On the Theory of Positional Differential Games for Neutral-Type Systems

Abstract

For a dynamical system whose motion is described by neutral-type differential equations in Hale’s form, we consider a minimax–maximin differential game with a quality index evaluating the motion history realized up to the terminal time. The control actions of the players are subject to geometric constraints. The game is formalized in classes of pure positional strategies with a memory of the motion history. It is proved that the game has a value and a saddle point. The proof is based on the choice of an appropriate Lyapunov–Krasovskii functional for the construction of control strategies by the method of an extremal shift to accompanying points.

INTRODUCTION

This paper is devoted to the development of the theory of positional differential games [1, 2, 3] for systems of neutral type. We consider a zero-sum differential game in which the motion of a dynamical system is described by differential equations of neutral type in Hale’s form [4]. There are geometric constraints on the control actions of the players. The quality of the control process is estimated in terms of the motion history of the system that has formed by the terminal time. The game is formalized in classes of pure positional strategies within the approach of [1, 2, 3]. The result of the paper is a theorem on the existence of a value and a saddle point in the differential game under consideration.

Issues of the existence of a value and optimal strategies in positional differential games for neutral-type systems were studied earlier in [5, 6, 7, 8]. Linear neutral-type systems were considered in [8]. Differential games for nonlinear systems formalized in classes of control strategies with a guide were the subject of [5, 7]. The result closest to the present paper was established in [6], where a differential game in classes of pure positional strategies was considered for nonlinear neutral-type systems of a fairly general form. However, because of a special proof technique based on the constructions of two guides [9, 10], additional considerable constraints were imposed on the game in [6]: it was required that the functional defining the quality index and the functional under the derivative on the left-hand side of the motion equations should satisfy the Lipschitz condition, and the Lipschitz constant for the latter functional should be less than 1. These constraints are removed in the present paper, whereas the form of the system is slightly less general as compared to [5, 6, 7] but still quite typical. This result was obtained using the classical reasoning scheme from [3], in which an appropriate Lyapunov–Krasovskii functional [11, 12] was chosen.

1 DIFFERENTIAL GAME

Consider a zero-sum differential game in which the motion of the system is described by a differential equation of neutral type in Hale’s form [4]

$$\begin{array}[]{l}\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\Big{(}x(t)-g(t,x (t-h))\Big{)}=f(t,x(t),x(t-h),u(t),v(t)),\\ t\in[t_{0},\vartheta],\quad x(t)\in\mathbb{R}^{n},\quad u(t)\in\mathbb{U}, \quad v(t)\in\mathbb{V},\end{array}$$

(1.1)

and the quality index has the form

$$\gamma=\sigma(x_{\vartheta}(\cdot)).$$

(1.2)

Here $t$ is time; $x(t)$ is the state vector at a time $t$; $t_{0}$ and $\vartheta$ are fixed initial and terminal times; $h>0$ is the delay constant; $x_{\vartheta}(\cdot)$ is the motion history on the interval $[\vartheta-h,\vartheta]$: $x_{\vartheta}(\xi)=x(\vartheta+\xi)$ for $\xi\in[-h,0]$; $u(t)$ and $v(t)$ are the current control actions of the first and second players, respectively; and $\mathbb{U}\subset\mathbb{R}^{k}$ and $\mathbb{V}\subset\mathbb{R}^{l}$ are compact sets.

The first player aims to minimize the index (1.2); the second, to maximize it.

Throughout the paper, we denote the scalar product of vectors by $\langle\cdot,\cdot\rangle$, the Euclidean norm by $\|\cdot\|$, and the space of Lipschitz functions from $[a,b]$ to $\mathbb{R}^{n}$ equipped with the uniform norm by Lip$([a,b],\mathbb{R}^{n})$; we also use the notation $\mathrm{Lip}=\mathrm{Lip}([-h,0],\mathbb{R}^{n})$. The uniform norm in Lip is denoted by $\|\cdot\|_{\infty}$. For $\alpha>0$, we write $B(\alpha)=\{x\in\mathbb{R}^{n}\colon\|x\|\leq\alpha\}$.

Assume that functions $g\colon[t_{0},\vartheta]\times\mathbb{R}^{n}\mapsto\mathbb{R}^{n}$ and $f\colon[t_{0},\vartheta]\times\mathbb{R}^{n}\times\mathbb{R}^{n}\times\mathbb{ U}\times\mathbb{V}\mapsto\mathbb{R}^{n}$ and a functional $\sigma\colon\mathrm{Lip}\mapsto\mathbb{R}$ satisfy the following conditions:

$(g)$  For any $\alpha>0$, there exists $\lambda_{g}=\lambda_{g}(\alpha)>0$ such that

$$\big{\|}g(t,x)-g(t^{\prime},x^{\prime})\big{\|}\leq\lambda_{g}\big{(}|t-t^{ \prime}|+\|x-x^{\prime}\|\big{)},\quad t,t^{\prime}\in[t_{0},\vartheta],\quad x ,x^{\prime}\in B(\alpha).$$

$(f_{1})$  The function $f$ is continuous.

