Multitime hybrid differential games with curvilinear integral functional

TL;DR

This paper advances the theory of multitime differential games with curvilinear integral functionals, establishing fundamental properties, viscosity solutions, and representation formulas for associated PDEs in a multitime setting.

Contribution

It provides original results on upper and lower value functions, viscosity solutions, and representation formulas for multitime hybrid differential games constrained by an m-flow.

Findings

Established fundamental properties of multitime upper and lower values.

Derived viscosity solutions for multitime Hamilton-Jacobi-Isaacs PDEs.

Developed representation formulas for viscosity solutions of multitime PDEs.

Abstract

Multitime differential games are related to the modeling and analysis of cooperation or conflict in the context of a multitime dynamical systems. Their theory involves either a curvilinear integral functional or a multiple integral functional and an $m$-flow as constraint. The aim of this paper is to give original results regarding multitime hybrid differential games with curvilinear integral functional constrained by an $m$-flow: fundamental properties of multitime upper and lower values, viscosity solutions of multitime (HJIU) PDEs, representation formula of viscosity solutions for multitime (HJ) PDEs, and max-min representations.

Equations

$$\underset{v(\cdot )\in V}{\min }\underset{u(\cdot )\in U}{\max }J(u(\cdot ),v(\cdot ))={\int }_{{\Gamma }_{0T}}{L}_{\alpha }(s,x(s),u_{\alpha }(s),v_{\alpha }(s))d{s}^{\alpha }+g(x(T)),$$

$$\frac{\partial x^i}{\partial s^{\alpha }}(s)=X_{\alpha }^i(s,x(s),u_{\alpha }(s),v_{\alpha }(s)),$$

$$x(0)=x_0,s\in {\Omega }_{0T}\subset {\mathbb{R}}_{+}^m,x\in {\mathbb{R}}^n.$$

$$\left(\frac{\partial X_{\alpha }}{\partial u_{\lambda }^a}{\delta }_{\beta }^{\gamma }-\frac{\partial X_{\beta }}{\partial u_{\lambda }^a}{\delta }_{\alpha }^{\gamma }\right)\frac{\partial u_{\lambda }^a}{\partial s^{\gamma }}+\left(\frac{\partial X_{\alpha }}{\partial v_{\lambda }^b}{\delta }_{\beta }^{\gamma }-\frac{\partial X_{\beta }}{\partial v_{\lambda }^b}{\delta }_{\alpha }^{\gamma }\right)\frac{\partial v_{\lambda }^b}{\partial s^{\gamma }}=\left[X_{\alpha },X_{\beta }\right]+\frac{\partial X_{\beta }}{\partial s^{\alpha }}-\frac{\partial X_{\alpha }}{\partial s^{\beta }},$$

$$J_{x,t}(u(\cdot ),v(\cdot ))={\int }_{{\Gamma }_{tT}}{L}_{\alpha }(s,x(s),u_{\alpha }(s),v_{\alpha }(s))d{s}^{\alpha }+g(x(T)),$$

$$\frac{\partial x^i}{\partial s^{\alpha }}(s)=X_{\alpha }^i(s,x(s),u_{\alpha }(s),v_{\alpha }(s)),$$

$$x(t)=x,s\in {\Omega }_{tT}\subset {\mathbb{R}}_{+}^m,x\in {\mathbb{R}}^n.$$

$$\displaystyle\left\{\begin{array}[]{ll}\|X_{\alpha}(t,x,u_{\alpha},v_{\alpha})\|\leqslant A_{\alpha}\\ \|X_{\alpha}(t,x,u_{\alpha},v_{\alpha})-X_{\alpha}(t,\hat{x},u_{\alpha},v_{\alpha})\|\leqslant A_{\alpha}\|x-\hat{x}\|,\end{array}\right.$$

$$g:{\mathbb{R}}^n\to \mathbb{R},L_{\alpha }:{\Omega }_{0T}\times {\mathbb{R}}^n\times U\times V\to \mathbb{R}$$

$$\displaystyle\left\{\begin{array}[]{ll}|g(x)|\leqslant B\\ |g(x)-g(\hat{x})|\leqslant B\|x-\hat{x}\|,\end{array}\right.$$

