}=t_{0}^{m}<t_{1}^{m}<\cdots<t_{n(m)}^{m}=T$. For each $m\in{\mathbb{N}}$, we define an $x^{m}\in\mathcal{X}^{L}(t_{0},x_{0})$ as follows: First, note that, by (ii) there exists an $x_{1}^{m}\in\mathcal{X}^{L}(t_{0},x_{0})$ such that (3.3) holds with $t=t_{1}^{m}$ and $x=x^{m}$. Next, assuming that $x^{m}_{i}$ has already been chosen for some $i\in\{1,\ldots,n(m)-1\}$, note that, by (ii), there exists an $x_{i+1}^{m}\in\mathcal{X}^{L}(t_{i}^{m},x_{i}^{m})$ such that (3.3) holds with $(t_{0},x_{0})$ replaced by $(t_{i}^{m},x_{i}^{m})$ and with $t=t_{i+1}^{m}$ and $x=x_{i+1}^{m}$. Put

[TABLE]

By construction, $x^{m}\in\mathcal{X}^{L}(t_{0},x_{0})$ and $x^{m}$ satisfies (3.3) for all $t\in\{t_{i}^{m}\}_{i=0}^{m(n)}$. Since $\mathcal{X}^{L}(t_{0},x_{0})$ is compact and since $\cup_{m}\{t_{i}^{m}\}_{i=0}^{m(n)}$ is dense in $[t_{0},T]$, the sequence $(x^{m})_{m}$ has a subsequence $(x^{m_{k}})_{k}$ that converges to some $x^{0}$ in $\mathcal{X}^{L}(t_{0},x_{0})$. Since $u$ is upper semi-continuous,

[TABLE]

for every $t\in[t_{0},T]$. ∎

Proposition 3.7**.**

Fix $L\geq 0$ and a set $\tilde{\Omega}$ with $\Omega^{L}\subseteq\tilde{\Omega}\subseteq C([0,T],H)$. Then a mapping $u:[0,T]\times\tilde{\Omega}\to{\mathbb{R}}$ is a minimax $L$-solution of (TVP) on $[0,T]\times\tilde{\Omega}$ if and only if $u$ is a minimax $L$-super- as well as a minimax $L$-subsolution of (TVP) on $[0,T]\times\tilde{\Omega}$.

Proof.

If $u$ is a minimax $L$-solution of (TVP) on $[0,T]\times\tilde{\Omega}$, then it follows immediately from the definition that $u$ is a also a minimax $L$-super- and a minimax $L$-subsolution of (TVP) on $[0,T]\times\tilde{\Omega}$.

Let now $u$ be a minimax $L$-super- as well as a minimax $L$-subsolution of (TVP) on $[0,T]\times\tilde{\Omega}$. Clearly, $u$ is continuous and $u(T,\cdot)=h$. Thus it suffices, by Lemma 3.6, to show that, for every $(t_{0},x_{0},z)\in[0,T)\times\tilde{\Omega}\times H$ and every $t\in(t_{0},T]$, there exists an $(x,y)\in\mathcal{Y}^{L}(t_{0},x_{0},z)$ (see (5.1) for the definition of $\mathcal{Y}^{L}(t_{0},x_{0},z)$) such that $u(t,x)-y(t)=u(t_{0},x_{0})$. To this end, fix $(t_{0},x_{0},z)\in[0,T)\times\tilde{\Omega}\times H$ and let $t\in(t_{0},T]$. Since $u$ is a minimax $L$-super- and a minimax $L$-subsolution of (TVP) on $[0,T]\times\tilde{\Omega}$, there exist $(x_{1},y_{1})$, $(x_{2},y_{2})\in\mathcal{Y}^{L}(t_{0},x_{0},z)$ such that

[TABLE]

Since $\mathcal{Y}^{L}(t_{0},x_{0},z)$ is connected in $C([0,T],H)\times C([t_{0},T])$, which follows from Lemma A.5, and the mapping

[TABLE]

is continuous, there exists an $(x,y)\in\mathcal{Y}^{L}(t_{0},x_{0},z)$ such that

[TABLE]

This concludes the proof. ∎

3.2. Minimax solutions: Infinitesimal version of the notion

Proposition 3.8**.**

Fix $L\geq 0$ and some set $\tilde{\Omega}$ with $\Omega^{L}\subseteq\tilde{\Omega}\subseteq C([0,T],H)$.

(i) Let $u\in\mathrm{LSC}([0,T]\times\tilde{\Omega})$. Then $u$ is a minimax $L$-supersolution of (TVP) on $[0,T]\times\tilde{\Omega}$ if and only if $u(T,\cdot)\geq h$ on $\tilde{\Omega}$ and, for every $(t_{0},x_{0},z)\in[0,T)\times\tilde{\Omega}\times H$,

[TABLE]

(ii) Let $u\in\mathrm{USC}([0,T]\times\tilde{\Omega})$ Then $u$ is a minimax $L$-subsolution of (TVP) on $[0,T]\times\tilde{\Omega}$ if and only if $u(T,\cdot)\leq h$ on $\tilde{\Omega}$ and, for every $(t_{0},x_{0},z)\in[0,T)\times\tilde{\Omega}\times H$,

[TABLE]

Remark 3.9**.**

See [Lukoyanov03a, Theorem 8.1] for a corresponding result in the finite-dimensional case, where the outer infimum resp. supremum is taken over a slightly different class of paths $x$. Our choice however simplifies the proof and does not weaken our result because the compactness of $\mathcal{X}^{L}(t_{0},x_{0})$ implies that, for every ${\varepsilon}>0$, there exists a $\delta>0$ such that, for every $x\in\mathcal{X}^{L}(t_{0},x_{0})$,

[TABLE]

Proof of Proposition 3.8.

We prove only the non-obvious direction of part (i), i.e., we assume that, for every $(t_{0},x_{0},z)\in[0,T)\times\tilde{\Omega}\times H$, we have (3.4). For the sake of a contradiction, we assume that $u$ is not a minimax $L$-supersolution of (TVP) on $[0,T]$, i.e., by Lemma 3.6, there exist $(t_{0},x_{0},z)\in[0,T)\times\tilde{\Omega}\times H$, $t_{1}\in(t_{0},T]$ such that, for every $x\in\mathcal{X}^{L}(t_{0},x_{0})$,

[TABLE]

Furthermore, there exists a $\tilde{\delta}>0$ such that, for every $x\in\mathcal{X}^{L}(t_{0},x_{0})$,

[TABLE]

because otherwise lower semi-continuity of $u$, compactness of $\mathcal{X}(t_{0},x_{0})$, and continuity of $F$ would lead to a contradiction.

Denote by $\tilde{t}$ the supremum of all $t\in[t_{0},t_{1}]$ such that

[TABLE]

Clearly $\tilde{t}\in[t_{0},t_{1})$ and, due to the compactness of $\mathcal{X}^{L}(t_{0},x_{0})$, semicontinuity of $u$, and continuity of $F$, the supremum $\tilde{t}$ is actually a maximum, i.e., there exists an $\tilde{x}\in\mathcal{X}^{L}(t_{0},x_{0})$ such that

[TABLE]

We shall use (3.4) to derive the existence of a time $t\in(\tilde{t},t_{1})$ satisfying (3.5), which would be a contradiction. By (3.4), there exist an $x\in\mathcal{X}^{L}(\tilde{t},\tilde{x})$ and a $\delta\in(0,t_{1}-\tilde{t})$ such that

[TABLE]

Adding (3.6) and (3.7), we see that

[TABLE]

which concludes the proof. ∎

4. Comparison

Given $L\geq 0$, a set $\tilde{\Omega}$ with $\Omega^{L}\subseteq\tilde{\Omega}\subseteq C([0,T],H)$, a non-anticipating function $w:[0,T]\times\tilde{\Omega}\times\tilde{\Omega}\to{\mathbb{R}}$, and $(t_{0},x_{0},y_{0})\in[0,T)\times\tilde{\Omega}\times\tilde{\Omega}$, put

[TABLE]

Definition 4.1**.**

Fix $L\geq 0$ and a set $\tilde{\Omega}$ with $\Omega^{L}\subseteq\tilde{\Omega}\subseteq C([0,T],H)$. A function $w\in\mathrm{USC}([0,T]\times\tilde{\Omega}\times\tilde{\Omega})$ is a viscosity $L$-subsolution of

[TABLE]

on $[0,T]\times\tilde{\Omega}\times\tilde{\Omega}$ if, for every $(t_{0},x_{0},y_{0})\in[0,T)\times\tilde{\Omega}\times\tilde{\Omega}$ and for every test function $\varphi\in\underline{\mathcal{A}}^{L}w(t_{0},x_{0},y_{0})$, there exists a pair $(x,y)\in\mathcal{X}^{L}(t_{0},x_{0})\times\mathcal{X}^{L}(t_{0},y_{0})$ such that

[TABLE]

Theorem 4.2** (Comparison I).**

Fix $L\geq 0$ and a set $\tilde{\Omega}$ with $\Omega^{L}\subseteq\tilde{\Omega}\subseteq C([0,T],H)$. Let $w$ be a viscosity $L$-subsolution of (4.1) on $[0,T]\times\tilde{\Omega}\times\tilde{\Omega}$ with $w(T,x,x)\leq 0$ for all $x\in\tilde{\Omega}$. Then $w(t,x,x)\leq 0$ for all $(t,x)\in[0,T]\times\tilde{\Omega}$.

Proof.