$(f_{2})$  There exists a constant $c_{f}>0$ such that

$$\big{\|}f(t,x,y,u,v)\big{\|}\leq c_{f}\big{(}1+\|x\|+\|y\|\big{)},\quad(t,x,y, u,v)\in[t_{0},\vartheta]\times\mathbb{R}^{n}\times\mathbb{R}^{n}\times\mathbb{ U}\times\mathbb{V}.$$

$(f_{3})$  For any $\alpha>0$, there exists $\lambda_{f}=\lambda_{f}(\alpha)>0$ such that

$$\begin{array}[]{l}\big{\|}f(t,x,y,u,v)-f\big{(}t,x^{\prime},y^{\prime},u,v) \big{\|}\leq\lambda_{f}\big{(}\|x-x^{\prime}\|+\|y-y^{\prime}\|\big{)},\\ t\in[t_{0},\vartheta],\quad x,y,x^{\prime},y^{\prime}\in B(\alpha),\quad u\in \mathbb{U},\quad v\in\mathbb{V}.\end{array}$$

($f_{4}$)  For any $t\in[t_{0},\vartheta]$ and $x,y,s\in\mathbb{R}^{n}$,

$$\min\limits_{u\in\mathbb{U}}\,\max\limits_{v\in\mathbb{V}}\,\langle f(t,x,y,u, v),s\rangle=\max\limits_{v\in\mathbb{V}}\,\min\limits_{u\in\mathbb{U}}\, \langle f(t,x,y,u,v),s\rangle.$$

($\sigma$)  The functional $\sigma$ is continuous.

Fix some numbers $\alpha_{0},\lambda_{0}>0$ and define the set of initial positions

$$G_{0}=\big{\{}t_{0}\big{\}}\times\big{\{}w(\cdot)\in\mathrm{Lip}\colon\|w(\xi) \|\leq\alpha_{0},\ \|w(\xi)-w(\xi^{\prime})\|\leq\lambda_{0}|\xi-\xi^{\prime}| ,\ \xi,\xi^{\prime}\in[-h,0]\big{\}}.$$

Taking the number $c_{f}$ from condition $(f_{2})$, define the set of admissible positions

$$\begin{array}[]{l}G=\Big{\{}(t,x_{t}(\cdot))\in[t_{0},\vartheta]\times\mathrm{ Lip}\colon x(\cdot)\in\mathrm{Lip}([t_{0}-h,\vartheta],\mathbb{R}^{n}),\,(t_{0 },x_{t_{0}}(\cdot))\in G_{0},\\ \Big{\|}\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\Big{(}x(t)-g(t,x(t-h))\Big {)}\Big{\|}\leq c_{f}\big{(}1+\|x(t)\|+\|x(t-h)\|\big{)}\text{ for a.a. }t\in[ t_{0},\vartheta]\Big{\}}.\end{array}$$

(1.3)

Hereinafter, $x_{t}(\cdot)$ is a function such that $x_{t}(\xi)=x(t+\xi)$, $\xi\in[-h,0]$.

Let a position $(\tau,w(\cdot))\in G$, $\tau<\vartheta$, be chosen. Admissible realizations of the control actions $u(t)$ and $v(t)$ on the interval $[\tau,\vartheta]$ are measurable functions $u(\cdot):$$[\tau,\vartheta)\mapsto\mathbb{U}$ and $v(\cdot)\colon[\tau,\vartheta)\mapsto\mathbb{V}$. Proceeding, for example, by the scheme from [13] (see also [14, P1]), one can show that, under conditions $(g)$ and $(f_{1})$–$(f_{3})$, any pair of admissible realizations $u(\cdot)$ and $v(\cdot)$ uniquely generates a motion $x(\cdot)$ of system (1.1) from the position $(\tau,w(\cdot))$, which is a function from $\mathrm{Lip}([\tau-h,\vartheta],\mathbb{R}^{n})$ satisfying the initial condition $x(\tau+\xi)=w(\xi)$, $\xi\in[-h,0]$, and, together with $u(t)$ and $v(t)$, equation (1.1) almost everywhere on $[\tau,\vartheta]$. In addition, by the definition (1.3) of the set $G$, the motion $x(\cdot)$ is such that

$$(t,x_{t}(\cdot))\in G,\quad\tau\in[t,\vartheta].$$

(1.4)

The differential game (1.1), (1.2) will be formalized in classes of positional strategies of the players’ controls in accordance with the approach [1, 3, 2]. In view of condition ($f_{4}$), we can restrict ourselves to the class of pure positional strategies [3, Sect. 8].

A control strategy of the first player is understood as a mapping

$$U=U(t,w(\cdot),\varepsilon)\in\mathbb{U},\quad(t,w(\cdot))\in G,\quad \varepsilon>0,$$

where $\varepsilon$ is an accuracy parameter [3, p. 68].

Fix a position $(\tau,w(\cdot))\in G$, a number $\varepsilon>0$, and a partition of the interval $[\tau,\vartheta]$:

$$\Delta_{\delta}=\big{\{}t_{j}:0<t_{j+1}-t_{j}\leq\delta,\,j\in\overline{1,J-1} ,\,t_{1}=\tau,\,t_{J}=\vartheta\big{\}}.$$

(1.5)

The triple $\{U,\varepsilon,\Delta_{\delta}\}$ defines a feedback control law of the first player, who forms a piecewise constant (hence, admissible) realization $u(\cdot)$ by the rule

$$u(t)=U(t_{j},x_{t_{j}}(\cdot),\varepsilon),\quad t\in[t_{j},t_{j+1}),\quad j \in\overline{1,J-1}.$$

(1.6)

This law, together with an admissible realization of the second player’s control $v(\cdot)$, uniquely generates a motion $x(\cdot)$ of system (1.1) from the position $(\tau,w(\cdot))$. Denote the corresponding value of the quality index (1.2) by $\gamma(\tau,w(\cdot);U,\varepsilon,\Delta_{\delta};v(\cdot))$.