$$\displaystyle\left\{\begin{array}[]{ll}|L_{\alpha}(t,x,u_{\alpha},v_{\alpha})|\leqslant C_{\alpha}\\ |L_{\alpha}(t,x,u_{\alpha},v_{\alpha})-L_{\alpha}(t,\hat{x},u_{\alpha},v_{\alpha})|\leqslant C_{\alpha}\|x-\hat{x}\|,\end{array}\right.$$

$$\mathcal{U}(t)=\left\{u_{\alpha }(\cdot ):{\mathbb{R}}_{+}^m\to U|u_{\alpha }(\cdot )\mathrm{is}\mathrm{measurable}\mathrm{and}\mathrm{satisfies}\mathrm{CIC}\right\}$$

$$\mathcal{V}(t)=\left\{v_{\alpha }(\cdot ):{\mathbb{R}}_{+}^m\to V|v_{\alpha }(\cdot )\mathrm{is}\mathrm{measurable}\mathrm{and}\mathrm{satisfies}\mathrm{CIC}\right\}$$

$$m(t,x)=\underset{\Psi \in \mathcal{V}}{\min }\underset{u(\cdot )\in U}{\max }{J}_{t,x}(u(\cdot ),\Psi [u](\cdot ))$$

$$M(t,x)=\underset{\Phi \in \mathcal{U}}{\max }\underset{v(\cdot )\in V}{\min }{J}_{t,x}(\Phi [v](\cdot ),v(\cdot ))$$

$$\begin{split}m(t,x)\ &=\min_{\Psi\in\mathcal{B}(t)}\max_{u_{\alpha}\in\mathcal{U}(t)}\bigg{\{}\int_{\Gamma_{tt+h}}L_{\alpha}(s,x(s),u_{\alpha}(s),\Psi[u_{\alpha}](s))ds^{\alpha}\\ &+m(t+h,x(t+h))\bigg{\}}\end{split}$$

$$\begin{split}M(t,x)\ &=\max_{\Phi\in\mathcal{A}(t)}\min_{v_{\alpha}\in\mathcal{V}(t)}\bigg{\{}\int_{\Gamma_{tt+h}}L_{\alpha}(s,x(s),\Phi[v_{\alpha}](s),v_{\alpha}(s))ds^{\alpha}\\ &+M(t+h,x(t+h))\bigg{\}},\end{split}$$

$$\begin{split}w(t,x)\ &=\min_{\Psi\in\mathcal{B}(t)}\max_{u_{\alpha}\in\mathcal{U}(t)}\bigg{\{}\int_{\Gamma_{tt+h}}L_{\alpha}(s,x(s),u_{\alpha}(s),\Psi[u_{\alpha}](s))ds^{\alpha}\\ &+m(t+h,x(t+h))\bigg{\}}.\end{split}$$

$$\begin{split}w(t,x)\ &\geqslant\max_{u_{\alpha}\in\mathcal{U}(t)}\bigg{\{}\int_{\Gamma_{tt+h}}L_{\alpha}(s,x(s),u_{\alpha}(s),\Upsilon[u_{\alpha}](s))ds^{\alpha}\\ &+m(t+h,x(t+h))\bigg{\}}-\varepsilon.\end{split}$$

$$\begin{split}m(t+h,\overline{x})\ &=\min_{\Psi\in\mathcal{B}(t+h)}\max_{u_{\alpha}\in\mathcal{U}(t+h)}\bigg{\{}\int_{\Gamma_{t+hT}}L_{\alpha}(s,x(s),u_{\alpha}(s),\Psi[u_{\alpha}](s))ds^{\alpha}\\ &+g(x(T))\bigg{\}}.\end{split}$$

$$\begin{split}m(t+h,\overline{x})\ &\geqslant\max_{u_{\alpha}\in\mathcal{U}(t+h)}\bigg{\{}\int_{\Gamma_{t+hT}}L_{\alpha}(s,x(s),u_{\alpha}(s),\Upsilon_{\overline{x}}[u_{\alpha}](s))ds^{\alpha}\\ &+g(x(T))\bigg{\}}-\varepsilon.\end{split}$$