Assume that there exists a $(t_{0},x_{0})\in[0,T)\times\tilde{\Omega}$ such that

[TABLE]

For every ${\varepsilon}>0$, define a mapping $\Phi_{\varepsilon}:[t_{0},T]\times\tilde{\Omega}\times\tilde{\Omega}\to{\mathbb{R}}$ by

[TABLE]

where

[TABLE]

Since $\Phi_{\varepsilon}\in\mathrm{USC}([t_{0},T]\times\tilde{\Omega}\times\tilde{\Omega})$ and $\mathcal{X}^{L}(t_{0},x_{0})$ is compact, there exists a point

[TABLE]

such that

[TABLE]

Following the proof of [UserGuide, Proposition 3.7], we can deduce that $\lim_{{\varepsilon}\to 0}\frac{1}{{\varepsilon}}\Psi(k_{\varepsilon})=0$ and that

[TABLE]

for every limit point $\hat{k}=(\hat{t},\hat{x},\hat{y})$ of $\{k_{\varepsilon}\}$ in $K$ as ${\varepsilon}\to 0$. Moreover, $\hat{x}=\hat{y}$ on $[0,\hat{t}]$. Note that the last equality in (4.3) follows from $w$ being non-anticipating because $\Psi(t,x,y)=0$ together with $x$, $y\in\mathcal{X}^{L}(t_{0},x_{0})$ yields $x=y$ on $[0,t]$. Now, from (4.3) we can infer that the maxima $M_{\varepsilon}$ are attained at interior points, i.e., we have $t_{\varepsilon}<T$ for sufficiently small ${\varepsilon}$; in particular, $w(T,x,x)-(T-t_{0})^{-1}(T-T)M_{0}/2\leq 0<M_{0}/2$ yields $\hat{t}<T$.

Next, we construct test functions. For every ${\varepsilon}>0$, define a function $\varphi_{\varepsilon}:[t_{\varepsilon},T]\times C([0,T],H)\times C([0,T],H)\to{\mathbb{R}}$ by $\varphi_{\varepsilon}(t,x,y):=\Theta_{\varepsilon}(t,x,y)-M_{\varepsilon}$. Clearly, $\varphi_{\varepsilon}\in\underline{\mathcal{A}}^{L}w(k_{\varepsilon})$ with

[TABLE]

Hence, there exists an $(x,y)\in\mathcal{X}^{L}(t_{\varepsilon},x_{\varepsilon})\times\mathcal{X}^{L}(t_{\varepsilon},y_{\varepsilon})$ such that, for small ${\varepsilon}$,

[TABLE]

Here, we obtained the last inequality from condition H($F$)(iii). Now, fix some limit point $\hat{k}=(\hat{t},\hat{x},\hat{y})$ of $\{k_{\varepsilon}\}$. Then $\hat{k}\in K$ and, with slight abuse of notation, we can assume that $(t_{\varepsilon},x_{\varepsilon},y_{\varepsilon})\to\hat{k}$ as ${\varepsilon}\downarrow 0$. Recall also that $\lim_{{\varepsilon}\to 0}\frac{1}{{\varepsilon}}\Psi(k_{\varepsilon})=0$ and that $\hat{x}=\hat{y}$ on $[0,\hat{t}]$. Thus, letting ${\varepsilon}\downarrow 0$ in (4.4) yields a contradiction. (Note that $\sup_{s\leq t_{\varepsilon}}\left|{x_{\varepsilon}(s)-y_{\varepsilon}(s)}\right|\to 0$ as ${\varepsilon}\downarrow 0$ can also be deduced from Remark 2.9 together with $\lim_{{\varepsilon}\to 0}\frac{1}{{\varepsilon}}\Psi(k_{\varepsilon})=0$. For more details, we refer the reader to (A.9) in the proof of Lemma A.2.) ∎

Theorem 4.3** (Doubling theorem).**

Fix $L\geq 0$ and a set $\tilde{\Omega}$ with $\Omega^{L}\subseteq\tilde{\Omega}\subseteq C([0,T],H)$. Let $u$ be a minimax $L$-sub- and let $v$ be a minimax $L$-supersolution of (TVP) on $[0,T]\times\tilde{\Omega}$. Then $w$ defined by $w(t,x,y):=u(t,x)-v(t,y)$ is a viscosity $L$-subsolution of (4.1) on $[0,T]\times\tilde{\Omega}\times\tilde{\Omega}$.

Proof.

Fix $(t_{0},x_{0},y_{0})\in[0,T)\times\tilde{\Omega}\times\tilde{\Omega}$. Let $\varphi\in\underline{\mathcal{A}}^{L}(t_{0},x_{0},y_{0})$, i.e., there exists a $T_{0}\in(t_{0},T]$ such that, for all $(t,x,y)\in[t_{0},T_{0}]\times\mathcal{X}^{L}(t_{0},x_{0})\times\mathcal{X}^{L}(t_{0},y_{0})$,

[TABLE]

Put $z:=\partial_{x}\varphi(t_{0},x_{0},y_{0})$ and $\zeta:=\partial_{y}\varphi(t_{0},x_{0},y_{0})$. Since $u$ is a minimax $L$-sub- and $v$ is a minimax $L$-supersolution of (TVP) on $[0,T]\times\tilde{\Omega}$, there exists an $(x,y)\in\mathcal{X}^{L}(t_{0},x_{0})\times\mathcal{X}^{L}(t_{0},y_{0})$ such that, for all $t\in[t_{0},T_{0}]$,

[TABLE]

Thus, by the functional chain-rule applied to (4.5) and by (4.6),

[TABLE]

which yields (4.2), i.e., $w$ is a viscosity $L$-subsolution of (4.1) on $[0,T]\times\tilde{\Omega}\times\tilde{\Omega}$. ∎

Immediately, from the doubling theorem (Theorem  4.3) together with Theorem 4.2, we obtain a comparison principle for minimax $L$-semisolutions.

Theorem 4.4** (Comparison II).**

Fix $L\geq 0$ and a set $\tilde{\Omega}$ with $\Omega^{L}\subseteq\tilde{\Omega}\subseteq C([0,T],H)$. Let $u$ be a minimax $L$-sub- and let $v$ be a minimax $L$-supersolution of (TVP) on $[0,T]\times\tilde{\Omega}$. Then $u\leq v$ on $[0,T]\times\tilde{\Omega}$.

Theorem 4.5** (Comparison III).**

Fix a set $\tilde{\Omega}$ with $\Omega\subseteq\tilde{\Omega}\subseteq C([0,T],H)$. Let $u$ be a minimax sub- and let $v$ be a minimax supersolution of (TVP) on $[0,T]\times\tilde{\Omega}$. Then $u\leq v$ on $[0,T]\times\tilde{\Omega}$.

Proof.

There exist $L_{1}$, $L_{2}\geq 0$ such that $u$ is a minimax $L_{1}$-sub- and $v$ is a minimax $L_{2}$-supersolution of (TVP) on $[0,T]\times\tilde{\Omega}$. By Remark 3.3, we can replace $L_{1}$ and $L_{2}$ with $L_{1}\vee L_{2}$. Since $\Omega=\cup_{L\geq 0}\Omega^{L}\subseteq\tilde{\Omega}$, we have $\Omega^{L_{1}\vee L_{2}}\subseteq\tilde{\Omega}$. Thus we can apply Theorem 4.3 to deduce that the function $w$ defined by $w(t,x,y):=u(t,x)-v(t,y)$ is a viscosity $(L_{1}\vee L_{2})$-subsolution of (4.1) on $[0,T]\times\tilde{\Omega}$. By Theorem 4.2, $u\leq v$ on $[0,T]\times\tilde{\Omega}$. ∎

5. Existence and uniqueness

Thanks to our previous results concerning the topological properties of the spaces $\mathcal{X}^{L}(t_{0},x_{0})$, we are able to follow the approach of Lukoyanov [Lukoyanov03a, Section 7] (see also Subbotin [Subbotin_book, Section 8, pp. 69-80] for the non-path-dependent case).

Given $(t_{0},x_{0},z)\in[0,T)\times\Omega\times H$ and $L\geq 0$, put

[TABLE]

Definition 5.1**.**

Let $\tilde{\Omega}\subseteq C([0,T],H)$ and put $S(t_{0}):=\tilde{\Omega}\times C([t_{0},T])$, $t_{0}\in[0,T)$. Consider a non-anticipating function $u:[0,T]\times\tilde{\Omega}\to[-\infty,\infty]$.

(i) We say that $u$ is an improper minimax $L$-supersolution of (TVP) on $[0,T]\times\tilde{\Omega}$ if $u(T,\cdot)\geq h$ on $\tilde{\Omega}$ and if, for every $(t_{0},x_{0},z)\in[0,T)\times\tilde{\Omega}\times H$ and every $t\in(t_{0},T]$,

[TABLE]

(ii) We say that $u$ is an improper minimax $L$-subsolution of (TVP) on $[0,T]\times\tilde{\Omega}$ if $u(T,\cdot)\leq h$ on $\tilde{\Omega}$ and if, for every $(t_{0},x_{0},z)\in[0,T)\times\tilde{\Omega}\times H$ and every $t\in(t_{0},T]$,

[TABLE]

Note that we use here $\inf$ and $\sup$ instead of $\min$ and $\max$ because we do not require any semicontinuity for $u$.

Next, we show that the set of improper minimax $L_{0}$-supersolutions of (TVP) is not empty. To this end, define functions $u_{+}^{z}:[0,T]\times\Omega\to[-\infty,\infty]$, $z\in H$, by

[TABLE]

Lemma 5.2**.**

Let $z\in H$. The functional $u_{+}^{z}$ is an improper minimax $L_{0}$-supersolution of (TVP) on $[0,T]\times\Omega$ with values in $[-\infty,\infty)$.

Proof.

First, note that $u^{z}_{+}(T,\cdot)=h$.

Next, fix $(t_{0},x_{0},\tilde{z})\in[0,T)\times\Omega\times H$. By Proposition 2.13, the initial value problem

[TABLE]

where $\tilde{f}:[t_{0},T]\times C([0,T],H)\to H$ is defined by

[TABLE]

has a unique solution $\tilde{x}\in\mathcal{X}^{L_{0}}(t_{0},x_{0})$. Moreover, put

[TABLE]

Then $(\tilde{x},\tilde{y})\in\mathcal{Y}^{L_{0}}(t_{0},x_{0},z)\cap\mathcal{Y}^{L_{0}}(t_{0},x_{0},\tilde{z})$. Since, for every $t\in(t_{0},T]$,

[TABLE]

we can conclude that $u^{z}_{+}$ is an improper minimax $L_{0}$-supersolution of (TVP) on $[0,T]\times\Omega$. The pointwise boundedness from above ($u^{z}_{+}<\infty$) follows from the continuity of $h$ and $F$ in $x_{0}$ and from the compactness of the sets $\mathcal{X}^{L_{0}}(t_{0},x_{0})$ (Proposition 2.10). ∎

Next, we will show that the functions $u^{L}_{0}:[0,T]\times\Omega^{L}\to[-\infty,\infty]$, $L\geq L_{0}$, defined by

[TABLE]

are both, improper minimax $L_{0}$-super- as well as a improper minimax $L_{0}$-subsolutions of (TVP) on $[0,T]\times\Omega^{L}$, $L\geq L_{0}$.