Define the guaranteed result of the strategy $U$:

$$\rho_{u}(\tau,w(\cdot),U)=\overline{\lim_{\varepsilon\downarrow 0}}\,\lim_{ \delta\downarrow 0}\,\sup_{\Delta_{\delta}}\,\sup_{v(\cdot)}\gamma(\tau,w( \cdot);U,\varepsilon,\Delta_{\delta};v(\cdot)).$$

(1.7)

Then the optimal guaranteed result of the first player is the value

$$\rho_{u}^{\circ}(\tau,w(\cdot))=\inf_{U}\rho_{u}(\tau,w(\cdot),U).$$

(1.8)

A strategy $U^{\circ}$ is called optimal if

$$\rho_{u}(\tau,w(\cdot),U^{\circ})=\rho_{u}^{\circ}(\tau,w(\cdot)),\quad(\tau,w (\cdot))\in G.$$

Similarly, with obvious changes, we consider for the second player a control strategy $V=V(t,w(\cdot),\varepsilon)\in\mathbb{V}$, $(t,w(\cdot))\in G$, $\varepsilon>0$, the control law $\{V,\varepsilon,\Delta_{\delta}\}$ defining a piecewise constant realization $v(\cdot)$ by the rule

$$v(t)=V(t_{j},x_{t_{j}}(\cdot),\varepsilon),\quad t\in[t_{j},t_{j+1}),\quad j \in\overline{1,J-1},$$

the guaranteed result of the strategy $V$

$$\rho_{v}(\tau,w(\cdot),V)=\varliminf_{\varepsilon\downarrow 0}\,\lim_{\delta \downarrow 0}\,\inf_{\Delta_{\delta}}\,\inf_{u(\cdot)}\gamma(\tau,w(\cdot);u( \cdot);V,\varepsilon,\Delta_{\delta}),$$

(1.9)

and the optimal guaranteed result of the second player

$$\rho_{v}^{\circ}(\tau,w(\cdot))=\sup_{V}\rho_{v}(\tau,w(\cdot),V).$$

(1.10)

A control strategy of the second player $V^{\circ}$ is optimal if

$$\rho_{v}(\tau,w(\cdot),V^{\circ})=\rho_{v}^{\circ}(\tau,w(\cdot)),\quad(\tau,w (\cdot))\in G.$$

It follows from relations (1.8) and (1.10) that

$$\rho_{u}^{\circ}(\tau,w(\cdot))\geq\rho_{v}^{\circ}(\tau,w(\cdot)),\quad(\tau, w(\cdot))\in G.$$

(1.11)

In the case where

$$\rho_{u}^{\circ}(\tau,w(\cdot))=\rho_{v}^{\circ}(\tau,w(\cdot)),\quad(\tau,w( \cdot))\in G,$$

the differential game (1.1), (1.2) is said to have a value, and the pair of optimal strategies $\{U^{\circ},V^{\circ}\}$ is called a saddle point of the game.

Theorem

The differential game (1.1), (1.2) has a value and a saddle point $\{U^{\circ},V^{\circ}\}$.

A key role in the proof of this theorem is played by the following auxiliary Lyapunov–Krasovskii functional [11, 12].

2 LYAPUNOV–KRASOVSKII FUNCTIONAL

Based on relation (1.3), we can show the existence of $\alpha_{G},\lambda_{G}>0$ such that

$$\|w(\xi)\|\leq\alpha_{G},\quad\|w(\xi)-w(\xi^{\prime})\|\leq\lambda_{G}|\xi- \xi^{\prime}|,\quad\xi,\xi^{\prime}\in[-h,0],\quad(t,w(\cdot))\in G.$$

(2.1)

Then, by conditions $(g)$ and $(f_{3})$, for $\lambda_{g}=\lambda_{g}(\alpha_{G})$ and $\lambda_{f}=\lambda_{f}(\alpha_{G})$, we have

$$\|g(t,w(-h))-g(t^{\prime},w^{\prime}(-h))\|\leq\lambda_{g}\big{(}|t-t^{\prime} |+\|w(-h)-w^{\prime}(-h)\|\big{)},$$

$$\|f(t,w(0),w(-h),u,v)-f(t,w^{\prime}(0),w^{\prime}(-h),u,v)\|$$

$$\leq\lambda_{f}\big{(}\|w(0)-w^{\prime}(0)\|+\|w(-h)-w^{\prime}(-h)\|\big{)},$$

(2.2)

$$(t,w(\cdot)),(t^{\prime},w^{\prime}(\cdot))\in G,\quad u\in\mathbb{U},\quad v \in\mathbb{V}.$$

Define the functional

$$V_{\varepsilon}(t,p,w(\cdot))=\kappa_{\varepsilon}(t,p,w(\cdot))e^{-2(\lambda_ {f}+\lambda_{g}/h)(t-t_{0})},\quad(t,p,w(\cdot))\in[t_{0},\vartheta]\times \mathbb{R}^{n}\times\mathrm{Lip},$$

