Minimax and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations for Time-Delay Systems

TL;DR

This paper investigates Hamilton-Jacobi-Bellman equations for time-delay systems, establishing the existence, uniqueness, and equivalence of minimax and viscosity solutions to the value functional in a delay differential control problem.

Contribution

It introduces a framework for analyzing Hamilton-Jacobi-Bellman equations with delay, proving the equivalence of minimax and viscosity solutions for such systems.

Findings

Existence of minimax and viscosity solutions

Uniqueness of these solutions

Solutions coincide with the value functional

Abstract

The paper deals with a Bolza optimal control problem for a dynamical system which motion is described by a delay differential equation under an initial condition defined by a piecewise continuous function. For the value functional in this problem, the Cauchy problem for the Hamilton-Jacobi-Bellman equation with coinvariant derivatives is considered. Minimax and viscosity solutions of this problem are studied. It is proved that both of these solutions exist, are unique and coincide with the value functional.

Equations

$$\mathrm{PC}=\mathrm{PC}([-h,0],{\mathbb{R}}^n),\mathbb{G}=[t_0,\vartheta ]\times {\mathbb{R}}^n\times \mathrm{PC}.$$

$$\Vert w(\cdot ){\Vert }_1=\underset{-h}{\overset{0}{\int }}\Vert w(\xi )\Vert \mathrm{d}\xi ,\Vert w(\cdot ){\Vert }_{\infty }=\underset{\xi \in [-h,0]}{\sup }\Vert w(\xi )\Vert ,w(\cdot )\in \mathrm{PC}.$$

$$\dot{x}(\tau )=f(\tau ,x(\tau ),x(\tau -h),u(\tau )),\tau \in [t_0,\vartheta ],x(\tau )\in {\mathbb{R}}^n,u(\tau )\in \mathbb{U}.$$

$$\begin{array}[]{rl}\Lambda(t,z,w(\cdot))=\big{\{}x(\cdot)\in\mathrm{PC}([t-h,\vartheta],\mathbb{R}^{n})\colon x(\tau)=w(\tau-t),\ \tau\in[t-h,t),\\[5.69046pt] x(\tau)=y(\tau),\ \tau\in[t,\vartheta],\ y(\cdot)\in\mathrm{Lip}([t,\vartheta],\mathbb{R}^{n}),\ y(t)=z\big{\}}.\end{array}$$

$$J(t,z,w(\cdot ),u(\cdot ))=\sigma (x(\vartheta ),x_{\vartheta }(\cdot ))+\underset{t}{\overset{\vartheta }{\int }}{f}^0(\xi ,x(\xi ),x(\xi -h),u(\xi ))\mathrm{d}\xi ,$$

$$\begin{array}[]{c}\big{\|}f(t,x,y,u)-f(t,x^{\prime},y^{\prime},u)\big{\|}+\big{|}f^{0}(t,x,y,u)-f^{0}(t,x^{\prime},y^{\prime},u)\big{|}\\[8.5359pt] \leq\lambda_{f}\big{(}\|x-x^{\prime}\|+\|y-y^{\prime}\|\big{)}\end{array}$$

$$\big{\|}f(t,x,y,u)\big{\|}+\big{|}f^{0}(t,x,y,u)\big{|}\leq c_{f}(1+\|x\|+\|y\|)$$

$$\big{|}\sigma(z,w(\cdot))-\sigma(z^{\prime},w^{\prime}(\cdot))\big{|}\leq\lambda_{\sigma}\big{(}\|z-z^{\prime}\|+\|w(\cdot)-w^{\prime}(\cdot)\|_{1}\big{)}$$

$$P(\alpha)=\big{\{}(z,w(\cdot))\in\mathbb{R}^{n}\times\mathrm{PC}\colon\|z\|\leq\alpha,\,\|w(\cdot)\|_{\infty}\leq\alpha\big{\}}.$$

$$\rho (t,z,w(\cdot ))=\underset{u(\cdot )\in \mathfrak{U}(t)}{\inf }J(t,z,w(\cdot ),u(\cdot )),(t,z,w(\cdot ))\in \mathbb{G}.$$