Lemma 5.3**.**

Let $L\geq L_{0}$. Then the function $u_{0}^{L}$ is an improper minimax $L_{0}$-super as well as an improper minimax $L_{0}$-subsolution of (TVP) on $[0,T]\times\Omega^{L}$. Moreover, $u_{0}$ is real-valued.

Proof.

Fix $(t_{0},x_{0},z)\in[0,T)\times\Omega^{L}\times H$. As every improper minimax $L_{0}$-supersolution $u$ of (TVP) on $[0,T]\times\Omega^{L}$ satisfies

[TABLE]

we have

[TABLE]

By Lemma 5.2, $u_{0}^{L}(t_{0},x_{0})<\infty$. Thus $u_{0}^{L}$ is real-valued.

By the definition of $u_{0}^{L}$, $u_{0}^{L}(T,x_{0})\geq h(x_{0})$. Since $u_{0}^{L}(T,x_{0})\leq u^{z}_{+}(T,x_{0})=h(x_{0})$, we can deduce that

[TABLE]

Next, fix $t\in(t_{0},T]$. Let $(u_{n})_{n}$ be a sequence of improper minimax $L_{0}$-supersolutions of (TVP) on $[0,T]\times\Omega^{L}$ such that $\left|{u_{n}(t_{0},x_{0})-u_{0}^{L}(t_{0},x_{0})}\right|\leq n^{-1}$ and that

[TABLE]

for some $(x_{n},y_{n})\in\mathcal{Y}^{L_{0}}(t_{0},x_{0},z)$, $n\in{\mathbb{N}}$. Then

[TABLE]

for every $n\in{\mathbb{N}}$, i.e., $u_{0}^{L}$ is an improper minimax $L_{0}$-supersolution of (TVP) on $[0,T]\times\Omega^{L}$.

To show that $u_{0}^{L}$ is also an improper minimax $L_{0}$-subsolution of (TVP) on $[0,T]\times\Omega^{L}$ it suffices (because of (5.3)) to verify that

[TABLE]

To this end, we prove that the function $u^{z}:[0,T]\times\Omega^{L}\to[-\infty,\infty]$ defined by

[TABLE]

is an improper minimax $L_{0}$-supersolution of (TVP) on $[0,T]\times\Omega^{L}$. Then, according to the definition of $u_{0}^{L}$, we have $u_{0}^{L}(\tilde{t},\tilde{x})\leq u^{z}(\tilde{t},\tilde{x})$, $(\tilde{t},\tilde{x})\in[0,T)\times\Omega^{L}$, which immediately yields (5.4).

To show that $u^{z}$ is an improper minimax $L_{0}$-supersolution of (TVP) on $[0,T]\times\Omega^{L}$, let us first check whether the terminal condition is satisfied. Let $\tilde{x}\in\Omega^{L}$. If $T>t$, then $u^{z}(T,\tilde{x})=u_{0}^{L}(T,\tilde{x})\geq h(\tilde{x})$ because $u_{0}^{L}$ is an improper minimax $L_{0}$-supersolution of (TVP) on $[0,T]\times\Omega^{L}$. If $T=t$, then we also get immediately the same inequality. Next, note that, by (5.2), $u^{z}(\tilde{t},\tilde{x})>-\infty$ for all $(\tilde{t},\tilde{x})\in[0,T]\times\Omega^{:}$. Thus, given $(\tilde{t},\tilde{x},\tilde{z})\in[0,T)\times\Omega^{L}\times H$, $\tau\in(\tilde{t},T]$, and ${\varepsilon}>0$, it suffices to show that there exists an $(x,y)\in\mathcal{Y}^{L_{0}}(\tilde{t},\tilde{x},\tilde{z})$ such that

[TABLE]

We shall distinguish among the following three cases.

Case 1. Let $\tilde{t}\geq t$. Then $u^{z}(\tilde{t},\tilde{x})=u_{0}^{L}(\tilde{t},\tilde{x})$ and $u^{z}(\tau,x)=u_{0}(\tau,x)$ for all $x\in\mathcal{X}^{L_{0}}(\tilde{\tau},\tilde{x})$, i.e., we can deduce (5.5) from $u_{0}$ being an improper minimax $L_{0}$-supersolution of $\eqref{E:PPDE2}$ on $[0,T]\times\Omega^{L}$.

Case 2. Let $\tilde{t}<t$ and $\tau<t$. By Proposition 2.13, the initial value problem

[TABLE]

where $\hat{f}:[\tilde{t},T]\times C([0,T],H)\to H$ is defined by

[TABLE]

has a unique solution $\hat{x}\in\mathcal{X}^{L_{0}}(\tilde{t},\tilde{x})$. Next, put

[TABLE]

so that $(\hat{x},\hat{y})\in\mathcal{Y}^{L_{0}}(\tilde{t},\tilde{x},z)\cap\mathcal{Y}^{L_{0}}(\tilde{t},\tilde{x},\tilde{z})$. Since $\hat{x}\in\mathcal{X}^{L_{0}}(t_{0},x_{0})$,

[TABLE]

i.e., (5.5) holds (even for ${\varepsilon}=0$).

Case 3. Let $\tilde{t}<t$ and $\tau\geq t$. By Case 2, there exists an $x_{2}\in\mathcal{X}^{L_{0}}(\tilde{t},\tilde{x})$ such that

[TABLE]

By Case 1, there exists an $x_{1}\in\mathcal{X}^{L_{0}}(\tilde{t},x_{2})$ such that

[TABLE]

Consequently, with $x:={\mathbf{1}}_{[0,t)}\cdot x_{2}+{\mathbf{1}}_{[t,T]}\cdot x_{1}$, we have

[TABLE]

which completes the proof. ∎

Let $L\geq L_{0}$. Since $u_{0}^{L}$ is possibly not semicontinuous, consider the envelopes $u^{L,-}_{0}$, $u^{L,+}_{0}:[0,T]\times\Omega^{L}\to\ [-\infty,\infty]$ defined by

[TABLE]

where

[TABLE]

Clearly,

[TABLE]

Lemma 5.4**.**

Let $L\geq L_{0}$. Then $u_{0}^{L,-}>-\infty$ and $u^{L,+}_{0}<\infty$.

Proof.

Fix $\delta>0$. Then

[TABLE]

To show that $u^{L,+}_{0}<\infty$, one can proceed similarly as in the proof of Lemma 5.4 but one should use Lemma 5.2 instead of (5.2). ∎

Remark 5.5**.**

By (5.6) and Lemma 5.4, $u_{0}^{L,-}$ and $u_{0}^{L,+}$ are real-valued. Moreover,

[TABLE]

The proof of this statement is standard. One can, e.g., follow the argument in [Subbotin_book, Proof of Proposition 8.7, p. 77].

Lemma 5.6**.**

Let $L\geq L_{0}$. The function $u^{L,-}_{0}$ is a minimax $L_{0}$-supersolution of (TVP) on $[0,T]\times\Omega^{L}$ and the function $u^{L,+}_{0}$ is a minimax $L_{0}$-subsolution of (TVP) on $[0,T]\times\Omega^{L}$.

Proof.

We only show that $u^{L,-}_{0}$ is a minimax $L_{0}$-supersolution of (TVP) on $[0,T]\times\Omega^{L}$. To this end, fix $(t_{0},x_{0})\in[0,T)\times\Omega^{L}$ and let $(t,z)\in(t_{0},T]\times H$. By the definition of $u_{0}^{L,-}$, there exists a sequence $(t_{n},x_{n})_{n}$ in $[0,T]\times\Omega^{L}$ such that

[TABLE]

Since $t>t_{0}$, we can assume that $t_{n}<t$ for all $n\in{\mathbb{N}}$. Recall that $u_{0}^{L}$ is an improper minimax $L_{0}$-supersolution of (TVP) on $[0,T]\times\Omega^{L}$. Thus

[TABLE]

for some $(\tilde{x}_{n},\tilde{y}_{n})\in\mathcal{Y}^{L_{0}}(t_{n},x_{n},z)$. By Proposition 2.12, we can assume that $(\tilde{x}_{n},\tilde{y}_{n})_{n}$ converges to some $(\tilde{x}_{0},\tilde{y}_{0})\in\mathcal{Y}^{L_{0}}(t_{0},x_{0},z)$. Hence

[TABLE]

and thus, by Lemma 3.6, $u_{0}^{L,-}$ satisfies the minimax $L_{0}$-supersolution property on $[0,T)\times\Omega^{L}$. The minimax $L_{0}$-supersolution property for the terminal condition can be verified similarly. Together with Remark 5.5, this concludes the proof. ∎

Theorem 5.7** (Existence and uniqueness I).**

Let $L\geq L_{0}$. Then $u_{0}^{L}$ is the unique minimax $L_{0}$-solution of (TVP) on $[0,T]\times\Omega^{L}$.

Proof.

By Theorem 4.4 and Lemma 5.6, $u^{L,+}_{0}\leq u^{L,-}_{0}$ on $[0,T]\times\Omega^{L}$. Together with (5.6), we can conclude that $u_{0}^{L}=u_{0}^{L,+}=u_{0}^{L,-}$. Therefore, by Proposition 3.7, the function $u_{0}^{L}$ is a minimax $L_{0}$-solution of (TVP) on $[0,T]\times\Omega^{L}$. Uniqueness follows from Theorem 4.4. ∎

From Theorem 4.5 and Theorem 5.7 together with Remark 3.3 and Remark 3.4, we obtain the following result.

Theorem 5.8** (Existence and uniqueness II).**

There exists a function $u_{0}:[0,T]\times\Omega\to{\mathbb{R}}$ that is the unique minimax solution on $[0,T]\times\Omega$. Moreover, $u_{0}=u_{0}^{L}$ on $[0,T]\times\Omega^{L}$ for every $L\geq L_{0}$.