(2.3)

where

$$\kappa_{\varepsilon}(t,p,w(\cdot))=\sqrt{\varepsilon^{2}+\|p\|^{2}}+\lambda_{f }\displaystyle\intop\limits_{-h}^{0}\Big{(}1-\frac{2\lambda_{g}\xi}{h}\Big{)} \|w(\xi)\|\mathrm{d}\xi,\quad\varepsilon>0.$$

(2.4)

Lemma 1

Assume that $\tau\in[t_{0},\vartheta],$$\varepsilon>0,$ and functions $s(\cdot)\in\mathrm{Lip}([\tau,\vartheta],\mathbb{R}^{n})$ and $z(\cdot)\in\mathrm{Lip}([\tau-h,\vartheta],\mathbb{R}^{n})$ satisfy the bounds

$$\begin{array}[]{l}\|s(t)-z(t)\|\leq\lambda_{g}\|z(t-h)\|,\quad t\in[\tau, \vartheta],\\ \displaystyle\Big{\langle}\frac{\mathrm{d}{s}(t)}{\mathrm{d}t},s(t)\Big{ \rangle}\leq\lambda_{f}\big{(}\|z(t)\|+\|z(t-h)\|\big{)}\|s(t)\|+\varepsilon^{ 2}\text{ for a.a. }t\in[\tau,\vartheta].\end{array}$$

(2.5)

Then

$$V_{\varepsilon}(t,s(t),z_{t}(\cdot))\leq V_{\varepsilon}(\tau,s(\tau),z_{\tau} (\cdot))+(t-\tau)\varepsilon,\quad t\in[\tau,\vartheta].$$

(2.6)

Proof

Based on relations (2.3) and (2.4), since the functions $s(\cdot)$ and $z(\cdot)$ are Lipschitz, we can show that the functions $\omega_{1}(t)=\kappa_{\varepsilon}(t,s(t),z_{t}(\cdot))$ and $\omega_{2}(t)=V_{\varepsilon}(t,s(t),z_{t}(\cdot))$, $t\in[\tau,\vartheta]$, are also Lipschitz. Then, using bounds (2.5), we get

$$\begin{array}[]{l}\displaystyle\frac{\mathrm{d}\omega_{1}(t)}{\mathrm{d}t}= \frac{\langle\mathrm{d}s(t)/\mathrm{d}t,s(t)\rangle}{\sqrt{\varepsilon^{2}+\|s (t)\|^{2}}}+\lambda_{f}\|z(t)\|-\lambda_{f}(1+2\lambda_{g})\|z(t-h)\|+\frac{2 \lambda_{f}\lambda_{g}}{h}\intop\limits_{t-h}^{t}\|z(\xi)\|\mathrm{d}\xi\\ \displaystyle\leq\varepsilon+2\lambda_{f}\|z(t)\|-2\lambda_{f}\lambda_{g}\|z(t -h)\|+\frac{2\lambda_{f}\lambda_{g}}{h}\intop\limits_{t-h}^{t}\|z(\xi)\| \mathrm{d}\xi\\ \leq\varepsilon+2\lambda_{f}\|s(t)\|+\displaystyle\frac{2\lambda_{f}\lambda_{g }}{h}\displaystyle\intop\limits_{t-h}^{t}\|z(\xi)\|\mathrm{d}\xi\end{array}$$

for almost all $t\in[\tau,\vartheta]$. Hence, we obtain the bound

$$\displaystyle\frac{\mathrm{d}\omega_{2}(t)}{\mathrm{d}t}=\bigg{(}\varepsilon-2 (\lambda_{g}/h)\|s(t)\|-2\lambda_{f}^{2}\intop\limits_{t-h}^{t}\|z(\xi)\| \mathrm{d}\xi\bigg{)}e^{-2(\lambda_{f}+\lambda_{g}/h)(t-t_{0})}\leq\varepsilon,$$

which implies inequality (2.6).   $\square$

Lemma 2

There exists a number $\lambda_{V}>0$ such that$,$ for all $\varepsilon>0,$$\tau\in[t_{0},\vartheta],$$p\in\mathbb{R}^{n},$ and $w(\cdot)\in\mathrm{Lip},$ under the condition

$$\|w(\xi)\|\leq 2\alpha_{G},\quad\|w(\xi)-w(\xi^{\prime})\|\leq 2\lambda_{G}| \xi-\xi^{\prime}|,\quad\xi,\xi^{\prime}\in[-h,0],$$

(2.7)

the following inequality holds:

$$\|w(\cdot)\|_{\infty}^{2}\leq\lambda_{V}V_{\varepsilon}(t,p,w(\cdot)).$$

(2.8)

Proof

Under condition (2.7), there exists $\lambda_{*}>0$ such that

$$\|w(\cdot)\|_{\infty}^{2}\leq\lambda_{*}\intop\limits_{-h}^{0}\|w(\xi)\| \mathrm{d}\xi.$$

Therefore, setting $\lambda_{V}=e^{2(\lambda_{f}+\lambda_{g}/h)(\vartheta-t_{0})}\lambda_{*}/ \lambda_{f}$ and using (2.3), we obtain inequality (2.8).   $\square$

3 PROOF OF THE THEOREM

For the right-hand side of system (1.1), we define the Hamiltonian

$$H(t,x,y,s)=\min\limits_{u\in\mathbb{U}}\max\limits_{v\in\mathbb{V}}\langle f(t ,x,y,u,v),s\rangle,\quad t\in[t_{0},\vartheta],\quad x,y,s\in\mathbb{R}^{n},$$

(3.1)

and the set-valued mappings

$$\begin{array}[]{l}F_{+}(t,x,y,v)=\text{co}\big{\{}f(t,x,y,u,v)\,|\,u\in\mathbb {U}\big{\}}\subset\mathbb{R}^{n},\\ F_{-}(t,x,y,u)=\text{co}\big{\{}f(t,x,y,u,v)\,|\,v\in\mathbb{V}\big{\}}\subset \mathbb{R}^{n},\end{array}t\in[t_{0},\vartheta],\ x,y\in\mathbb{R}^{n},\ u\in \mathbb{U},\ v\in\mathbb{V},$$

(3.2)

where co means a convex hull in $\mathbb{R}^{n}$. By conditions $(f_{1})$–$(f_{4})$, we have the following facts.