$$\rho(t,z,w(\cdot))=\inf\limits_{u(\cdot)\in\mathfrak{U}(t)}\bigg{(}\rho(\tau,x(\tau),x_{\tau}(\cdot))+\int\limits_{t}^{\tau}f^{0}(\xi,x(\xi),x(\xi-h),u(\xi))\mathrm{d}\xi\bigg{)},$$

$$\begin{array}[]{c}\varphi(\tau,v,x_{\tau}(\cdot))-\varphi(t,z,w(\cdot))=\partial^{ci}_{t,w}\varphi(t,z,w(\cdot))(\tau-t)\\[8.5359pt] +\langle v-z,\nabla_{z}\varphi(t,z,w(\cdot))\rangle+o(|\tau-t|+\|v-z\|),\end{array}$$

$$\begin{array}[]{c}H(t,x,y,s)=\min\limits_{u\in\mathbb{U}}\big{(}\langle f(t,x,y,u),s\rangle+f^{0}(t,x,y,u)\big{)},\\[8.5359pt] t\in[t_{0},\vartheta],\quad x,y,s\in\mathbb{R}^{n}.\end{array}$$

$$\begin{array}[]{c}\partial^{ci}_{t,w}\varphi(t,z,w(\cdot))+H(t,z,w(-h),\nabla_{z}\varphi(t,z,w(\cdot)))=0,\\[8.5359pt] (t,z,w(\cdot))\in\mathbb{G},\quad t<\vartheta,\end{array}$$

$$\phi (\vartheta ,z,w(\cdot ))=\sigma (z,w(\cdot )),(\vartheta ,z,w(\cdot ))\in \mathbb{G}.$$

$$|\varphi(t,z,w(\cdot))-\varphi(t,z^{\prime},w^{\prime}(\cdot))|\leq\lambda_{\varphi}\big{(}\|z-z^{\prime}\|+\|w(\cdot)-w^{\prime}(\cdot)\|_{1}\big{)}$$

$$F(x,y)=\big{\{}l\in\mathbb{R}^{n}\colon\|l\|\leq c_{f}(1+\|x\|+\|y\|)\big{\}}\subset\mathbb{R}^{n},\quad x,y\in\mathbb{R}^{n}.$$

$$\dot{x}(\tau )\in F(x(\tau ),x(\tau -h))\text{ for a.e. }\tau \in [t,\vartheta ].$$

$$x(\cdot )\in X(t,z,w(\cdot )).$$

$$\displaystyle\begin{array}[]{rl}\inf\limits_{x(\cdot)\in X(t,z,w(\cdot))}&\bigg{(}\varphi(\tau,x(\tau),x_{\tau}(\cdot))-\varphi(t,z,w(\cdot))\\[5.69046pt] &\displaystyle\quad+\int\limits_{t}^{\tau}\big{(}H(\xi,x(\xi),x(\xi-h),s)-\langle\dot{x}(\xi),s\rangle\big{)}\mathrm{d}\xi\bigg{)}\leq 0,\end{array}$$

$$\displaystyle\begin{array}[]{rl}\sup\limits_{x(\cdot)\in X(t,z,w(\cdot))}&\bigg{(}\varphi(\tau,x(\tau),x_{\tau}(\cdot))-\varphi(t,z,w(\cdot))\\[5.69046pt] &\displaystyle\quad+\int\limits_{t}^{\tau}\big{(}H(\xi,x(\xi),x(\xi-h),s)-\langle\dot{x}(\xi),s\rangle\big{)}\mathrm{d}\xi\bigg{)}\geq 0,\end{array}$$

$${\partial }_l^{-}\phi (t,z,w(\cdot ))$$

$${\partial }_l^{+}\phi (t,z,w(\cdot ))$$

$$\displaystyle\begin{array}[]{rl}D^{-}\varphi(t,z,w(\cdot))=&\big{\{}(p_{0},p)\in\mathbb{R}\times\mathbb{R}^{n}\colon\\[2.84544pt] &\quad p_{0}+\langle l,p\rangle\leq\partial^{-}_{l}\varphi(t,z,w(\cdot)),\,l\in\mathbb{R}^{n}\big{\}},\end{array}$$