$$\displaystyle\Psi\in\mathcal{B}(t),\Psi[u_{\alpha}](s)\equiv\left\{\begin{array}[]{ll}\Upsilon[u_{\alpha}](s)&s\in\Omega_{tt+h}\\ \Upsilon_{\overline{x}}[u_{\alpha}](s)&s\in\Omega_{tT}\setminus\Omega_{tt+h},\end{array}\right.$$

$$w(t,x)⩾{\int }_{{\Gamma }_{tT}}{L}_{\alpha }(s,x(s),u_{\alpha }(s),\Psi [u_{\alpha }](s))d{s}^{\alpha }+g(x(T))-2\epsilon .$$

$$\underset{u_{\alpha }\in \mathcal{U}(t)}{\max }\left\{{\int }_{{\Gamma }_{tT}}{L}_{\alpha }(s,x(s),u_{\alpha }(s),\Psi [u_{\alpha }](s))d{s}^{\alpha }+g(x(T))\right\}\le w(t,x)+2\epsilon .$$

$$m(t,x)\le w(t,x)+2\epsilon .$$

$$m(t,x)⩾\underset{u_{\alpha }\in \mathcal{U}(t)}{\max }\left\{{\int }_{{\Gamma }_{tT}}{L}_{\alpha }(s,x(s),u_{\alpha }(s),\Psi [u_{\alpha }](s))d{s}^{\alpha }+g(x(T))\right\}-\epsilon .$$

$$\begin{split}w(t,x)\ &\leqslant\max_{u_{\alpha}\in U(t)}\bigg{\{}\int_{\Gamma_{tt+h}}L_{\alpha}(s,x(s),u_{\alpha}(s),\Psi[u_{\alpha}](s))ds^{\alpha}\\ &+m(t+h,x(t+h))\bigg{\}}\end{split}$$

$$\begin{array}{rl}w(t,x)& ⩽{\int }_{{\Gamma }_{tt+h}}{L}_{\alpha }(s,x(s),u_{\alpha }^1(s),\Psi [u_{\alpha }^1](s))d{s}^{\alpha }\\ & +m(t+h,x(t+h))+\epsilon .\end{array}$$

$$\displaystyle{u_{\alpha}^{\star}}\in\mathcal{U}(t),{u_{\alpha}^{\star}}(s)\equiv\left\{\begin{array}[]{ll}u^{1}_{\alpha}(s)&s\in\Omega_{tt+h}\\ u_{\alpha}(s)&s\in\Omega_{tT}\setminus\Omega_{tt+h},\end{array}\right.$$

$$\begin{array}{rl}& m(t+h,x(t+h))\\ & \le \underset{u_{\alpha }\in \mathcal{U}(t+h)}{\max }\left\{{\int }_{{\Gamma }_{tt+h}}{L}_{\alpha }(s,x(s),u_{\alpha }(s),{\Psi }^{\star }[u_{\alpha }](s))d{s}^{\alpha }+g(x(T))\right\}\end{array}$$

Full text

Multitime hybrid differential games with

curvilinear integral functional

Constantin Udrişte, Elena-Laura Otobîcu, Ionel Ţevy

Multitime differential games are related to the modeling and analysis of cooperation or conflict in the context of a multitime dynamical systems. Their theory involves either a curvilinear integral functional or a multiple integral functional and an $\displaystyle m$-flow as constraint. The aim of this paper is to give original results regarding multitime hybrid differential games with curvilinear integral functional constrained by an $\displaystyle m$-flow: fundamental properties of multitime upper and lower values, viscosity solutions of multitime (HJIU) PDEs, representation formula of viscosity solutions for multitime (HJ) PDEs, and max-min representations.

Mathematics Subject Classification 2010: 49L20, 91A23, 49L25, 35F21.

Key words: multitime hybrid differential games, curvilinear integral cost, multitime dynamic programming, multitime viscosity solutions.