$(H)$  The following bound holds for the number $\lambda_{f}>0$ from (2.2) and any $(t,w(\cdot)),(t,w^{\prime}(\cdot))\in G$ and $s\in\mathbb{R}^{n}$:

$$|H(t,w(0),w(-h),s)-H(t,w^{\prime}(0),w^{\prime}(-h),s)|$$

$$\leq\lambda_{f}\big{(}\|w(0)-w^{\prime}(0)\|+\|w(-h)-w^{\prime}(-h)\|\big{)}\| s\|.$$

$(F_{1})$  The set-valued mappings $F_{+}$ and $F_{-}$ have convex compact values and are continuous in the Hausdorff metric.

$(F_{2})$  The following inequality holds for the number $c_{f}>0$ from condition $(f_{2})$ for any $t\in[t_{0},\vartheta]$, $x,y\in\mathbb{R}^{n}$, $u\in\mathbb{U}$, $v\in\mathbb{V}$, and $l\in F_{+}(t,x,y,v)\cup F_{-}(t,x,y,u)$:

$$\|l\|\leq c_{f}\big{(}1+\|x\|+\|y\|\big{)}.$$

$(F_{3})$  For any $t\in[t_{0},\vartheta]$ and $x,y,s\in\mathbb{R}^{n}$,

$$\max\limits_{v\in\mathbb{V}}\min\limits_{l\in F_{+}(t,x,y,v)}\langle l,s \rangle=H(t,x,y,s)=\min\limits_{u\in\mathbb{U}}\max\limits_{l\in F_{-}(t,x,y,u )}\langle l,s\rangle.$$

For $(\tau,w(\cdot))\in G$, $u\in\mathbb{U}$, and $v\in\mathbb{V}$, denote by $X_{+}(\tau,w(\cdot),v)$ and $X_{-}(\tau,w(\cdot),u)$ the sets of functions $x(\cdot)\in\mathrm{Lip}([\tau-h,\vartheta],\mathbb{R}^{n})$ satisfying the condition $x(\tau+\xi)=w(\xi)$ for $\xi\in[-h,0]$ and the following differential inclusions for almost all $t\in[\tau,\vartheta]$, respectively:

$$\frac{\mathrm{d}}{\mathrm{d}t}\Big{(}x(t)-g(t,x(t-h))\Big{)}\in F_{+}(t,x(t),x (t-h),v),$$

$$\frac{\mathrm{d}}{\mathrm{d}t}\Big{(}x(t)-g(t,x(t-h))\Big{)}\in F_{-}(t,x(t),x (t-h),u).$$

Proceeding by the scheme from [14, P2], we can show that the sets $X_{+}(\tau,w(\cdot),v)$ and $X_{-}(\tau,w(\cdot),u)$ are compact in $\mathrm{Lip}([\tau-h,\vartheta],\mathbb{R}^{n})$ and, in view of (1.3),

$$(t,x_{t}(\cdot))\in G,\quad t\in[\tau,\vartheta],\quad x(\cdot)\in X_{+}(\tau, w(\cdot),v)\cup X_{-}(\tau,w(\cdot),u).$$

(3.3)

The results of [7] imply the following statement.

Assertion

There exists a continuous functional $\varphi\colon G\mapsto\mathbb{R}$ such that

$$\varphi(\vartheta,w(\cdot))=\sigma(w(\cdot)),\quad(\vartheta,w(\cdot))\in G;$$

(3.4)

$$\begin{array}[]{l}\varphi(\tau,w(\cdot))\geq\min\limits_{x(\cdot)\in X_{+}( \tau,w(\cdot),v)}\varphi(t,x_{t}(\cdot)),\\ \varphi(\tau,w(\cdot))\leq\max\limits_{x(\cdot)\in X_{-}(\tau,w(\cdot),u)} \varphi(t,x_{t}(\cdot)),\end{array}$$

(3.5)

where $(\tau,w(\cdot))\in G,\quad t\in[\tau,\vartheta],\quad u\in\mathbb{U},\quad v \in\mathbb{V}.$

Suppose that $\varepsilon>0$, $(t,w(\cdot))\in G$, and a functional $V_{\varepsilon}$ is defined according to (2.3). Denote by $O_{\varepsilon}(t,w(\cdot))$ the set of positions $(t,r(\cdot))\in G$ satisfying the inequality

$$V_{\varepsilon}(t,w(0)-g(t,w(-h))-r(0)+g(t,r(-h)),w(\cdot)-r(\cdot))\leq \varepsilon(1+t-t_{0}).$$

(3.6)

Using the definition (1.3) of the set $G$, we can show that $O_{\varepsilon}(t,w(\cdot))$ is compact in $\{t\}\times\mathrm{Lip}$.