$$\displaystyle\begin{array}[]{rl}D^{+}\varphi(t,z,w(\cdot))=&\big{\{}(q_{0},q)\in\mathbb{R}\times\mathbb{R}^{n}\colon\\[2.84544pt] &\quad q_{0}+\langle l,q\rangle\geq\partial^{+}_{l}\varphi(t,z,w(\cdot)),\,l\in\mathbb{R}^{n}\big{\}}.\end{array}$$

$${\partial }_l^{-}\phi (t,z,w(\cdot ))={\partial }_l^{+}\phi (t,z,w(\cdot ))={\partial }_{t,w}^{ci}\phi (t,z,w(\cdot ))+⟨l,{\nabla }_z\phi (t,z,w(\cdot ))⟩,$$

$$\displaystyle D^{-}\varphi(t,z,w(\cdot))=\big{\{}(p_{0},p)\colon p_{0}\leq\partial^{ci}_{t,w}\varphi(t,z,w(\cdot)),\,p=\nabla_{z}\varphi(t,z,w(\cdot))\big{\}},$$

$$\displaystyle D^{+}\varphi(t,z,w(\cdot))=\big{\{}(q_{0},q)\colon q_{0}\geq\partial^{ci}_{t,w}\varphi(t,z,w(\cdot)),\,q=\nabla_{z}\varphi(t,z,w(\cdot))\big{\}}.$$

$$p_0+H(t,z,w(-h),p)$$

$$q_0+H(t,z,w(-h),q)$$

$$\displaystyle\inf\limits_{l\in F(z,w(-h))}\big{(}\partial^{-}_{l}\varphi(t,z,w(\cdot))+H(t,z,w(-h),s)-\langle l,s\rangle\big{)}$$

$$\displaystyle\sup\limits_{l\in F(z,w(-h))}\big{(}\partial^{+}_{l}\varphi(t,z,w(\cdot))+H(t,z,w(-h),s)-\langle l,s\rangle\big{)}$$

$$(x(\tau ),x_{\tau }(\cdot ))\in P({\alpha }_X),\Vert x(\tau )-x({\tau }^{\prime })\Vert \le {\lambda }_X|\tau -{\tau }^{\prime }|,\tau ,{\tau }^{\prime }\in [t,\vartheta ]$$

$$\|x(\tau)\|\leq\|z\|+c_{f}\int\limits_{t}^{\tau}\big{(}1+\|x(\xi)\|+\|x(\xi-h)\|\big{)}\mathrm{d}\xi\leq\alpha_{*}+2c_{f}\int\limits_{t}^{\tau}\|x(\xi)\|\mathrm{d}\xi.$$

Full text

††journal: JOTA

1 Introduction

In optimal control problems for dynamical systems which motions are described by ordinary differential equations, studies of infinitesimal properties of a value function lead to a Hamilton-Jacobi-Bellman (HJB) equation, which is a particular case of Hamilton-Jacobi (HJ) equations with partial derivatives. In the case when an optimal control problem is considered on a finite time interval and has a cost functional of Bolza type, a value function satisfies the corresponding natural terminal condition, which, together with the HJB equation, determine the Cauchy problem. Since in many cases Cauchy problems for HJ equations do not have a classical (continuously differentiable) solution, various approaches to a notion of a generalized solution were developed. The main of them are minimax and viscosity approaches. The minimax approach Subbotin_1980; Subbotin_1984; Subbotin_1995 originates in the positional differential game theory Krasovskii_Subbotin_1988; Krasovskii_Krasovskii_1995. According to this approach, a generalized (minimax) solution is a function that satisfies the pair of stability conditions with respect to characteristic differential inclusions. In infinitesimal form, these conditions reduce to the pair of inequalities for directional derivatives. In the viscosity approach Crandall_Lions_1983; Crandall_Evans_Lions_1984, a HJ equation is replaced by the pair of inequalities for sub- and supergradients, and a generalized (viscosity) solution is a function satisfying these inequalities. In investigations of minimax and viscosity solutions of Cauchy problems for HJB equations, it was shown (see, e.g., Subbotin_1995; Barbu_1986; Bardi_Capuzzo-Dolcetta_1997; Evans_1998) that both of these solutions exist, are unique and coincide with the value function in the corresponding optimal control problems. The goal of the paper is to obtain the similar result in the case when a motion of a dynamical system is described by delay differential equations.