6. Stability

Given $n\in{\mathbb{N}}$, consider the terminal value problem

[TABLE]

where $h_{n}:C([0,T],H)\to{\mathbb{R}}$ and $F_{n}:[0,T]\times C([0,T],H)\times H\to{\mathbb{R}}$, respectively, are supposed to satisfy H$(h)$ and H$(F)$, respectively, with the same Lipschitz constants and moduli of continuity.

Our approach for the following basic stability result is similar as in the finite-dimensional case (regarding PDEs, see [Subbotin_minimaxSol, pp. 36 ff.]; regarding PPDEs, see [Lukoyanov01minimax, Theorem 6.1]).

Theorem 6.1**.**

Let $L\geq 0$.

(i) For every $n\in{\mathbb{N}}$, let $u^{L}_{n}$ be a minimax supersolution of (TVP $n$) on $[0,T]\times\Omega^{L}$. Then the function $u^{L}:[0,T]\times\Omega^{L}\to{\mathbb{R}}$ defined by

[TABLE]

is a minimax supersolution of (TVP) on $[0,T]\times\Omega^{L}$.

(ii) For every $n\in{\mathbb{N}}$, let $u^{L}_{n}$ be a minimax subsolution of (TVP $n$) on $[0,T]\times\Omega^{L}$. Then the function $u^{L}:[0,T]\times\Omega^{L}\to{\mathbb{R}}$ defined by

[TABLE]

is a minimax subsolution of (TVP) on $[0,T]\times\Omega^{L}$.

(iii) For every $n\in{\mathbb{N}}$, let $u^{L}_{n}$ be a minimax solution of (TVP $n$) on $[0,T]\times\Omega^{L}$. Then $(u_{n}^{L})_{n}$ converges to the function $u^{L,+}$, which is a minimax solution of (TVP) on $[0,T]\times\Omega^{L}$.

Proof of Theorem 6.1.

We prove only (i) and (iii).

(i) First, note that $u^{L,-}\in\mathrm{LSC}([0,T]\times\Omega^{L})$ (cf. Remark 5.5). It suffices to verify that $u^{L,-}$ satisfies the minimax $L$-supersolution property for the terminal condition on $\Omega^{L}$ and for the PPDE in $[0,T)\times\Omega^{L}$. We shall do only the latter as the former can be done similarly.

Fix $(t_{0},x_{0})\in[0,T)\times\Omega^{L}$. Let $t\in(t_{0},T)$ and $z\in H$. Consider a sequence $(t_{n},x_{n})_{n}$ in $[0,T)\times\Omega^{L}$ such that $\mathbf{d}_{\infty}((t_{n},x_{n}),(t_{0},x_{0}))<n^{-1}$ and

[TABLE]

Without loss of generality, assume that $t_{n}<t$ for every $n\in{\mathbb{N}}$. Since $u_{n}^{L}$ is a minimax $L$-supersolution of (TVP $n$) on $[0,T)\times\Omega^{L}$ (cf. Remark 3.5), there exists an $\tilde{x}_{n}\in\mathcal{X}^{L}(t_{n},x_{n})$ such that

[TABLE]

We can also assume that $(\tilde{x}_{n})_{n}$ converges in $\Omega^{L}$ to some $\tilde{x}_{0}\in\mathcal{X}^{L}(t_{0},x_{0})$ (see Proposition 2.12). Therefore,

[TABLE]

(iii) Let $(t_{0},x_{0})\in[0,T)\times\Omega^{L}$. Then

[TABLE]

Moreover, by Comparison (Theorem 4.4), $u^{L,+}(t_{0},x_{0})\leq u^{L,-}(t_{0},x_{0})$. Thus $u_{n}^{L}\to u^{L,+}$ and $u^{L,+}=u^{L,-}$. Consequently, noting (i), (ii), and Proposition 3.7, we can infer that $u^{L,+}$ is a minimax $L$-solution of (TVP) on $[0,T]\times\Omega^{L}$. ∎

7. Applications to optimal control

Consider a controlled evolution equation of the form

[TABLE]

Here, $t_{0}\in[0,T]$, $x_{0}\in\Omega$, $a\in\mathcal{A}^{t_{0}}$ with

[TABLE]

for some compact topological space $P$. A solution of (7.1) is to be understood as an element of $\mathcal{X}^{L}(t_{0},x_{0})$ for some $L\geq 0$ and is denoted by $x^{t_{0},x_{0},a}$.

Goal of the controller is to minimize a cost functional of the form

[TABLE]

over all $a\in\mathcal{A}^{t_{0}}$.

Next, we state the hypotheses for the data of our control problem. To this end, let $L_{f}\geq 0$.

H$(f)_{1}$: The function $f:[0,T]\times C([0,T],H)\times P\to H$ satisfies the following:

(i) $f$ is continuous.

(ii) For every $(t,x,{\mathbf{p}})\in[0,T]\times C([0,T],H)\times P$,

[TABLE]

(iii) For a.e. $t\in(0,T)$, for every $x$, $y\in\Omega$, and for every ${\mathbf{p}}\in P$,

[TABLE]

H$(\ell)_{1}$: The function $\ell:[0,T]\times C([0,T],H)\times P\to{\mathbb{R}}$ satisfies the following:

(i) $\ell$ is continuous.

(ii) For every $(t,x,{\mathbf{p}})\in[0,T]\times C([0,T],H)\times P$,

[TABLE]

(iii) For a.e. $t\in(0,T)$, for every $x$, $y\in\Omega$, and for every ${\mathbf{p}}\in P$,

[TABLE]

H$(h)_{1}$: The function $h:C([0,T],H)\to{\mathbb{R}}$ satisfies the following:

(i) $h$ is continuous.

(ii) For every $x$, $y\in\Omega$,

[TABLE]

Immediately from Proposition 2.13, we obtain the following statement.

Proposition 7.1**.**

Suppose that H$(f)_{1}$ holds. Let $(t_{0},x_{0})\in[0,T)\times\Omega$ and $a\in\mathcal{A}^{t_{0}}$. Then the initial-value problem (7.1) has a unique solution in $\mathcal{X}^{L_{f}}(t_{0},x_{0})$.

We also need to establish continuous dependence properties of our controlled evolution equation with respect to the initial data.

Proposition 7.2**.**

Suppose that H$(f)_{1}$ holds. Let $t_{0}$, $t_{1}\in[0,T]$ and $x_{0}$, $y_{0}\in\Omega$. Fix $a\in\mathcal{A}^{0}$ and denote its restrictions to subintervals also by $a$.

(i) For every $t\in[t_{0},T]$,

[TABLE]

(ii) Let $L\geq 0$. If $x_{0}\in\Omega^{L}$, then

[TABLE]

for some constant $C$ that depends only on $L$, $L_{f}$, and $x_{\ast}$.

Proof.

(i) Put $x:=x^{t_{0},x_{0},a}$, $y:=x^{t_{0},y_{0},a}$, $z:=x-y$, ${\mathbf{z}}:={\mathbf{x}}-{\mathbf{y}}$, and

[TABLE]

By integration-by-parts (Proposition 2.4), for every $t\in[t_{0},T]$,

[TABLE]

Here, we used the monotonicity of $A$ and H$(f)_{1}$(iii). Finally, Gronwall’s inequality yields, $m(t)^{2}\leq e^{2L_{f}(t-t_{0})}m(t_{0})^{2}$, i.e., we have (7.2).

(ii) Without loss of generality, let $L\geq L_{f}$. Put $x:=x^{t_{0},x_{0},a}$, $z:=x-x_{0}$, and ${\mathbf{z}}:={\mathbf{x}}-{\mathbf{x}}_{0}$, and

[TABLE]

Recall that $x$, $x_{0}\in\Omega^{L}=\mathcal{X}^{L}(0,x_{\ast})$. Thus, by the a-priori estimates of Lemma A.1, there exists a constant $C$ depending only on $L$ and $x_{\ast}$ such that

[TABLE]

Therefore, we can apply integration-by-parts (Proposition 2.4) and obtain thanks to the monotonicity of $A$ and to H$(f)_{1}$(ii)

[TABLE]

for every $t\in[t_{0},T]$. This yields

[TABLE]

Next, without loss of generality, let $t_{1}\geq t_{0}$. Fix $t\in[t_{1},T]$. By (7.2) and (7.4),

[TABLE]

which together with (7.4) yields (7.3). ∎

The value function $v:[0,T]\times\Omega\to{\mathbb{R}}$ for our control problem is defined by

[TABLE]

where

[TABLE]

Similarly as in the finite-dimensional case, one can prove a dynamic programming principle.

Theorem 7.3**.**

Suppose that H$(f)_{1}$, H$(\ell)_{1}$, and H$(h)$ hold. Let $(t_{0},x_{0})\in[0,T]\times\Omega$. Then, for every $t\in[t_{0},T]$,

[TABLE]

In order to show that the value function $v$ is a minimax solution of the corresponding Bellman equation, we need to establish that $v$ is continuous.

Theorem 7.4** (Regularity of the value function).**

Suppose that H$(f)_{1}$, H$(\ell)_{1}$, and H$(h)_{1}$ hold. Then $v$ is continuous. In particular, the following holds:

(i) For every $t_{0}\in[0,T]$, $x_{0}$, $y_{0}\in\Omega$,

[TABLE]

(ii) Let $L\geq 0$. If $x_{0}\in\Omega^{L}$, then, for every $t_{0}$, $t_{1}\in[0,T]$,

[TABLE]

for some constant $C$ that depends only on $L$, $L_{f}$, $x_{\ast}$, and $T$.

Proof.

Fix $a\in\mathcal{A}^{0}$.

(i) By H$(\ell)_{1}$(iii), H$(h)_{1}$(ii), and H$(f)_{1}$ together with Proposition 7.2 (i),

[TABLE]

from which we get (7.5).

(ii) Without loss of generality, let $t_{1}\geq t_{0}$. By H$(\ell)_{1}$, H$(h)_{1}$, Lemma A.1, and H$(f)_{1}$ together with Proposition 7.2 (ii),

[TABLE]

for some constant $C$ that depends only on $L$, $L_{f}$, $x_{\ast}$, and $T$. This yields (7.6).

By parts (i) and (ii) of this proof together with $t\mapsto x_{0}(t)$ being continuous for each $x_{0}\in\Omega$, we have continuity of $v$ under $\mathbf{d}_{\infty}$. ∎

Remark 7.5**.**

Note that we have not established that $v$ is (locally) Lipschitz continuous under $\mathbf{d}_{\infty}$ on the sets $\Omega^{L}$.