Let $\varphi$ be the functional from the assertion. Define

$$U^{*}(t,w(\cdot),\varepsilon)\in\operatorname*{argmin}_{u\in\mathbb{U}}\max \limits_{v\in\mathbb{V}}\langle f(t,w(0),w(-h),u,v),$$

$$w(0)-g(t,w(-h))-r_{*}(0)+g(t,r_{*}(-h))\rangle,$$

(3.7)

where

$$(t,r_{*}(\cdot))\in\operatorname*{argmin}_{(t,r(\cdot))\in O_{\varepsilon}(t,w (\cdot))}\varphi(t,r(\cdot)).$$

Lemma 3

The inequality $\rho_{u}(\tau,w(\cdot),U^{*})\leq\varphi(\tau,w(\cdot))$ holds for $(\tau,w(\cdot))\in G,$ where $\rho_{u}$ is defined according to (1.7).

Proof

By the definition (1.7) of the value $\rho_{u}(\tau,w(\cdot),U^{*})$, it is sufficient to show that, for any $\zeta>0$, there exist a number $\varepsilon_{*}=\varepsilon_{*}(\zeta)>0$ and a function $\delta_{*}(\varepsilon)=\delta_{*}(\zeta,\varepsilon)>0$, $\varepsilon\in(0,\varepsilon_{*})$, such that, for any position $(\tau,w(\cdot))\in G$, numbers $\varepsilon\in(0,\varepsilon_{*})$ and $\delta\in(0,\delta_{*}(\varepsilon))$, partition $\Delta_{\delta}$ (1.5), and admissible realization $v(\cdot)$, the motion $x(\cdot)$ of system (1.1) generated from the position $(\tau,w(\cdot))$ by the control law $\{U^{*},\varepsilon,\Delta_{\delta}\}$ and the realization $v(\cdot)$ satisfies the inequality

$$\gamma(\tau,w(\cdot);U^{*},\varepsilon,\Delta_{\delta};v(\cdot))=\sigma(x_{ \vartheta}(\cdot))\leq\varphi(\tau,w(\cdot))+\zeta.$$

(3.8)

By conditions $(f_{1})$ and $(F_{1})$ and bounds (2.1), there exists a function $\delta_{f}(\varepsilon)$, $\varepsilon\in[0,+\infty)$, such that, for any $\varepsilon>0$, $u\in\mathbb{U}$, $v\in\mathbb{V}$, and $(t,w(\cdot)),(t^{\prime},w^{\prime}(\cdot))\in G$, under the conditions

$$|t-t^{\prime}|\leq\delta_{f}(\varepsilon),\quad\|w(0)-w^{\prime}(0)\|\leq \delta_{f}(\varepsilon),\quad\|w(-h)-w^{\prime}(-h)\|\leq\delta_{f}( \varepsilon),$$

the following bounds hold:

$$\begin{array}[]{l}\|f(t,w(0),w(-h),u,v)-f(t^{\prime},w^{\prime}(0),w^{\prime}( -h),u,v)\|\leq\varepsilon,\\ \max\limits_{l\in F_{+}(t,w(0),w(-h),v)}\ \min\limits_{l^{\prime}\in F_{+}(t^{ \prime},w^{\prime}(0),w^{\prime}(-h),v)}\|l-l^{\prime}\|\leq\varepsilon.\end{array}$$

(3.9)

Let $\zeta>0$. By condition $(\sigma)$ and bounds (2.1), there exists a number $\varepsilon_{\sigma}>0$ such that, for any $(\vartheta,w(\cdot)),(\vartheta,w^{\prime}(\cdot))\in G$, under the condition $\|w(\cdot)-w^{\prime}(\cdot)\|_{\infty}\leq\varepsilon_{\sigma}$, the following inequality holds:

$$|\sigma(w(\cdot))-\sigma(w^{\prime}(\cdot))|\leq\zeta.$$

(3.10)

Taking the numbers $\alpha_{G}$ and $\lambda_{G}$ from (2.1) and the numbers $\lambda_{g}$ and $\lambda_{f}$ from (2.2), define

$$\alpha_{s}=2(1+\lambda_{g})\alpha_{G},\quad\lambda_{s}=2\lambda_{g}(1+\lambda_ {G}),\quad c_{*}=c_{f}(1+2\alpha_{G}).$$

(3.11)

Take the number $\lambda_{V}>0$ from Lemma 2 and define

$$\begin{array}[]{l}\varepsilon_{*}=\displaystyle\frac{\varepsilon^{2}_{\sigma}} {\lambda_{V}^{2}(1+\vartheta-t_{0})},\\ \delta_{*}(\varepsilon)=\min\Big{\{}\displaystyle\frac{\delta_{f}(\varepsilon^ {2}/(8\alpha_{s}))}{1+\lambda_{G}},\ \displaystyle\frac{\varepsilon^{2}}{8c_{* }\lambda_{s}},\ \displaystyle\frac{\varepsilon^{2}}{16\alpha_{s}\lambda_{f} \lambda_{G}},\ \displaystyle\frac{\varepsilon^{2}}{16\alpha_{G}\lambda_{f} \lambda_{s}}\Big{\}}.\end{array}$$

(3.12)