The first investigations of control problems for time-delay systems showed (see Krasovskii_1962; Osipov_1971 and also Oguztoreli_1966; Banks_1968; Banks_Manitius_1974) that an analogue of a value function in such problems is a value functional on a space of motion histories. It raises natural questions about the suitable notion of the differentiability of such functionals, the corresponding notions of directional derivatives, sub- and supergradients, and definitions of generalized solutions of the corresponding HJB equations.

The viscosity solution theory for HJ equations with Frechet derivatives began with Crandall_Lions_1985; Crandall_Lions_1986a. In these papers, the definition of the viscosity solution in terms of inequalities for Frechet sub- and supergradients was given, and existence and uniqueness of such solution were proved. After that, a lot of investigations (see, e.g., Barbu_Barron_Jensen_1988; Soner_1988; Cannarsa_Da_Prato_1990; Cannarsa_Frankowska_1992; Li_1995) dealt with applications of the viscosity approach to control problems for abstract evolution systems in Hilbert or Banach spaces. In particular, in Soner_1988; Cannarsa_Da_Prato_1990, for Bolza optimal control problems for evolution systems, modified definitions of viscosity solutions of Cauchy problem for HJB equations were given, their existence, uniqueness and coincidence with the value functional were shown. Note that the conditions in these papers allow to interpret some class of time-delay systems as evolution systems, however, this class is not general enough, since it does not contain systems with discrete delay. The optimal control problem for systems with discrete delay was considered in Barron_1990. It was proved that the value functional is a viscosity solution of the Cauchy problem for the HJB equation, but the uniqueness question of the viscosity solution was not investigated. One could also mention papers Wolenski_1994; Clarke_Wolenski_1996 in which optimization problems for quite general delay differential inclusions (which cover the case of discrete delay) were considered and various necessary optimality conditions were given.

In Kim_1999, for the description of infinitesimal properties of a value functional in optimal control problems for time-delay systems, the notion of coinvariant derivatives was used. Note that such derivatives and their close analogues were applied later to a wide range of control problems for various functional differential systems (see, e.g., Lukoyanov_2000; Lukoyanov_2001; Lukoyanov_2010a; Lukoyanov_2010b; Aubin_Haddad_2002; Pepe_Ito_2012; Lukoyanov_Gomoyunov_Plaksin_2017; Bayraktar_Keller_2018). The theory of minimax and viscosity solutions of Cauchy problems for HJ equations with coinvariant derivatives and its application to differential games for time-delay systems were developed in Lukoyanov_2000; Lukoyanov_2001; Lukoyanov_2010a; Lukoyanov_2010b. In these papers, the class of time-delay systems under consideration is quite general and includes systems with discrete delay. In Lukoyanov_2000; Lukoyanov_2001, it was shown that the value functional is the unique minimax solution. In Lukoyanov_2010a, the description of the value functional in terms of suitable directional derivatives was given. In Lukoyanov_2010b, similar to Soner_1988, the modified definition of a viscosity solution based on a sequence of compact sets is considered. It allows to prove that the viscosity solution exists, is unique and coincides with the minimax solution, however, such definition is not reduced to the classical definition of a viscosity solution in the particular case without delay. For more natural definitions of a viscosity solution, the uniqueness questions is still open.

This paper is aimed to solve this question and to develop the theory of minimax and viscosity solutions of HJB equations for time-delay systems, which generalizes in a natural way the classical theory of both minimax and viscosity solutions of HJB equations for systems of ordinary differential equations.