1 Multitime hybrid differential game with

curvilinear integral functional

Let $\displaystyle t=(t^{\alpha})\in\Omega_{0T}\subset\mathbb{R}^{m}_{+}$, $\displaystyle\alpha=1,...,m,$ be an evolution multi-parameter, called multitime. Consider an arbitrary $\displaystyle C^{1}$ curve $\displaystyle\Gamma_{0T}$ joining the diagonal opposite points $\displaystyle 0=(0,\ldots,0)$ and $\displaystyle T=(T^{1},\ldots,T^{m})$ in the $\displaystyle m$-dimensional parallelepiped $\displaystyle\Omega_{0T}=[0,T]$ (multitime interval) in $\displaystyle\mathbb{R}^{m}_{+}$ endowed with the product order, a $\displaystyle C^{2}$ state vector $\displaystyle x:\Omega_{0T}\rightarrow\mathbb{R}^{n},x(t)=(x^{i}(t)),$ $\displaystyle i=1,...,n$, a $\displaystyle C^{1}$ control vector $\displaystyle u(t)=(u_{\alpha}(t)):\Omega_{0T}\rightarrow U\subset\mathbb{R}^{qm},$ for the first equip of $\displaystyle m$ players (who wants to maximize), a $\displaystyle C^{1}$ control vector $\displaystyle v(t)=(v_{\alpha}(t)):\Omega_{0T}\rightarrow V\subset\mathbb{R}^{qm},$ for the second equip of $\displaystyle m$ players (who wants to minimize), $\displaystyle u_{\alpha}(\cdot)=\Phi(\cdot,\eta_{1}(\cdot)),v_{\alpha}(\cdot)=\Psi(\cdot,\eta_{2}(\cdot)),$ a running cost $\displaystyle L_{\alpha}(t,x(t),u_{\alpha}(t),v_{\alpha}(t))dt^{\alpha}$ as a nonautonomous closed Lagrangian $\displaystyle 1$-form (satisfies $\displaystyle D_{\beta}L_{\alpha}=D_{\alpha}L_{\beta}),$ a terminal cost $\displaystyle g(x(T))$ and the $\displaystyle C^{1}$ vector fields $\displaystyle X_{\alpha}=(X_{\alpha}^{i})$ satisfying the complete integrability conditions (CIC) $\displaystyle D_{\beta}X_{\alpha}=D_{\alpha}X_{\beta}$ (m-flow type problem).

In our paper, a multitime hybrid differential game is given by a multitime dynamics, as a PDE system controlled by two controllers (first equip, second equip) and a target including a curvilinear integral functional. The approach we follow below is those in the paper [2], but we must be more creative since our theory is multitemporal one (see also [8]-[20]). More precisely, we introduce and analyze a multitime differential game whose Bolza payoff is the sum between a path independent curvilinear integral (mechanical work) and a function of the final event (the terminal cost, penalty term), and whose evolution PDE is an m-flow: Find

[TABLE]

subject to the Cauchy problem

[TABLE]

[TABLE]

Let $\displaystyle D_{\alpha}$ be the total derivative operator and $\displaystyle[X_{\alpha},X_{\beta}]$ be the bracket of vector fields. Suppose the piecewise complete integrability conditions (CIC)

[TABLE]

where $\displaystyle a,b=1,...,q$, are satisfied throughout.

To simplify, suppose that the curve $\displaystyle\Gamma_{0T}$ is an increasing curve in the multitime interval $\displaystyle\Omega_{0T}$. If we vary the starting multitime and the initial point, then we obtain a larger family of similar multitime problems containing the functional

[TABLE]

and the evolution constraint

[TABLE]

[TABLE]

We assume that each vector field $\displaystyle X_{\alpha}:\Omega_{0T}\times\mathbb{R}^{n}\times U\times V\rightarrow\mathbb{R}^{n}$ is uniformly continuous, satisfying

[TABLE]

for some constant 1-form $\displaystyle A=(A_{\alpha})$ and all $\displaystyle t\in\Omega_{0T},\,\,x,\hat{x}\in\mathbb{R}^{n},u\in U,v\in V.$

Suppose the functions

[TABLE]

are uniformly continuous and satisfy the boundedness conditions

[TABLE]

[TABLE]

for constant $\displaystyle 1$-form $\displaystyle C=(C_{\alpha})$ and all $\displaystyle t\in\Omega_{0T},\,\,x,\hat{x}\in\mathbb{R}^{n},\,\,u\in U,\,\,v\in V.$

Definition 1.1**.**

(i) The set

[TABLE]

is called the control set for the first equip of players. (ii) The set

[TABLE]

is called the control set for the second equip of players.