Now, we are in the position to prove the main result of this section.

The Bellman equation associated to our control problem is given by (TVP) with

[TABLE]

Theorem 7.6**.**

Suppose that H$(f)_{1}$, H$(\ell)_{1}$, and H$(h)_{1}$ hold. Let $F$ be given by (7.7). Then the value function $v$ is a minimax $L_{f}$-solution of (TVP) on $[0,T]\times\Omega$.

Proof.

Note that H$(f)_{1}$ and H$(\ell)_{1}$ imply H$(F)$.

(i) First, we show that $v$ is a minimax $L_{f}$-supersolution of (TVP) on $[0,T]\times\Omega$. To this end, fix $(t_{0},x_{0},z)\in[0,T]\times\Omega\times H$ and $t\in[t_{0},T]$. For every ${\varepsilon}>0$, there exists, by Theorem 7.3, a control $a^{\varepsilon}\in\mathcal{A}^{t_{0}}$ such that, with $x^{\varepsilon}:=x^{t_{0},x_{0},a^{\varepsilon}}$,

[TABLE]

Since by (7.7) also

[TABLE]

holds, we have

[TABLE]

Since $\mathcal{X}^{L_{f}}(t_{0},x_{0})$ is compact and $F$ is continuous, there exists an $x\in\mathcal{X}^{L_{f}}(t_{0},x_{0})$ depending on $t$ such that

[TABLE]

Recall that $v$ is continuous (Theorem 7.4). Thus we can apply Lemma 3.6 to deduce the existence of an $x\in\mathcal{X}^{L_{f}}(t_{0},x_{0})$ such that, for all $t\in[t_{0},T]$, (7.8) holds. Since the definition of $v$ yields $v(T,\cdot)=h$, we can conclude that $v$ is a minimax $L_{f}$-supersolution of (TVP) on $[0,T]\times\Omega$.

(ii) Next, we show that $v$ is a minimax $L_{f}$-solution of (TVP) on $[0,T]\times\Omega$.

Note that, by Theorem 5.8, there exists a unique minimax solution $u$ of (TVP) on $[0,T]\times\Omega$. Moreover, for every $L\geq L_{f}$, the restriction of $u$ to $[0,T]\times\Omega^{L}$ is, according to Theorem 5.7, the unique minimax $L_{f}$-solution of (TVP) on $[0,T]\times\Omega^{L}$. Thus it suffices to show that $u=v$ on $\Omega^{L}$ for each $L\geq L_{f}$. To this end, fix $L\geq L_{f}$.

(ii-a) First, note that $u\leq v$ on $\Omega^{L}$. This follows from part (i) of this proof together with Theorem 4.4.

(ii-b) Now, we show that $v\leq u$ on $\Omega^{L}$. To this end, we verify that the function $w:[0,T]\times\Omega^{L}\times\Omega^{L}\to{\mathbb{R}}$ defined by $w(t,x,y):=v(t,x)-u(t,y)$ is a viscosity $L_{f}$-subsolution of (4.1) on $[0,T]\times\Omega^{L}\times\Omega^{L}$. We proceed similarly as in the proof of the doubling theorem (Theorem 4.3). Let $(t_{0},x_{0},y_{0})\in[0,T]\times\Omega^{L}\times\Omega^{L}$ and $\varphi\in\underline{\mathcal{A}}^{L_{f}}w(t_{0},x_{0},y_{0})$. Then there exists a $T_{0}\in(t_{0},T]$ such that, for every $(t,x,y)\in[t_{0},T]\times\mathcal{X}^{L_{f}}(t_{0},x_{0})\times\mathcal{X}^{L_{f}}(t_{0},x_{0})$,

[TABLE]

Put $z:=-\partial_{y}\varphi(t_{0},x_{0},y_{0})$. Since $u$ is a minimax $L_{f}$-solution of (TVP) on $[0,T]\times\Omega^{L}$, there exists a $y\in\mathcal{X}^{L_{f}}(t_{0},y_{0})$ such that, for every $t\in[t_{0},T]$,

[TABLE]

Next, let ${\mathbf{p}}\in P$ such that

[TABLE]

which is possible because $P$ is compact and $F$ is continuous. Also put $x:=x^{t_{0},x_{0},a}$, where $a(t):={\mathbf{p}}$, $t\in[t_{0},T]$. Then, by Theorem 7.3,

[TABLE]

Applying the functional chain-rule to the left-hand side of (7.9) yields together with (7.10) and (7.12)

[TABLE]

which in turn implies

[TABLE]

Noting (7.11), we get (4.2), i.e., $w$ is $L_{f}$-subsolution of (4.1) on $[0,T]\times\Omega^{L}\times\Omega^{L}$. Thus, by Theorem 4.2, $v\leq u$ on $\Omega^{L}$. This concludes the proof. ∎

8. Applications to differential games

It will be demonstrated that the Krasovskiĭ-Subbotin approach for differential games (see [KrasovksiiSubbotin]) can also be applied to infinite-dimensional settings. We are able to adapt the methods by Lukoyanov [Lukoyanov03b], where the finite-dimensional PPDE case is treated. Note that in this section, we require stronger conditions for our data in contrast to the previous section.

Consider an evolution equation controlled by two players, which we call the controller $a$ and the disturbance $b$, of the form

[TABLE]

where $t_{0}\in[0,T]$, $x_{0}\in\Omega$, $a\in\mathcal{A}^{t_{0}}$, and $b\in\mathcal{B}^{t_{0}}$ with

[TABLE]

for some compact topological spaces $P$ and $Q$. A solution of (8.1) is denoted by $x^{t_{0},x_{0},a,b}$ and is to be understood as an element of $\mathcal{X}^{L}(t_{0},x_{0})$ for some $L\geq 0$.

Goal of the controller is to minimize via $a\in\mathcal{A}^{t_{0}}$ a cost functional of the form

[TABLE]

while assuming this functional is to be maximized by the disturbance via $b\in\mathcal{B}^{t_{0}}$, thus representing uncertainty. The precise description of the admissible actions of the controller and the corresponding notions for the disturbance will be given shortly.

We proceed by stating the additional hypotheses for the data of our game. To this end, fix a (global) constant $L_{f}>1$.

H$(f)_{2}$: The function $f:[0,T]\times C([0,T],H)\times P\times Q\to H$ satisfies the following:

(i) $f$ is continuous.

(ii) For every $(t,x,{\mathbf{p}},{\mathbf{q}})\in[0,T]\times C([0,T],H)\times P\times Q$,

[TABLE]

(iii) For a.e. $t\in(0,T)$ and for every $x$, $y\in\Omega$, every ${\mathbf{p}}\in P$, and every ${\mathbf{q}}\in Q$,

[TABLE]

H$(\ell)_{2}$: The function $\ell:[0,T]\times C([0,T],H)\times P\times Q\to{\mathbb{R}}$ satisfies the following:

(i) $\ell$ is continuous.

(ii) For every $(t,x,{\mathbf{p}},{\mathbf{q}})\in[0,T]\times C([0,T],H)\times P\times Q$,

[TABLE]

(iii) For a.e. $t\in(0,T)$, and for every $x$, $y\in\Omega$, every ${\mathbf{p}}\in P$, and every ${\mathbf{q}}\in Q$,

[TABLE]

By Proposition 2.13, we have immediately the following result.

Proposition 8.1**.**

Suppose that H$(f)_{2}$ holds. Let $(t_{0},x_{0})\in[0,T)\times\Omega$, $a\in\mathcal{A}^{t_{0}}$, and $b\in\mathcal{B}^{t_{0}}$. Then the initial-value problem (8.1) has a unique solution in $\mathcal{X}^{L_{f}}(t_{0},x_{0})$.

Next, we present the precise setup for our differential game. To this end, fix $t_{0}\in[0,T]$. The spaces of feedback strategies for the controller and the disturbance are defined as follows:

[TABLE]

Remark 8.2**.**

Note that elements of $\mathbb{A}^{t_{0}}$ and $\mathbb{B}^{t_{0}}$ do not need to be measurable functions.

Let us first consider the controller’s point of view. For every $\delta>0$, every partition $\pi:t_{0}<t_{1}<\cdots<t_{n}=T$ with mesh size $\left|{\pi}\right|\leq\delta$, every $x_{0}\in\Omega$, every feedback strategy $\mathbf{a}\in\mathbb{A}^{t_{0}}$, and every control $b\in\mathcal{B}^{t_{0}}$, define a step-by-step feedback control $a=a(\cdot;\pi,x_{0},\mathbf{a},b)\in\mathcal{A}^{t_{0}}$ recursively by

[TABLE]

Remark 8.3**.**

Note that $a(t)$, $t\in[t_{i},t_{i+1})$, depends only on $b$ via the state trajectory $\{x^{t_{0},x_{0},a,b}_{s}\}_{s\in[0,t_{i}]}$.

The guaranteed result of the controller given a strategy $\mathbf{a}\in\mathbb{A}^{t_{0}}$ and a state $x_{0}\in\Omega$ is defined by

[TABLE]

The optimal guaranteed result of the controller $v_{a}:[0,T]\times\Omega\to{\mathbb{R}}$ is defined by

[TABLE]

Let us next consider the disturbance’s point of view. For every $\delta>0$, every partition $\pi:t_{0}<t_{1}<\cdots<t_{n}=T$ with mesh size $\left|{\pi}\right|\leq\delta$, every $x_{0}\in\Omega$, every feedback strategy $\mathbf{b}\in\mathbb{B}^{t_{0}}$, and every control $a\in\mathcal{A}^{t_{0}}$, define a step-by-step feedback control $b=b(\cdot;\pi,x_{0},a,\mathbf{b})\in\mathcal{B}^{t_{0}}$ recursively by

[TABLE]

The guaranteed result of the disturbance given a strategy $\mathbf{b}\in\mathbb{B}^{t_{0}}$ and a state $x_{0}\in\Omega$ is defined by

[TABLE]

The optimal guaranteed result of the disturbance $v_{b}:[0,T]\times\Omega\to{\mathbb{R}}$ is defined by

[TABLE]

Definition 8.4**.**

If $v_{a}=v_{b}$, then we call $v:=v_{a}=v_{b}$ the value of our game.