In the paper, a Bolza optimal control problem for a time-delay system with discrete delay is considered. For the value functional of this problem, a HJB equation with coinvariant derivatives is investigated. Definitions of minimax and viscosity solutions (which are consistent with the classical definitions) of the Cauchy problems for this equation are studied. It is proved that both of these solutions exist, are unique and coincide with the value functional. Besides, the feedback scheme for constructing the optimal control by the minimax (viscosity) solution is given (see the proof of Theorem 2.2 $(b)\Rightarrow(a)$).

A principle idea for obtaining these results is to use the space of piecewise continuous functions as the state space in which the optimal control problem and the HJB equation are considered. As already noted earlier (see Wolenski_1994), the choice of a suitable state space plays an important role for an application of the viscosity approach to HJB equations for time-delay systems. The space of measurable functions can be used as the state space. But such choice significantly narrows the class of the corresponding time-delay systems and excludes important for applications systems with discrete delay (see Soner_1988). The space of continuous functions can also be used as the state space. It allows to cover the case of systems with discrete delay, but, as mentioned above, it makes it possible to prove the uniqueness only of the modified viscosity solution (see Lukoyanov_2010b). In Barron_1990; Wolenski_1994, other functional spaces were considered as the state space, but the uniqueness question of the viscosity solutions was not investigated. Presented in this paper choice of the space of piecewise continuous functions allows on the one hand to consider the case of systems with discrete delays, and on the other hand, to prove the uniqueness of the viscosity solutions in the classical sense. Note that this proof is based on Lemma LABEL:lem_MVI, which is an analogue of the theorem about ”Mean value inequality” Clarke_Ledyaev_1994; Clarke_Ledyaev_Stern_Wolenski_1998 (see also Subbotin_1993) for functionals defined on the space of piecewise continuous functions.

2 Formulation of Results

Let $\mathbb{R}^{n}$ be the $n$-dimensional Euclidian space with the inner product $\langle\cdot,\cdot\rangle$ and the norm $\|\cdot\|$. A function $x(\cdot)\colon[a,b]\mapsto\mathbb{R}^{n}$ is called piecewise continuous if there exist numbers $a=\xi_{1}<\xi_{2}<\ldots<\xi_{k}=b$ such that, for each $i\in\overline{1,k-1}$, the function $x(\cdot)$ is continuous on the interval $[\xi_{i},\xi_{i+1})$ and there exist a finite limit of $x(\xi)$ as $\xi$ approaches $\xi_{i+1}$ from left. Denote by $\mathrm{PC}([a,b],\mathbb{R}^{n})$ and $\mathrm{Lip}([a,b],\mathbb{R}^{n})$ the linear spaces of piecewise continuous and Lipschitz continuous functions $x(\cdot)\colon[a,b]\mapsto\mathbb{R}^{n}$.

Let $t_{0}<\vartheta$ and $h>0$. Let us denote

[TABLE]

Define the following norms on the space $\mathrm{PC}$:

[TABLE]

Consider a dynamical system which motion is described by the following delay differential equation:

[TABLE]

Here, $\tau$ is the time variable, $x(\tau)$ is the state vector at the time $\tau$, $\dot{x}(\tau)=\mathrm{d}x(\tau)/\mathrm{d}\tau$, $u(\tau)$ is the current control action, $\mathbb{U}\subset\mathbb{R}^{m}$ is a compact set.

Let $(t,z,w(\cdot))\in\mathbb{G}$. Define

[TABLE]

Denote by $\mathfrak{U}(t)$ the set of measurable functions $u(\cdot)\colon[t,\vartheta]\mapsto\mathbb{U}$. Let $u(\cdot)\in\mathfrak{U}(t)$. By a motion $x(\cdot)=x(\cdot\,|\,t,z,w(\cdot),u(\cdot))$ of system (1), we mean a function $x(\cdot)\in\Lambda(t,z,w(\cdot))$ that satisfies equation (1) for almost every $\tau\in[t,\vartheta]$.