Let a position $(\tau,w(\cdot))\in G$, numbers

$$\varepsilon\in(0,\varepsilon_{*}),\quad\delta\in(0,\delta_{*}(\varepsilon)),$$

(3.13)

a partition $\Delta_{\delta}$ (1.5), and an admissible realization $v(\cdot)$ be fixed. Consider the motion $x(\cdot)$ of system (1.1) generated by the control law $\{U^{*},\varepsilon,\Delta_{\delta}\}$ and realization $v(\cdot)$. Let us prove by induction the following bound:

$$\varphi(t_{j},r^{j}(\cdot))\leq\varphi(\tau,w(\cdot)),\quad j\in\overline{1,J},$$

(3.14)

where

$$(t_{j},r^{j}(\cdot))\in\operatorname*{argmin}_{(t_{j},r(\cdot))\in O_{ \varepsilon}(t_{j},x_{t_{j}}(\cdot))}\varphi(t_{j},r(\cdot)).$$

(3.15)

For $j=1$, the bound holds by the choice (3.15) of the position $(\tau,r^{1}(\cdot))$. Assume now that (3.14) holds for $j=k$ and prove it for $j=k+1$. Let

$$v^{k}\in\operatorname*{argmax}_{v\in\mathbb{V}}\min\limits_{l\in F_{+}(t_{k},r ^{k}(\cdot),v)}\langle l,x(t_{k})-g(t_{k},x(t_{k}-h))-r^{k}(0)+g(t_{k},r^{k}(- h))\rangle.$$

(3.16)

According to inequalities (3.5), there exists a function $y^{k}(\cdot)\in X_{+}(t_{k},r^{k}(\cdot),v^{k})$ such that

$$\varphi(t_{k+1},y^{k}_{t_{k+1}}(\cdot))\leq\varphi(t_{k},r^{k}(\cdot)).$$

(3.17)

Define the functions

$$z^{k}(t)=x(t)-y^{k}(t),\ s^{k}(t)=z^{k}(t)-g(t,x(t-h))+g(t,y^{k}(t-h)),\ t\in[ t_{k},\vartheta].$$

(3.18)

Then, in view of the choice (3.12), (3.13) of the number $\delta$, notation (3.11), relations (1.4) and (3.3), bounds (2.1) and (2.2), and conditions $(f_{2})$ and $(F_{2})$, we derive the following inequalities for any $t\in[t_{k},t_{k+1}]$, $u\in\mathbb{U}$, and $v\in\mathbb{V}$:

$$\begin{array}[]{l}\max\big{\{}|t-t_{k}|,\|x_{t}(\cdot)-x_{t_{k}}(\cdot)\|_{ \infty},\|y^{k}_{t}(\cdot)-y^{k}_{t_{k}}(\cdot)\|_{\infty}\big{\}}\leq\delta_{ f}(\varepsilon^{2}/(8\alpha_{s})),\\ \|z^{k}_{t}(\cdot)\|_{\infty}\leq 2\alpha_{G},\ \|z^{k}_{t}(\cdot)-z^{k}_{t_{k }}(\cdot)\|_{\infty}\leq\varepsilon^{2}/(8\lambda_{f}\alpha_{s}),\\ \|s^{k}(t)-z^{k}(t)\|\leq\lambda_{g}\|z^{k}(t-h)\|,\\ \|s^{k}(t)\|\leq\alpha_{s},\ \|s^{k}(t)-s^{k}(t_{k})\|\leq\lambda_{s}|t-t_{k}| \leq\min\{\varepsilon^{2}/(8c_{*}),\varepsilon^{2}/(16\alpha_{G}\lambda_{f})\} ,\\ \|f(t,x(t),x(t-h),u,v)\|\leq c_{*},\ \sup\{\|l\|\colon l\in F_{+}(t,y^{k}(t),y ^{k}(t-h),v)\}\leq c_{*}.\end{array}$$

(3.19)

By equation (1.1) and the inclusion $y^{k}(\cdot)\in X_{+}(t_{k},r^{k}(\cdot),v^{k})$, we have

$$\Big{\langle}\frac{\mathrm{d}s^{k}(t)}{\mathrm{d}t},s^{k}(t)\Big{\rangle}= \langle f(t,x(t),x(t-h),u(t),v(t))-l^{k}(t),s^{k}(t)\rangle,$$

$$l^{k}(t)\in F(t,y^{k}(t),y^{k}(t-h),v_{k})$$

for almost all $t\in[t_{k},t_{k+1}]$. Hence, using (3.9) and (3.19), we derive

$$\displaystyle\Big{\langle}\frac{\mathrm{d}s^{k}(t)}{\mathrm{d}t},s^{k}(t)\Big{ \rangle}\leq\langle f(t_{k},x(t_{k}),x(t_{k}-h),u(t),v(t))-l^{k}_{*}(t),s^{k}( t)\rangle+\varepsilon^{2}/4$$

$$\leq\langle f(t_{k},x(t_{k}),x(t_{k}-h),u(t),v(t))-l^{k}_{*}(t),s^{k}(t_{k}) \rangle+\varepsilon^{2}/2,$$

where $l^{k}_{*}(t)\in F_{+}(t_{k},r^{k}(0),r^{k}(-h),v^{k})$. Further, using the definition (3.7) of the strategy $U^{*}$, the rule (1.6) of forming the realization $u(\cdot)$ corresponding to this strategy, the choice (3.16) of the value $v^{k}$, properties $(H)$ and $(F_{3})$, the definition (3.18) of the function $z^{k}(\cdot)$, and bounds (3.19), we obtain