Let us also mention a perhaps more intuitive definition for the optimal guaranteed results. Let $(t_{0},x_{0})\in[0,T]\times\Omega$. Denoting by $\Pi^{t_{0}}$ the set of all (finite) partitions of $[t_{0},T]$, put

[TABLE]

Proposition 8.5**.**

We have $v_{b}\leq v^{\ast}_{b}\leq v^{\ast}_{a}\leq v_{a}$, i.e., if our game has value, both definitions for the optimal guaranteed results coincide.

Proof.

We show only $v_{b}\leq v_{a}$. The inequality $v_{b}^{\ast}\leq v_{a}^{\ast}$ can be shown similarly and the remaining inequalities $v_{b}\leq v_{b}^{\ast}$ and $v_{a}^{\ast}\leq v_{a}$ follow immediately from the definitions. Fix $(t_{0},x_{0})\in[0,T]\times\Omega$. Let $\mathbf{a}\in\mathbb{A}^{t_{0}}$, $\mathbf{b}\in\mathbb{B}^{t_{0}}$. Since for every partition $\pi$ of $[t_{0},T]$,

[TABLE]

we have

[TABLE]

Thus $J_{b}(t_{0},x_{0};\mathbf{b})\leq J_{a}(t_{0},x_{0};\mathbf{a})$, and since $\mathbf{a}$ and $\mathbf{b}$ were arbitrary feedback strategies, we obtain $v_{b}(t_{0},x_{0})\leq v_{a}(t_{0},x_{0})$. ∎

The Isaacs equation for our differential game is given by (TVP) with

[TABLE]

Assumption 8.6** (Isaacs condition).**

For every $(t,x,z)\in[0,T]\times C([0,T],H)\times H$,

[TABLE]

It can easily be shown that then H$(\ell)_{2}$ and H$(f)_{2}$ imply H($F$) (i) and (ii) with $L_{0}=L_{f}$. Moreover, similarly as in [Subbotin_book, Section 11.1, pp. 116ff.], one can show that in this case, for a.e. $t\in(0,T)$, and for every $x$, $y\in\Omega$, and every $z\in H$,

[TABLE]

i.e., we also have H($F$) (iii).

Theorem 8.7**.**

Suppose that H$(f)_{2}$, H$(\ell)_{2}$, and H$(h)_{1}$ hold. Let $u$ be the minimax solution of (TVP) on $[0,T]\times\Omega$ with $F$ given by (8.2). Then our game has value and $u=v$.

Proof.

First, recall that, by Theorem 5.8, the function $u$ is actually well-defined. Next, note that $v_{b}\leq v_{a}$ (see Proposition 8.5), Thus, we only need to show that $v_{a}\leq u\leq v_{b}$. Fix $(t_{0},x_{0})\in[0,T)\times\Omega$. We prove only $v_{a}(t_{0},x_{0})\leq u(t_{0},x_{0})$. To this end, it suffices to specify a family $(\mathbf{a}^{\varepsilon})_{{\varepsilon}>0}$ of strategies in $\mathbb{A}^{t_{0}}$ to show that $J_{a}(t_{0},x_{0};\mathbf{a}^{\varepsilon})\leq u(t_{0},x_{0})+m_{a}({\varepsilon})$, ${\varepsilon}>0$ for some modulus of continuity $m_{a}$. In order to construct such a family, we shall use auxiliary functions $\alpha^{\varepsilon}:[t_{0},T]\to{\mathbb{R}}$, $\beta^{\varepsilon}:[t_{0},T]\times C([0,T],H)\to{\mathbb{R}}$, and $\nu^{\varepsilon}:[t_{0},T]\times C([0,T],H)\to{\mathbb{R}}$, ${\varepsilon}>0$, defined by

[TABLE]

Moreover, put

[TABLE]

Then, for every ${\varepsilon}\in(0,{\varepsilon}_{0})$, we claim that $\nu^{\varepsilon}\in\mathcal{C}_{V}^{1,1}([t_{0},T]\times C([0,T],H))$ with

[TABLE]

To show this claim, write $\nu^{\varepsilon}(t,x)$ in the form $\psi(t,\xi(t),x(t))$, where

[TABLE]

Then, one can readily verify that $\psi\in\mathcal{C}^{1,1,1}_{V}([t_{0},T]\times{\mathbb{R}}\times H)$ and an application of Proposition 2.15 yields our claim.

Now we are in the position to specify our strategies. For every ${\varepsilon}\in(0,{\varepsilon}_{0})$, define $u^{\varepsilon}_{a}:[t_{0},T]\times\Omega\to{\mathbb{R}}$ by

[TABLE]

For every $(t,x)\in[t_{0},T]\times\Omega$, choose a path $x^{{\varepsilon},t,x}_{a}\in\mathcal{X}^{L_{f}}(t_{0},x_{0})$ for which the minimum in (8.4) is attained. Then we can define strategies $\mathbf{a}^{\varepsilon}\in\mathbb{A}^{t_{0}}$ as follows. For every $(t,x)\in[t_{0},T]\times\Omega$, choose an $\mathbf{a}^{\varepsilon}(t,x)\in P$ such that

[TABLE]

Given $\delta>0$, a partition $\pi$ of $[t_{0},T]$ with $\left|{\pi}\right|\leq\delta$, and a control $b\in\mathcal{B}^{t_{0}}$, put $x^{\pi,b}:=x^{t_{0},x_{0},a(\cdot;\pi,x_{0},\mathbf{a}^{\varepsilon},b),b}$.

We shall show the following statement:

S. For every $i\in\{0,\ldots,n-1\}$,

[TABLE]

for some modulus of continuity $m$.

According to (8.4), for every $\tilde{x}\in\mathcal{X}^{L_{f}}(t_{0},x_{0})$,

[TABLE]

where $x^{\varepsilon}_{a,i}:=x^{{\varepsilon},t_{i},x^{\pi,b}}_{a}\in\mathcal{X}^{L_{f}}(t_{0},x_{0})$ is a path for which the minimum in (8.4) with $(t,x)$ replaced by $(t_{i},x^{\pi,b})$ is attained.

Note that $u$ is also a minimax $L_{f}$-solution of (TVP) on $[0,T]\times\Omega$. Thus we can assume that $\tilde{x}\in\mathcal{X}^{L_{f}}(t_{i},{x}^{\varepsilon}_{a,i})$ and that

[TABLE]

Then (8.5) becomes

[TABLE]

i.e.,

[TABLE]

Since $f$ and $\ell$ as well as suitable compositions of $\nu^{\varepsilon}$, $\partial_{t}\nu^{\varepsilon}$, and $\partial_{x}\nu^{\varepsilon}$ are uniformly continuous on the compact set $\Omega^{L_{f}}$ (and also on finite Cartesian products of said set), since $\mathcal{X}^{L_{f}}(t_{0},x_{0})$ is equicontinuous (see [Kelley, Theorem 7.16, p. 233]), since $A$ is monotone, and since

[TABLE]

there exists a modulus $m$ such that

[TABLE]

where $y:=x^{\pi,b}-x_{a,i}^{\varepsilon}$, and thus Statement S has been proven.

Next, note that

[TABLE]

Assuming for a moment that there exists a modulus of continuity $m_{a}$ independent from $\pi$ and $b$ such that

[TABLE]

is valid, we can from the definition of $\nu^{\varepsilon}$ and from S infer that, for every $\delta>0$,

[TABLE]

which would imply that

[TABLE]

and this would conclude the proof.

Therefore it remains to find a modulus $m_{a}$ such that (8.6) holds. We have

[TABLE]

where $x^{\varepsilon}$ is an appropriate minimizer. Thus

[TABLE]

which implies, by the definition of $\nu^{\varepsilon}$,

[TABLE]

and therefore, proceeding along the lines of (A.9), we obtain

[TABLE]

where $C=C(t_{0},x_{0},L_{f})$ is the constant from the a-priori estimates of Lemma A.1. Hence, together with $h$ being uniformly continuous on the compact set $\Omega^{L_{f}}$, we can infer the existence of a modulus $m_{a}$ such that

[TABLE]

i.e., we have (8.6) and thus our proof is complete. ∎

Appendix A Properties of solution sets of evolution equations

We shall use the Nemytsky operator $\hat{A}:L^{p}(t_{0},T;V)\to L^{q}(t_{0},T;V^{\ast})$ corresponding to $A$, which is defined by

[TABLE]

The derivation of a-priori estimates like the following is standard in the literature. Nevertheless, for the convenience of the reader, we provide a complete proof.

Lemma A.1**.**

Fix $(t_{0},x_{0})\in[0,T)\times C([0,T],H)$ and $L\geq 0$. There exists a constant $C=C(t_{0},x_{0},L)>0$ such that, for all $x\in\mathcal{X}^{L}(t_{0},x_{0})$,

[TABLE]

Proof.

Let $x\in\mathcal{X}^{L}(t_{0},x_{0})$. Then

[TABLE]

By integration-of-parts, the coercivity of $A$, and Young’s inequality, for every ${\varepsilon}>0$,

[TABLE]

Here, the constant $C_{1}>0$ comes from the continuous embedding of $V$ into $H$, i.e., $\left|{v}\right|\leq C_{1}\lVert{v}\rVert$ for all $v\in V$. Now, choose ${\varepsilon}>0$ such that $c_{2}=p^{-1}\,{\varepsilon}^{p}\,C_{1}^{p}$. Then

[TABLE]

Put $m(t):=\sup_{s\leq t}\left|{x(s)}\right|$, $t\in[t_{0},T]$. Then integrating (A.3) yields

[TABLE]

Noting that $p\geq 2$ implies $q\leq 2$ and thus $[1+m(s)]^{q}\leq[1+m(s)]^{2}$, we get

[TABLE]

By Gronwall’s inequality,

[TABLE]

i.e., $\lVert{x}\rVert_{\infty}\leq C_{2}=C_{2}(t_{0},x_{0},L)$. Furthermore,

[TABLE]

i.e.,

[TABLE]

Moreover, by the boundedness of $A$ and since $(p-1)q=p$,

[TABLE]

Finally,

[TABLE]

and

[TABLE]

Here, the constant $C_{5}>0$ comes from the continuous embedding of $H$ into $V^{\ast}$, i.e., $\lVert{h}\rVert_{\ast}\leq C_{5}\left|{h}\right|$ for all $h\in H$. ∎

Lemma A.2**.**

Fix $(t_{0},x_{0})\in[0,T)\times C([0,T],H)$ and $L\geq 0$. Consider a sequence $(x_{n})_{n}$ in $\mathcal{X}^{L}(t_{0},x_{0})$. Then there exist a pair $({\mathbf{x}},f)\in W_{pq}(t_{0},T)\times L^{2}(t_{0},T;H)$ with ${\mathbf{x}}^{\prime}+\hat{A}{\mathbf{x}}=f$ and a subsequence $(x_{n_{k}})_{k}$ of $(x_{n})_{n}$ such that ${\mathbf{x}}_{n_{k}}\xrightarrow{w}{\mathbf{x}}$ in $W_{pq}(t_{0},T)$, $x_{n_{k}}\to x$ in $C([0,T],H)$, and $f^{x_{n}}\xrightarrow{w}f$ in $L^{2}(t_{0},T;H)$. Here, $x$ is the unique element of $C([0,T],H)$ such that $x={\mathbf{x}}$ a.e. on $(t_{0},T)$ and $x=x_{0}$ on $[0,t_{0}]$.