Consider the following optimal control problem: for each $(t,z,w(\cdot))\in\mathbb{G}$, minimize the Bolza cost functional

[TABLE]

over all $u(\cdot)\in\mathfrak{U}(t)$, where $x(\cdot)=x(\cdot\,|\,t,z,w(\cdot),u(\cdot))$ is the motion of system (1), $x_{\vartheta}(\cdot)$ is the function defined by $x_{\vartheta}(\xi)=x(\vartheta+\xi)$, $\xi\in[-h,0]$.

We assume that the following conditions hold:

$(f_{1})$

The functions $f(t,x,y,u)\in\mathbb{R}^{n}$, $f^{0}(t,x,y,u)\in\mathbb{R}$, $t\in[t_{0},\vartheta]$, $x,y\in\mathbb{R}^{n}$, $u\in\mathbb{U}$ are continuous.

$(f_{2})$

For every $\alpha>0$, there exists a number $\lambda_{f}=\lambda_{f}(\alpha)>0$ such that

[TABLE]

for any $t\in[t_{0},\vartheta]$, $x,y,x^{\prime},y^{\prime}\in O(\alpha)=\{x\in\mathbb{R}^{n}\colon\|x\|\leq\alpha\}$ and $u\in\mathbb{U}$.

$(f_{3})$

There exists a constant $c_{f}>0$ such that

[TABLE]

for any $t\in[t_{0},\vartheta]$, $x,y\in\mathbb{R}^{n}$ and $u\in\mathbb{U}$.

($\sigma$)

For every $\alpha>0$, there exists a number $\lambda_{\sigma}=\lambda_{\sigma}(\alpha)>0$ such that

[TABLE]

for any $(z,w(\cdot)),(z^{\prime},w^{\prime}(\cdot))\in P(\alpha)$, where

[TABLE]

It is known that, under such conditions, for each $(t,z,w(\cdot))\in\mathbb{G}$ and $u(\cdot)\in\mathfrak{U}(t)$, there exists a unique motion $x(\cdot)=x(\cdot\,|\,t,z,w(\cdot),u(\cdot))$ of system (1).

The value functional $\rho\colon\mathbb{G}\mapsto\mathbb{R}$ in optimal control problem (1), (2) is defined by

[TABLE]

One can show (following, e.g., the scheme from (Evans_1998, p. 553)) that, for every $(t,z,w(\cdot))\in\mathbb{G}$ and $\tau\in[t,\vartheta]$, the functional $\rho$ satisfies the following equation (a dynamic programming principle):

[TABLE]

where $x(\cdot)=x(\cdot\,|\,t,z,w(\cdot),u(\cdot))$ is the motion of system (1).

In order to obtain a Hamilton-Jacobi-Bellman (HJB) equation as infinitesimal form of equation (4), we will use the following definition of differentiability of functionals. Following Kim_1999; Lukoyanov_2000; Lukoyanov_2001, a functional $\varphi\colon\mathbb{G}\mapsto\mathbb{R}$ is called coinvariantly differentiable (ci-differentiable) at a point $(t,z,w(\cdot))\in\mathbb{G}$, $t<\vartheta$ if there exist a number $\partial^{ci}_{t,w}\varphi(t,z,w(\cdot))\in\mathbb{R}$ and a vector $\nabla_{z}\varphi(t,z,w(\cdot))\in\mathbb{R}^{n}$ such that, for any $v\in\mathbb{R}^{n}$, $x(\cdot)\in\Lambda(t,z,w(\cdot))$ and $\tau\in[t,\vartheta]$, the following relation holds:

[TABLE]

where the function $x_{\tau}(\cdot)\in\mathrm{PC}$ is defined by $x_{\tau}(\xi)=x(\tau+\xi)$, $\xi\in[-h,0]$, the value $o(\cdot)$ depends on the triplet $\{t,z,x(\cdot)\}$, and $o(\delta)/\delta\to 0$ as $\delta\to+0$. Then $\partial^{ci}_{t,w}\varphi(t,z,w(\cdot))$ is called the ci-derivative of $\varphi$ with respect to $\{t,w(\cdot)\}$ and $\nabla_{z}\varphi(t,z,w(\cdot))$ is the gradient of $\varphi$ with respect to $z$. Let us note that if $\varphi$ does not depend on the functional variable $w(\cdot)$, then the definition of ci-differentiability coincides with the definition of differentiability of functions.