$$\displaystyle\Big{\langle}\frac{\mathrm{d}s^{k}(t)}{\mathrm{d}t},s^{k}(t)\Big{ \rangle}\leq\max\limits_{v\in\mathbb{V}}\langle f(t_{k},x(t_{k}),x(t_{k}-h),u( t),v),s^{k}(t_{k})\rangle$$

$${}-\min\limits_{l\in F(t_{k},r^{k}(0),r^{k}(-h),v^{k})}\langle l,s^{k}(t_{k}) \rangle+\varepsilon^{2}/2$$

$$=H(t_{k},x(t_{k}),x(t_{k}-h),s^{k}(t_{k}))-H(t_{k},y^{k}(t_{k}),y^{k}(t_{k}-h) ,s^{k}(t_{k}))+\varepsilon^{2}/2$$

$$\leq\lambda_{f}\big{(}\|z^{k}(t_{k})\|+\|z^{k}(t_{k}-h)\|\big{)}\|s^{k}(t_{k}) \|+\varepsilon^{2}/2\leq\lambda_{f}\big{(}\|z^{k}(t)\|+\|z^{k}(t-h)\|\big{)}\| s^{k}(t)\|+\varepsilon^{2}.$$

Hence, in view of the fourth inequality in (3.19), we conclude that all conditions of Lemma 1 hold for $z(\cdot)=z^{k}(\cdot)$ and $s(\cdot)=s^{k}(\cdot)$. Therefore, we have the inequality

$$V_{\varepsilon}(t_{k+1},s^{k}(t_{k+1}),z^{k}_{t_{k+1}}(\cdot))\leq V_{ \varepsilon}(t_{k},s^{k}(t_{k}),z^{k}_{t_{k}}(\cdot))+(t_{k+1}-t_{k})\varepsilon,$$

which, in view of the inclusion $r^{k}(\cdot)\in O_{\varepsilon}(t_{k},x_{t_{k}}(\cdot))$ and inequality (3.6), means that $y^{k}_{t_{k+1}}(\cdot)\in O_{\varepsilon}(t_{k+1},x_{t_{k+1}}(\cdot))$. Thus, according to relations (3.15) and (3.17) and the induction assumption, we derive

$$\varphi(t_{k+1},r^{k+1}_{t_{k+1}}(\cdot))\leq\varphi(t_{k+1},y^{k}_{t_{k+1}}( \cdot))\leq\varphi(t_{k},r^{k}(\cdot))\leq\varphi(\tau,w(\cdot)).$$

Thus, inequality (3.14) is proved for all $j\in\overline{1,J}$.

By Lemma 2, using first the inclusion $(\vartheta,r^{J}(\cdot))\in O_{\varepsilon}(\vartheta,x_{\vartheta}(\cdot))$ together with inequality (3.6) and then relations (3.12) and (3.13), we obtain

$$\begin{array}[]{l}\|x_{\vartheta}(\cdot)-r^{J}(\cdot)\|_{\infty}^{2}\leq \lambda_{V}V_{\varepsilon}(\vartheta,x(\vartheta)-g(\vartheta,x(\vartheta-h))- r^{J}(0)+g(\vartheta,r^{J}(-h)),x_{\vartheta}(\cdot)-r^{j}(\cdot))\\ \leq\lambda_{V}(1+\vartheta-t_{0})\varepsilon\leq\lambda_{V}(1+\vartheta-t_{0} )\varepsilon_{*}=\varepsilon_{\sigma}^{2}.\end{array}$$

Hence, using relations (3.10), (3.14), and (3.4), we conclude that

$$\sigma(x_{\vartheta}(\cdot))\leq\sigma(r^{J}(\cdot))+\zeta=\varphi(\vartheta,r ^{J}(\cdot))+\zeta\leq\varphi(t,w(\cdot))+\zeta.$$

The lemma is proved.   $\square$

Similarly, we can prove the following lemma for the second player, setting

$$V^{*}(t,w(\cdot),\varepsilon)\in\operatorname*{argmax}_{v\in\mathbb{V}}\min \limits_{u\in\mathbb{U}}\langle f(t,w(0),w(-h),u,v),$$

$$r_{*}(0)-g(t,r_{*}(-h))-w(0)+g(t,w(-h))\rangle,$$

where

$$(t,r_{*}(\cdot))\in\operatorname*{argmax}_{(t,r(\cdot))\in O_{\varepsilon}(t,w (\cdot))}\varphi(t,r(\cdot)).$$

Lemma 4

The inequality $\rho_{v}(\tau,w(\cdot),V^{*})\geq\varphi(\tau,w(\cdot))$ holds for $(\tau,w(\cdot))\in G,$ where $\rho_{v}$ is defined in (1.9).

Proof of the theorem

From Lemmas 3 and 4 and relations (1.8), (1.10), and (1.11), we obtain the equality

$$\rho^{\circ}_{u}(\tau,w(\cdot))=\rho_{u}(\tau,w(\cdot),U^{*})=\rho_{v}(\tau,w( \cdot),V^{*})=\rho^{\circ}_{v}(\tau,w(\cdot)),$$

which shows that the differential game (1.1), (1.2) has a value and the pair of strategies $\{U^{*},V^{*}\}$ forms a saddle point of the game. The theorem is proved.   $\square$