Definition 1.2**.**

(i) A map $\displaystyle\Phi:\mathcal{V}(t)\rightarrow\mathcal{U}(t)$ is called a strategy for the first equip of players, if the equality $\displaystyle v(\tau)=\widehat{v}(\tau),t\leq\tau\leq s\leq T$ implies $\displaystyle\Phi[v](\tau)=\Phi[\widehat{v}](\tau).$ (ii) A map $\displaystyle\Psi:\mathcal{U}(t)\rightarrow\mathcal{V}(t)$ is called a strategy for the second equip of players, if the equality $\displaystyle u(\tau)=\widehat{u}(\tau),t\leq\tau\leq s\leq T$ implies $\displaystyle\Psi[u](\tau)=\Psi[\widehat{u}](\tau).$

Let $\displaystyle\mathcal{A}(t)$ be ** the set of strategies for the first equip of players** and $\displaystyle\mathcal{B}(t)$ be ** the set of strategies for the second equip of players**.

Definition 1.3**.**

(i) The function

[TABLE]

is called the multitime lower value function. (ii) The function

[TABLE]

is called the multitime upper value function.

The multitime lower value function $\displaystyle m(t,x)$ and the multitime upper value function $\displaystyle M(t,x)$ are piecewise continuously differentiable (see below, the boundedness and continuity of the values functions).

2 Properties of lower and upper values

Theorem 2.1**.**

(multitime dynamic programming optimality conditions) For each pair of strategies $\displaystyle(\Phi,\Psi),$ the lower and upper value functions can be written respectively in the form

[TABLE]

and

[TABLE]

for all $\displaystyle(t,x)\in\Omega_{tT}\times\mathrm{R}^{n}$ and all $\displaystyle h\in\Omega_{0T-t}.$

Proof.

First we recognize the Bellman principle (we write the value of a decision problem at a certain point in multitime in terms of the payoff from some initial choices and the value of the remaining decision problem that results from those initial choices).

To confirm the first statement, we shall use the function

[TABLE]

We will show that, for all $\displaystyle\varepsilon>0,$ the lower value function $\displaystyle m(t,x)$ will satisfies two inequalities, $\displaystyle m(t,x)\leq w(t,x)+2\varepsilon$ and $\displaystyle m(t,x)\geq w(t,x)-3\varepsilon.$ Since $\displaystyle\varepsilon>0$ is arbitrary, it follows $\displaystyle m(t,x)=w(t,x).$

1. i)

For $\displaystyle\varepsilon>0,$ there exists a strategy $\displaystyle\Upsilon\in\mathcal{B}(t)$ such that

[TABLE]

We shall use the state $\displaystyle x(\cdot)$ which solves the (PDE), with the initial condition $\displaystyle\overline{x}=x(t+h)$ (Cauchy problem) on the set $\displaystyle\Omega_{tT}\setminus\Omega_{tt+h},$ for each $\displaystyle\overline{x}\in\mathbb{R}^{n}$. We can write

[TABLE]

Thus there exists a strategy $\displaystyle\Upsilon_{\overline{x}}\in\mathcal{B}(t+h)$ for which

[TABLE]

Define a new strategy

[TABLE]

for each control $\displaystyle u_{\alpha}\in\mathcal{U}(t).$ For any $\displaystyle u_{\alpha}\in\mathcal{U}(t)$, replacing the inequality $\displaystyle\eqref{eq.8}$ in the inequality $\displaystyle\eqref{eq.7}$, we obtain

[TABLE]

Consequently

[TABLE]

Hence

[TABLE] 2. ii)

On the other hand, there exists a strategy $\displaystyle\Psi\in\mathcal{B}(t)$ for which we can write the inequality

[TABLE]

By the definition of $\displaystyle w(t,x),$ we have

[TABLE]

and consequently there exists a control $\displaystyle u^{1}_{\alpha}\in\mathcal{U}(t)$ such that

[TABLE]