Define the Hamiltonian of problem (1), (2) by

[TABLE]

Consider the following Cauchy problem for the HJB equation

[TABLE]

and the terminal condition

[TABLE]

Define the class of functionals in which we will search a solution of this problem. Denote by $\Phi$ the set of functionals $\varphi=\varphi(t,z,w(\cdot))\in\mathbb{R}$, $(t,z,w(\cdot))\in\mathbb{G}$ which are continuous with respect to $t$ and satisfy the following Lipschitz condition: for every $\alpha>0$, there exists a number $\lambda_{\varphi}=\lambda_{\varphi}(\alpha)>0$ such that

[TABLE]

for any $t\in[t_{0},\vartheta]$ and $(z,w(\cdot)),(z^{\prime},w^{\prime}(\cdot))\in P(\alpha)$. The choice of this class is motivated, in particular, the inclusion $\rho\in\Phi$, which will be shown in Lemma LABEL:lem_rho_Phi.

The following theorem establishes the relation between problem (7), (8) and the value functional $\rho$ in the case when $\rho$ is ci-differentiable.

Theorem 2.1

The following statements hold:

If a functional $\varphi\in\Phi$ is ci-differentiable at each point $(t,z,w(\cdot))\in\mathbb{G}$, $t<\vartheta$, satisfies HJB equation (7) at these points and satisfies terminal condition (8), then the identity $\varphi\equiv\rho$ holds.

• 2.

If the value functional $\rho$ is ci-differentiable at a point $(t,z,w(\cdot))\in\mathbb{G}$, $t<\vartheta$, then it satisfies HJB equation (7) at this point.

The proof of this theorem is described after the Theorem 2.2.

For the case when $\rho$ is not ci-differentiable, definitions of generalized (minimax and viscosity) solutions of problem (7), (8) are given below.

Taking the constant $c_{f}>0$ form $(f_{3})$, we denote

[TABLE]

Let $(t,z,w(\cdot))\in\mathbb{G}$. Denote by $X(t,z,w(\cdot))$ the set of the functions $x(\cdot)\in\Lambda(t,z,w(\cdot))$ that satisfy the following delay differential inclusion:

[TABLE]

Note that the set $X(t,z,w(\cdot))$ is not empty. In particular, for each $u(\cdot)\in\mathfrak{U}(t)$, the motion $x(\cdot)=x(\cdot\,|\,t,z,w(\cdot),u(\cdot))$ of system (1) satisfies the inclusion

[TABLE]

Definition 1

A functional $\varphi\colon\mathbb{G}\mapsto\mathbb{R}$ is called a minimax solution of problem (7), (8) if $\varphi$ satisfies the inclusion $\varphi\in\Phi$, terminal condition (8) and the following inequalities:

[TABLE]

for any $(t,z,w(\cdot))\in\mathbb{G}$, $t<\vartheta$, $\tau\in(t,\vartheta]$ and $s\in\mathbb{R}^{n}$.

By analogy with Lukoyanov_2010a, lower and upper right directional derivatives of a functional $\varphi\colon\mathbb{G}\mapsto\mathbb{R}$ along $l\in\mathbb{R}^{n}$ at $(t,z,w(\cdot))\in\mathbb{G}$, $t<\vartheta$ are defined by

[TABLE]

where $x^{l}(\cdot)\in\Lambda(t,z,w(\cdot))$ and $x^{l}(\tau)=z+l(\tau-t)$, $\tau\in[t,\vartheta]$.