Proof.

Major parts of this proof are slight modifications of the proofs of [ZeidlerIIB, Lemma 30.5 and Lemma 30.6, p. 776f.], [Tolstonogov99Sbornik, Assertion 2.1], and [TT1999NoDEA, Proposition 3.1] (cf. also the proofs of [ZCYC13, Theorem 4.1] and [LiuLiu14, Lemma 4.3]). First, recall that each $x_{n}$ satisfies $\mathbf{x}^{\prime}_{n}+\hat{A}\mathbf{x}_{n}=f_{n}$, where $f_{n}:=f^{x_{n}}$. Also note that $W_{pq}(t_{0},T)$ is reflexive (see, e.g. [HPHandbookII, Theorem 1.12 (a), p. 4]). Thus, using the the a-priori estimates from Lemma A.1, we can assume, by the Banach-Alaoglu theorem and the Eberlein-Šmulian theorem, that $(\mathbf{x}_{n})_{n}$ converges weakly to some $\mathbf{x}\in W_{pq}(t_{0},T)$. Then $(\mathbf{x}_{n})_{n}$ converges also strongly to $\mathbf{x}$ in $L^{p}(t_{0},T;H)$ because $W_{pq}(t_{0},T)$ is compactly embedded into $L^{p}(t_{0},T;H)$ (see [JLLions1969, Théorème 5.1, p. 58]). Similarly, we can also assume that $(\hat{A}\mathbf{x}_{n})_{n}$ converges weakly to some $w\in L^{q}(t_{0},T;V^{\ast})$, that $(f_{n})_{n}$ converges weakly to some $f\in L^{2}(t_{0},T;H)$, and that $(x_{n}(T))_{n}$ converges weakly to some $z$ in $H$.

Next, we show that

[TABLE]

To this end, let $\psi\in L^{p}(t_{0},T;V)$. Then

[TABLE]

as $n\to\infty$, i.e., (A.6) holds. In particular, we could use $f_{n}\xrightarrow{w}f$ in $L^{q}(t_{0},T;V^{\ast})$ because $L^{2}(t_{0},T;H)\subset L^{q}(t_{0},T;V^{\ast})$. To show (A.7), let $v\in V$ and consider $\psi\in C^{1}([t_{0},T],V)$ defined by $\psi(t):=\varphi(t)v$, $t\in[t_{0},T]$, for some $\varphi\in C^{\infty}([t_{0},T])$ with $\varphi(T)=1$ and $\varphi(t_{0})=0$. Then, by the integration-by-parts formula,

[TABLE]

Since $\psi(T)=v$, the vector $v$ was arbitrarily chosen in $V$, and $V$ is dense in $H$, we have (A.7).

Now, we show that $w=\hat{A}\mathbf{x}$. The proof of this statement is nearly identical to the proof of [ZeidlerIIB, Lemma 30.6, p. 777f.] up to very minor modifications. First, note that, by [ZeidlerIIB, Theorem 30.A (c), p. 771], the operator $\hat{A}$ is monotone and hemicontinuous. According to the monotonicity trick (see, e.g., [ZeidlerIIB, p. 474]), it suffices to show that

[TABLE]

To this end, we apply the integration-by-parts formula to obtain

[TABLE]

By (A.7), $x_{n}(T)\xrightarrow{w}x(T)$ in $H$ and thus together with [ZeidlerIIA, Proposition 21.23 (c)], $\left|{x(T)}\right|\leq\varliminf_{n}\left|{x_{n}(T)}\right|$. Moreover, since $\langle f_{n},\mathbf{x}_{n}\rangle_{L^{p}(t_{0},T;V)}=\langle f_{n},x_{n}\rangle_{L^{2}(t_{0},T;H)}$, $f_{n}\xrightarrow{w}f$ in $L^{2}(t_{0},T;H)$, and $x_{n}\to x$ in $L^{2}(t_{0},T;H)$, we have

[TABLE]

Thus, by (A.6) and again by the integration-by-parts formula,

[TABLE]

i.e., we have shown (A.8).

It remains to show that $x_{n}\to x$ in $C([0,T],H)$. Since $\mathbf{x}^{\prime}+\hat{A}\mathbf{x}=f$ and $\hat{A}$ is monotone, the integration-by-parts formula yields, for every $t\in(t_{0},T]$,

[TABLE]

As $\lVert{x_{n}-x}\rVert_{L^{2}(t_{0},T;H)}\to 0$ (by [JLLions1969, Théorème 5.1, p. 58]), we obtain $\lVert{x_{n}-x}\rVert_{\infty}\to 0$, which concludes the proof. ∎

Lemma A.3**.**

Fix $(t_{0},x_{0})\in[0,T)\times C([0,T],H)$ and $L\geq 0$. Then $\mathcal{X}^{L}(t_{0},x_{0})$ is closed in $C([0,T],H)$.

Proof.

Let $(x_{n})_{n}$ be a sequence in $\mathcal{X}^{L}(t_{0},x_{0})$ that converges to some $x\in C([0,T],H)$. By Lemma A.2, there exists a pair $({\mathbf{x}},f)\in W_{pq}(t_{0},T)\times L^{2}(t_{0},T;H)$ with ${\mathbf{x}}^{\prime}+\hat{A}{\mathbf{x}}=f$ such that $x={\mathbf{x}}$ a.e. on $(t_{0},T)$ and $f^{x_{n_{k}}}\xrightarrow{w}f$ in $L^{2}(t_{0},T;H)$ for some subsequence $(x_{n_{k}})_{k}$ of $(x_{n})_{n}$. Put $f_{k}:=f^{x_{n_{k}}}$, $k\in{\mathbb{N}}$. It suffices to show that

[TABLE]

By Lemma A.1, there exists a $C\geq 0$ such that, for every $k\in{\mathbb{N}}$,

[TABLE]

Since $\{h\in H:\left|{h}\right|\leq L(1+C)\}$ is weakly compact, non-empty, and convex, [P90JMAAOnAPaper, Theorem 3.1] yields

[TABLE]

(This result can also be found in [HPHandbookI, Proposition 3.9, p. 694] and in [P87IJMMS, Theorem 3.1].) In (A.12), \text{w$-$\varlimsup}\{f_{k}(t)\}_{k\in{\mathbb{N}}} is the set of all $h\in H$ for which some subsequence $(f_{k_{l}}(t))_{l}$ of $(f_{k}(t))_{k}$ converges weakly to $h$ in $H$ and \overline{\mathrm{conv}}\text{ w$-$\varlimsup}\{f_{k}(t)\}_{k\in{\mathbb{N}}} is the closure of the convex hull of \text{w$-$\varlimsup}\{f_{k}(t)\}_{k\in{\mathbb{N}}}. Now, fix an h\in\text{ w$-$\varlimsup}\{f_{k}(t)\}_{k\in{\mathbb{N}}} with a corresponding subsequence $(f_{k_{l}}(t))_{l}$. Then, by [ZeidlerIIA, Proposition 21.23 (c), p. 258] (Banach-Steinhaus), by (A.11), and since $x_{n}\to x$ in $C([0,T],H)$,

[TABLE]

Consequently, (A.12) yields (A.10). ∎

Remark A.4**.**

If, instead of $\mathcal{X}^{L}(t_{0},x_{0})$, we consider

[TABLE]

then the inequality corresponding to (A.10) can be obtained from Mazur’s lemma immediately (i.e., without invoking [P90JMAAOnAPaper, Theorem 3.1]) as follows. Let $f_{n}\xrightarrow{w}f$ in $L^{2}(t_{0},T;H)$. By Mazur’s lemma (see, e.g., [WernerFA, Korollar III.3.9, p. 108]), there exists, for each $n\in{\mathbb{N}}$, a convex linear combination $\tilde{f}_{n}$ of members of $(f_{m})_{m}$, say $\tilde{f}_{n}=\sum\nolimits_{i=1}^{N(n)}\lambda_{i}^{(n)}f_{i}$, where $\lambda_{i}^{(n)}\in(0,1)$, $i=1$, $\ldots$, $N(n)$, and $\sum_{i=1}^{N(n)}\lambda_{i}^{(n)}=1$, such that $\tilde{f}_{n}\to f$ in $L^{2}(t_{0},T;H)$. Then there exists a subsequence of $(\tilde{f}_{n})_{n}$, which we still denote by $(\tilde{f}_{n})_{n}$, that converges to $f$ a.e. on $(t_{0},T)$. Hence, for a.e. $t\in(t_{0},T)$ and for every ${\varepsilon}>0$, there exists an $n_{0}\in{\mathbb{N}}$ such that, for all $n\geq n_{0}$,

[TABLE]

For the next statement, recall that $\mathcal{Y}^{L}(t_{0},x_{0},\tilde{z})$ is defined by (5.1).

Lemma A.5**.**

Fix $(t_{0},x_{0},\tilde{z})\in[0,T)\times C([0,T],H)\times H$ and let $L\geq 0$. Then $\mathcal{X}^{L}(t_{0},x_{0})$ is path-connected in $C([0,T],H)$ and $\mathcal{Y}^{L}(t_{0},x_{0},\tilde{z})$ is path-connected in $C([0,T],H)\times C([t_{0},T])$.