Define a new control

[TABLE]

for each control $\displaystyle u_{\alpha}\in\mathcal{U}(t+h)$ and then define the strategy $\displaystyle{\Psi}^{\star}\in\mathcal{B}(t+h),\Psi^{\star}[u_{\alpha}](s)\equiv\Psi[{u_{\alpha}^{\star}}](s),s\in\Omega_{tT}\setminus\Omega_{tt+h}.$ We find the inequality

[TABLE]

and so there exists the control $\displaystyle u^{2}_{\alpha}\in\mathcal{U}(t+h)$ for which

[TABLE]

Define a new control

[TABLE]

Then the inequalities $\displaystyle\eqref{eq.10}$ and $\displaystyle\eqref{eq.11}$ yield

[TABLE]

and so $\displaystyle\eqref{eq.9}$ implies the inequality

[TABLE]

This inequality and $\displaystyle m(t,x)\leq w(x,t)+2\varepsilon$ complete the proof.

∎

Theorem 2.2**.**

(boundedness and continuity of the values functions) The lower, upper value function $\displaystyle m(t,x)$, $\displaystyle M(t,x)$ satisfy the boundedness conditions

[TABLE]

[TABLE]

for some constants $\displaystyle D,E,F$ and for all $\displaystyle t,\hat{t}\in\Omega_{0T},x,\hat{x}\in\mathbb{R}^{n}.$

Proof.

We prove only the statements for upper value function $\displaystyle M(t,x).$

Since $\displaystyle|g(x)|\leqslant B,|L_{\alpha}(t,x,u_{\alpha},v_{\alpha})|\leqslant C_{\alpha},\alpha=\overline{1,m}$, we find

[TABLE]

for all $\displaystyle u_{\alpha}(\cdot)\in\mathcal{U}(t),v_{\alpha}(\cdot)\in\mathcal{V}(t).$

Let $\displaystyle x_{1},x_{2}\in\mathbb{R}^{n},\,\,t_{1},t_{2}\in\Omega_{0T}.$ For $\displaystyle\varepsilon>0$ and the strategy $\displaystyle\Phi\in\mathcal{A}(t_{1}),$ we have

[TABLE]

Define the control

[TABLE]

for any $\displaystyle v_{\alpha}\in\mathcal{V}(t_{2})$ and some $\displaystyle v_{\alpha}^{1}\in V$ and for each $\displaystyle v_{\alpha}\in\mathcal{V}(t_{2}),\underline{\Phi}\in\mathcal{A}(t_{2})$ (the restriction of $\displaystyle\Phi$ over $\displaystyle\Omega_{0T}\setminus\Omega_{0t_{1}})$ by $\displaystyle\underline{\Phi}[v_{\alpha}]=\Phi[\overline{v}_{\alpha}],s\in\Omega_{0T}\setminus\Omega_{0t_{2}}.$

Choose the control $\displaystyle v_{\alpha}\in\mathcal{V}(t_{2})$ so that

[TABLE]

By the inequality $\displaystyle\eqref{eq:1},$ we have

[TABLE]

We know that the (unique, Lipschitz) solution $\displaystyle x(\cdot)$ of the Cauchy problem

[TABLE]

is the response to the controls $\displaystyle u_{\alpha}(\cdot),v_{\alpha}(\cdot)$ for $\displaystyle s\in\Omega_{0T}.$

We choose $\displaystyle x_{1}(\cdot)$ as solution of the Cauchy problem

[TABLE]

Equivalently, $\displaystyle x_{1}(\cdot)$ is solution of integral equation

[TABLE]

Take $\displaystyle x_{2}(\cdot)$ as solution of the Cauchy problem

[TABLE]

Equivalently, $\displaystyle x_{2}(\cdot)$ is solution of integral equation

[TABLE]

It follows that

[TABLE]

Since $\displaystyle v_{\alpha}=\overline{v}_{\alpha}$ and $\displaystyle\underline{\Phi}[v_{\alpha}]=\Phi[\overline{v}_{\alpha}],$ for $\displaystyle s\in\Omega_{0T}\setminus\Omega_{0t_{2}}$, we find the estimation

[TABLE]

Thus the inequalities $\displaystyle\eqref{eq:2}$ and $\displaystyle\eqref{eq:3}$ imply

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Since $\displaystyle\varepsilon$ is arbitrary, we obtain the inequality