The following sets are called the subdifferential $D^{-}\varphi(t,z,w(\cdot))$ and the superdifferential $D^{+}\varphi(t,z,w(\cdot))$ of the functional $\varphi$ at $(t,z,w(\cdot))\in\mathbb{G}$, $t<\vartheta$:

[TABLE]

Note that if a functional $\varphi$ is ci-differentiable at $(t,z,w(\cdot))\in\mathbb{G}$, $t<\vartheta$, then

[TABLE]

Definition 2

A functional $\varphi\colon\mathbb{G}\mapsto\mathbb{R}$ is called a viscosity solution of problem (7), (8) if $\varphi$ satisfies the inclusion $\varphi\in\Phi$, terminal condition (8) and the following inequalities:

[TABLE]

for any $(t,z,w(\cdot))\in\mathbb{G}$, $t<\vartheta$.

Theorem 2.2

For $\varphi\colon\mathbb{G}\mapsto\mathbb{R}$, the following statements are equivalent:

(a)

The identity $\varphi\equiv\rho$ holds.

(b)

$\varphi$* is a minimax solution of problem (7), (8).*

(c)

$\varphi\in\Phi$* satisfies terminal condition (8) and the following inequalities:*

[TABLE]

for any $(t,z,w(\cdot))\in\mathbb{G}$, $t<\vartheta$ and $s\in\mathbb{R}^{n}$.

(d)

$\varphi$* is a viscosity solution of problem (7), (8).*

In particular, this theorem establishes the existence and uniqueness of the minimax and viscosity solutions since the value functional $\rho$ is uniquely defined.

Note that Theorem 2.1 follows from the equivalence of statements $(a)$ and $(d)$ if we take into account (2). Below in the paper, auxiliary properties of system (1) and inclusion (11) will be given and Theorem 2.2 will be proved.

3 Properties of Time-Delay Systems

Proposition 1

For every $\alpha>0$, there exist numbers $\alpha_{X}=\alpha_{X}(\alpha)>\alpha$ and $\lambda_{X}=\lambda_{X}(\alpha)>0$ such that

[TABLE]

for each $t\in[t_{0},\vartheta]$, $(z,w(\cdot))\in P(\alpha)$ and $x(\cdot)\in X(t,z,w(\cdot))$.

Proof

Let $\alpha>0$. Put $\alpha_{*}=(1+c_{f}h)\alpha+c_{f}(\vartheta-t_{0})$, $\alpha_{X}=\alpha_{*}e^{2c_{f}(\vartheta-t_{0})}$ and $\lambda_{X}=c_{f}(1+2\alpha_{X})$. Let $t\in[t_{0},\vartheta]$, $(z,w(\cdot))\in P(\alpha)$ and $x(\cdot)\in X(t,z,w(\cdot))$. Then, according to (10), (11), we derive

[TABLE]

Therefore, applying Bellman-Gronwall lemma (see, e.g., (Bellman_Cooke_1963, p. 31)), we obtain $(x(\tau),x_{\tau}(\cdot))\in P(\alpha_{X})$, $\tau\in[t,\vartheta]$. Then, from (11), we deduce $\|\dot{x}(\tau)\|\leq\lambda_{X}$ for almost every $\tau\in[t,\vartheta]$, which concludes the proof. $\square$

For $(t,z,w(\cdot))\in\mathbb{G}$ and $u(\cdot)\in\mathfrak{U}(t)$, we denote

[TABLE]

where $x(\cdot)=x(\cdot\,|\,t,z,w(\cdot),u(\cdot))$ is the motion of system (1).

Proposition 2

For every $\alpha>0$, there exists a number $\lambda_{*}=\lambda_{*}(\alpha)>0$ such that, for each $t\in[t_{0},\vartheta]$, $(z,w(\cdot)),(z^{\prime},w^{\prime}(\cdot))\in P(\alpha)$ and $u(\cdot)\in\mathfrak{U}(t)$, the motions $x(\cdot)=x(\cdot\,|\,t,z,w(\cdot),u(\cdot))$ and $x^{\prime}(\cdot)=x(\cdot\,|\,t,z^{\prime},w^{\prime}(\cdot),u(\cdot))$ of system (1) satisfy the inequality

[TABLE]

[TABLE]