Proof.

Path-connectedness of $\mathcal{X}^{L}(t_{0},x_{0})$ follows immediately from [PS97AMO, Theorem 2] (see also [HPHandbookII, Theorem 3.5, p. 50f.]). For the sake of completeness, we provide a shortened proof in our less general setting.

Let $x$, $y\in\mathcal{X}^{L}(t_{0},x_{0})$. Given $s\in[t_{0},T]$, denote by $z_{s}$ the unique element of $\mathcal{X}(t_{0},x_{0})$ that satisfies

[TABLE]

It suffices to show that the mapping $s\mapsto z_{s}$, $[t_{0},T]\to C([0,T],H)$, is continuous. We show only right-continuity. Left-continuity can be shown similarly. To this end, let $s_{n}\downarrow s$ in $[t_{0},T)$. Since $\mathcal{X}^{L}(t_{0},x_{0})$ is compact (Lemmas A.2 and A.3), we can assume without loss of generality that $z_{s_{n}}\to z$ in $C([0,T],H)$ for some $z\in\mathcal{X}^{L}(t_{0},x_{0})$. Note that $z=z_{s}$ on $[0,s]$. Since $A$ is monotone, we have, for every $N\in{\mathbb{N}}$, a.e. $t\in[s_{N},T]$, and every $n\geq N$,

[TABLE]

hence $(1/2)(d/dt)\left|{z_{s_{n}}(t)-z_{s}(t)}\right|^{2}\leq 0$ and thus, for every $t\in[s_{N},T]$,

[TABLE]

Consequently, for every $t\in[s,T]$ and sufficiently large $N\in{\mathbb{N}}$,

[TABLE]

Thus we have shown that $\mathcal{X}^{L}(t_{0},x_{0})$ is path-connected in $C([0,T],H)$. To finish the proof, it suffices to verify that the mapping $s\mapsto(z_{s},u_{s})$, $[t_{0},T]\to C([0,T],H)\times C([t_{0},T])$, with $u_{s}$ being defined by

[TABLE]

is continuous. This can be done in the same manner as the proof of continuity for the mapping $s\mapsto z_{s}$, $[t_{0},T]\to C([0,T],H)$. ∎

Appendix B Other notions of solutions

B.1. Classical solutions

We introduce here a new notion of classical solution for our setting. This notion is compatible with our notion of minimax solution solutions. Moreover, it is a faithful extension of the usual classical solution in the case when $H=V$ and $A$ is continuous. We refer to Proposition B.3 for more details regarding the last point.

Definition B.1**.**

Let $u\in\mathcal{C}_{V}^{1,1}([0,T]\times C([0,T],H))$. Let $L\geq 0$ and let $\tilde{\Omega}$ be some set with $\Omega^{L}\subseteq\tilde{\Omega}\subseteq C([0,T],H)$.

(i) We say that the function $u$ is a classical $L$-supersolution of (TVP) on $[0,T]\times\tilde{\Omega}$ if $u(T,\cdot)\geq h$, and if, for every $(t_{0},x_{0},z)\in[0,T)\times\Omega\times H$, there exists a trajectory $x\in\mathcal{X}^{L}(t_{0},x_{0})$ such that

[TABLE]

(ii) We say that the function $u$ is a classical $L$-subsolution of (TVP) on $[0,T]\times\tilde{\Omega}$ if $u(T,\cdot)\leq h$, and if, for every $(t_{0},x_{0},z)\in[0,T)\times\Omega\times H$, there exists a trajectory $x\in\mathcal{X}^{L}(t_{0},x_{0})$ such that

[TABLE]

(iii) We say that $u$ is a classical $L$-solution of (TVP) if $u(T,\cdot)=h$, and if, for every $(t_{0},x_{0},z)\in[0,T)\times\Omega\times H$, there exists an $x\in\mathcal{X}^{L}(t_{0},x_{0})$ such that

[TABLE]

The following statement justifies our slightly involved notion of classical solutions, in particular, the reason that, in the case of classical $L$-supersolutions, we want (B.1) to hold for all $z\in H$ and not only for $\partial_{x}u(t_{0},x_{0})$.

Theorem B.2** (Compatibility).**

Let $u\in\mathcal{C}_{V}^{1,1}([0,T]\times C([0,T],H))$. Let $L\geq 0$ and let $\tilde{\Omega}$ be some set with $\Omega^{L}\subseteq\tilde{\Omega}\subseteq C([0,T],H)$. Then $u$ is a minimax $L$-solution (resp. minimax $L$-subsolution, resp. minimax $L$-supersolution) of (TVP) on $[0,T]\times\tilde{\Omega}$ if and only if $u$ is a classical $L$-solution (resp. classical $L$-subsolution, resp. classical $L$-supersolution) of (TVP) on $[0,T]\times\tilde{\Omega}$..

Proof.

We prove only the case of equivalence between minimax $L$-supersolutions and classical $L$-supersolutions. To this end, fix $(t_{0},x_{0},z)\in[0,T)\times\tilde{\Omega}\times H$.

(i) Let $u$ be a classical $L$-supersolution of (TVP) on $[0,T]\times\tilde{\Omega}$. Then there exists an $x\in\mathcal{X}^{L}(t_{0},x_{0})$ such that (B.1) holds. To show that $u$ is an upper solution, it suffices, by Proposition 3.8, to check that (3.4) holds. Indeed, applying the functional chain rule (2.5) as well as (B.1) yields

[TABLE]

(ii) Let $u$ be a minimax $L$-supersolution of (TVP). Then there exists an $x\in\mathcal{X}^{L}(t_{0},x_{0})$ such that, for every $\delta\in(0,T-t_{0}]$,

[TABLE]

Hence,

[TABLE]

i.e, (B.1) is satisfied. ∎

Proposition B.3**.**

Suppose that $V=H$ and that the map $(t,v)\mapsto A(t,v)$, $[0,T]\times H\to H$, is continuous. Let $u\in\mathcal{C}^{1,1}_{V}([0,T]\times\ C([0,T],H))$ with $u(T,\cdot)\geq h$. Let $\Omega^{L_{0}}\subseteq\tilde{\Omega}\subseteq C([0,T],H)$. Then $u$ is a classical $L_{0}$-supersolution of (TVP) on $[0,T]\times\tilde{\Omega}$ in the sense of Definition B.1 if and only if $u$ is a classical supersolution of (TVP) on $[0,T]\times\tilde{\Omega}$ in the usual sense, i.e., for every $(t_{0},x_{0})\in[0,T)\times\tilde{\Omega}$,

[TABLE]

Proof.

We show only the nontrivial direction. To this end, let $u$ be a classical supersolution of (TVP) in the usual sense on $[0,T]\times\tilde{\Omega}$ and fix $(t_{0},x_{0},z)\in[0,T)\times\tilde{\Omega}\times H$. Furthermore, let $\tilde{\mathbf{x}}\in W_{pq}(t_{0},T)$ be the solution of

[TABLE]

where $\tilde{f}:[t_{0},T]\times C([0,T],H)\to H$ is defined by

[TABLE]

Then $\tilde{x}\in\mathcal{X}^{L_{0}}(t_{0},x_{0})$ with $f^{\tilde{x}}=\tilde{f}(\cdot,\tilde{x})$ and

[TABLE]

I.e., $u$ is a classical $L_{0}$-supersolution of (TVP) on $[0,T]\times\tilde{\Omega}$ in the sense of Definition B.1 . ∎

B.2. Viscosity solutions

Given $L\geq 0$, a set $\tilde{\Omega}$ with $\Omega^{L}\subseteq\tilde{\Omega}\subseteq C([0,T],H)$, a non-anticipating function $u:[0,T]\times\tilde{\Omega}\times\to{\mathbb{R}}$, and $(t_{0},x_{0})\in[0,T)\times\tilde{\Omega}$, put

[TABLE]

Definition B.4**.**

Fix a set $\tilde{\Omega}$ with $\Omega^{L}\subseteq\tilde{\Omega}\subseteq C([0,T],H)$.

(i) Let $L\geq 0$. A function $u\in\mathrm{USC}([0,T]\times\tilde{\Omega})$ is a viscosity $L$-subsolution of (TVP) on $[0,T]\times\tilde{\Omega}$ if, for every $(t_{0},x_{0})\in[0,T)\times\tilde{\Omega}$ and for every test function $\varphi\in\underline{\mathcal{A}}^{L}u(t_{0},x_{0})$, there exists an $x\in\mathcal{X}^{L}(t_{0},x_{0})$ such that

[TABLE]

(ii) Let $L\geq 0$. A function $u\in\mathrm{LSC}([0,T]\times\tilde{\Omega})$ is a viscosity $L$-supersolution of (TVP) on $[0,T]\times\tilde{\Omega}$ if, for every $(t_{0},x_{0})\in[0,T)\times\tilde{\Omega}$ and for every test function $\varphi\in\underline{\mathcal{A}}^{L}u(t_{0},x_{0})$, there exists an $x\in\mathcal{X}^{L}(t_{0},x_{0})$ such that

[TABLE]

(iii) A function $u\in\mathrm{USC}([0,T]\times\tilde{\Omega})$ (resp. $\in\mathrm{LSC}([0,T]\times\tilde{\Omega})$ is a viscosity subsolution (resp. viscosity supersolution) of (TVP) on $[0,T]\times\tilde{\Omega}$ if it is a viscosity $L$-subsolution (resp. viscosity $L$-supersolution) of (TVP) on $[0,T]\times\tilde{\Omega}$ for some $L\geq 0$.

(iv) A function $u\in C([0,T]\times\tilde{\Omega})$ is a viscosity solution of (TVP) on $[0,T]\times\tilde{\Omega}$ if it is both, a viscosity sub- as well as a viscosity supersolution of (TVP) on $[0,T]\times\tilde{\Omega}$.

Similarly as in the proof of Theorem 4.3, one can show that a minimax solution is a viscosity solution. Thus, together with Theorem 5.8, we obtain the following result.

Theorem B.5** (Existence).**

There exists a viscosity solution of (TVP) on $[0,T]\times\Omega$.