[TABLE]

Let $\displaystyle\varepsilon>0$ and choose the strategy $\displaystyle\Phi\in\mathcal{A}(t_{2})$ such that

[TABLE]

For each control $\displaystyle v_{\alpha}\in\mathcal{V}(t_{1})$ and $\displaystyle s\in\Omega_{0T}\setminus\Omega_{0t_{2}},$ define the control $\displaystyle\underline{v}_{\alpha}\in\mathcal{V}(t_{2}),\underline{v}_{\alpha}(s)=v_{\alpha}(s).$

For some $\displaystyle u^{1}_{\alpha}\in U,$ we define the strategy $\displaystyle\overline{\Phi}\in\mathcal{A}(t_{1})$ (the restriction of $\displaystyle\Phi$ over $\displaystyle\Omega_{0T}\setminus\Omega_{0t_{2}}$) by

[TABLE]

Now choose a control $\displaystyle v_{\alpha}\in\mathcal{V}(t_{1})$ so that

[TABLE]

By the inequality $\displaystyle\eqref{eq:4},$ we have

[TABLE]

We choose $\displaystyle x_{1}(\cdot)$ as solution of the Cauchy problem (PDE system + initial condition)

[TABLE]

and $\displaystyle x_{2}(\cdot)$ as solution of the Cauchy problem (PDE system + initial condition)

[TABLE]

Using the associated integral equations, it follows that

[TABLE]

Also, for $\displaystyle s\in\Omega_{0T}\setminus\Omega_{0t_{2}},v_{\alpha}=\underline{v}_{\alpha}$ and $\displaystyle\overline{\Phi}[v_{\alpha}]=\Phi[\underline{v}_{\alpha}],$ we find

[TABLE]

Thus, the relations $\displaystyle\eqref{eq:5}$ and $\displaystyle\eqref{eq:6}$ imply

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Since $\displaystyle\varepsilon$ is arbitrary, we obtain the inequality

[TABLE]

By $\displaystyle 2.17$ and $\displaystyle 2.22$, we proved the continuity of the lower and upper value functions. ∎

3 Viscosity solutions of

multitime (HJIU) PDEs

Theorem 3.1**.**

(PDEs for multitime upper value function, resp. multitime lower value function)

The multitime upper value function $\displaystyle M(t,x)$ and the multitime lower value function $\displaystyle m(t,x)$ are the viscosity solutions of Hamilton-Jacobi-Isaacs-Udrişte (HJIU) PDEs:

• •

the multitime upper (HJIU) PDEs

[TABLE]

with the terminal condition $\displaystyle M(T,x)=g(x),$

• •

the multitime lower (HJIU) PDEs

[TABLE]

with the terminal condition $\displaystyle m(T,x)=g(x).$

Remark 3.2**.**

If we introduce the so-called upper and lower Hamiltonian $\displaystyle 1$-forms defined respectively by

[TABLE]

[TABLE]

then the multitime (HJIU) PDE systems can be written in the form

[TABLE]

and

[TABLE]

The proof will be given in another paper.

4 Representation formula of viscosity

solutions for multitime (HJ) PDEs

In this section, we want to obtain a representation formula for the viscosity solution $\displaystyle M(t,x)$ of the multitime (HJ) PDEs system

[TABLE]

[TABLE]

where the unique solution $\displaystyle M(t,x)$ satisfies the inequalities

[TABLE]

for some constants $\displaystyle D,E,F$ (for $\displaystyle m=1,$ see also [4]).

Also, we assume that $\displaystyle g:\mathbb{R}^{n}\rightarrow\mathbb{R},H_{\alpha}:\Omega_{0T}\times\mathbb{R}^{n}\times\mathbb{R}^{p}\rightarrow\mathbb{R},$ satisfy the inequalities

[TABLE]

and

[TABLE]

Max-min representation of a Lipschitz function as affine functions (for $\displaystyle m=1,$ see also [2], [3]).

Lemma 4.1**.**

For each $\displaystyle\alpha$, let

[TABLE]

Let $\displaystyle H_{\alpha}$ be a Lipschitz 1-form. For some constant $\displaystyle P>0$ and for each $\displaystyle t\in\Omega_{0T},\,\,x\in\mathbb{R}^{n},$ we have

[TABLE]

if $\displaystyle\|{p}\|\leq P$.

Proof.

In view of the assumption $\displaystyle H_{\alpha}(t,x,v_{\alpha})-H_{\alpha}(t,x,{p})\leq K_{\alpha}\|{p}-v_{\alpha}\|,$ by the Cauchy-Schwarz formula, and by the condition $\displaystyle||u||\leq 1$, we have for any $\displaystyle x\in\mathbb{R}^{n},$

[TABLE]

∎

Max-min representation of a Lipschitz function as positive homogeneous functions (for m=1, see also [2],[3]).

Lemma 4.2**.**

Let $\displaystyle H_{\alpha}$ be a Lipschitz $\displaystyle 1$-form which is homogeneous in $\displaystyle p,$ i.e.,

[TABLE]

Then there exist compact sets $\displaystyle U,V\subset\mathbb{R}^{2n}$ and vector fields

[TABLE]

satisfying

[TABLE]

and such that, for each $\displaystyle\alpha$,

[TABLE]

for all $\displaystyle t\in\Omega_{0T},x\in\mathbb{R}^{n},p\in\mathbb{R}^{n}.$

Proof.

Let $\displaystyle u_{\alpha}=(u^{1}_{\alpha},u^{2}_{\alpha}),v_{\alpha}=(v^{1}_{\alpha},v^{2}_{\alpha})$ ($\displaystyle 2n$-dimensional controls) and

[TABLE]

According to Lemma $\displaystyle\eqref{l-2}$ and the assumptions $\displaystyle\eqref{eq:9},$ if $\displaystyle\|\eta\|=1,$ we have

[TABLE]

for $\displaystyle U^{1}=V^{1}=B(0,1)\in\mathbb{R}^{n}$.

For any $\displaystyle p\neq 0$, we can write

[TABLE]

Then, if we choose $\displaystyle C_{\alpha}>0$ such that $\displaystyle|L_{\alpha}|\leq C_{\alpha},$ we find

[TABLE]

Now, interchanging $\displaystyle\min_{u_{\alpha}^{1}\in U^{1}}$ and $\displaystyle\max_{v_{\alpha}^{2}\in V^{1}}$, the result in Lemma follows. ∎

We are now in a position to give the main result of this section.

Theorem 4.3**.**

For each $\displaystyle t\in\Omega_{0T}$ and $\displaystyle x\in\mathbb{R}^{n},$ the upper value function $\displaystyle M(t,x)$ verifies the equality

[TABLE]

where for each pair of controls $\displaystyle v_{\alpha}\in V(T-t)$,   $\displaystyle u_{\alpha}=\Phi[v_{\alpha}]\in U(T-t),$ the state function $\displaystyle x(\cdot)$ solves the problem

[TABLE]

Proof.

Let

[TABLE]

$\displaystyle U=B(0,1)\subset\mathbb{R}^{pm},V=B(0,P)\subset\mathbb{R}^{qm}$ and $\displaystyle X^{i}_{\alpha},L_{\alpha}$ Lipschitz functions with the assumptions $\displaystyle\eqref{eq:10}.$

Then $\displaystyle H_{\alpha}(t,x,p)=H^{1}_{\alpha}(t,x,p)$ provided $\displaystyle|p|\leq P.$ Since $\displaystyle M(t,x)$ satisfies $\displaystyle\eqref{eq:11},$ it follows that $\displaystyle M(t,x)$ is also the unique viscosity solution of the multitime (HJ) PDEs system (for $\displaystyle m=1,$ see also [4])

[TABLE]

[TABLE]

If we take $\displaystyle M^{1}(t,x)=M(T-t,x),$ one observes that $\displaystyle M^{1}(t,x)$ is a viscosity solution of this system (for $\displaystyle m=1,$ see also [2])

[TABLE]

[TABLE]

and

[TABLE]

Using the above developments, we obtain

[TABLE]

where $\displaystyle x(\cdot)$ is the solution of the Cauchy problem

[TABLE]

for the control $\displaystyle u_{\alpha}(\cdot)=\Phi[v_{\alpha}].$ ∎