Game Control Problem for Systems of Distributed Equations

Abstract

We consider a game problem of guaranteed positional control for a distributed system described by the phase field equations under incomplete information on system’s phase states. This problem is investigated from the viewpoint of the first player (the partner). For this player, a procedure for forming feedback controls is specified. This procedure is stable with respect to informational noises and computational errors and is based on the method of extremal shift and the method of stable sets from the theory of guaranteed positional control. It uses the idea of stable dynamical inversion of controlled systems.

V. Maksimov—This work was supported by the Russian Science Foundation (project 14-01-00539).

1 Introduction

The control theory for distributed systems has been intensively developed in recent time as a part of mathematical control theory. At present, there exists a number of monographs devoted to control problems for distributed systems [1,2,3]. As a rule, the emphasis is on program control problems in the case when all system’s parameters are precisely specified. Along with this, the investigation of control problems for systems with uncontrollable disturbances (the problems of game control) is also reasonable. Similar problems have been poorly investigated. In the early 70es, N.N. Krasovskii suggested an effective approach to solving game (guaranteed) control problems, which is based on the formalism of positional strategies. The detailed description of this approach for dynamical systems described by ordinary differential equations is given in [4]. The goal of the present work is to illustrate possibilities of this approach for investigating a game problem for systems described by the phase field equations.

We consider a system modeling the solidification process and governed by the phase field equations (introduced in [5])

$$\begin{aligned} \begin{array}{l} \displaystyle \frac{\partial }{\partial t}\psi + l\displaystyle \frac{\partial }{\partial t}\varphi =k\varDelta _L \psi +Bu-Cv\quad \text{ in }\quad Q = \varOmega \times (t_0,\vartheta ],\\ \tau \displaystyle \frac{\partial }{\partial t}\varphi =\xi ^2\varDelta _L \varphi + g(\varphi )+\psi ,\quad \vartheta =\mathrm{const}<+\infty , \end{array} \end{aligned}$$

(1)

with the boundary condition $\frac{\partial }{\partial n} \psi =\frac{\partial }{\partial n}\varphi =0\quad \text{ on }\quad \partial \varOmega \times (t_0,\vartheta ]$ and the initial condition $\psi (t_0)=\psi _0,\quad \varphi (t_0)=\varphi _0\quad \text{ in }\quad \varOmega$. Here, $\varOmega \subset \mathbb {R}^n$ is a bounded domain with the sufficiently smooth boundary $\partial \varOmega$, $\varDelta _L$ is the Laplace operator, ${\partial }/{\partial n}$ is the outward normal derivative, $(U,|\cdot |_U)$ and $(V,|\cdot |_V)$ are Banach spaces, $B\in \mathcal{L}(U;H)$ and $C\in \mathcal{L}(V;H)$ are linear continuous operators, $H=L_2(\varOmega )$, and the function g(z) is the derivative of a so-called potential G(z). We assume that $g(z)=az+bz^2-cz^3$.

Systems of form (1) have been investigated by many authors. In what follows, for the sake of simplicity, we assume that $k=\xi =\tau =c=1$. Further, we assume that the following conditions are fulfilled: (A1) the domain $\varOmega \subset \mathbb {R}^n$, $n=2,3$, has the boundary of $C^2$-class; (A2) the coefficients a and b are elements of the space $L_{\infty }(T\times \varOmega ), T = [t_0,\vartheta ]$, and $\mathop {\mathrm {vrai}}\sup c(t,\eta )>0$ for $(t,\eta )\in [t_0,\vartheta ]\times \varOmega$; (A3) the initial functions $\psi _0$ and $\varphi _0$ are such that $\{\psi _0,\varphi _o\}\in \mathcal{R}=\{\psi ,\varphi \in W_{\infty }^2(\varOmega ):\quad \frac{\partial }{\partial n} \psi =\frac{\partial }{\partial n}\varphi =0\quad \text{ on }\quad \partial \varOmega \}.$

Introduce the notation: $W_p^{2,1}(Q)=\left\{ u \mid u, \frac{\partial u}{\partial \eta _i}, \frac{\partial ^2 u}{\partial \eta _i \partial \eta _j}, \frac{\partial u}{\partial t}\in L^p(Q)\right\}$ for $p\in [1,\infty )$ is the standard Sobolev space with the norm $\Vert u\Vert _{W_p^{2,1}(Q)};$ $(\cdot ,\cdot )_H$ and $|\cdot |_H$ are the scalar product and the norm in H, respectively. Let some initial state $x_0=\{\psi _0,\varphi _0\}$ and functions $u(\cdot )\in L_{\infty }(T;U)$ and $v(\cdot )\in L_{\infty }(T;V)$ be fixed. A solution of system (1), $x(\cdot ;t_0,x_0,u(\cdot ),v(\cdot ))=\{\psi (\cdot ;t_0, \psi _0, u(\cdot ),v(\cdot )), \varphi (\cdot ;t_0,\varphi _0,u(\cdot ),v(\cdot ))\}$, is a unique function $x(\cdot )=x(\cdot ; t_0,x_0,u(\cdot ), v(\cdot ))\in V_T^{(1)}=V_1\times V_1,\quad V_1=W_2^{2,1}(Q)$ satisfying relations (1). By virtue of the corresponding embedding theorem, without loss of generality, one can assume that the space $V_T^{(1)}$ is embedded into the space $C(T;H\times H)$. Therefore, the element $x(t)=\{\psi (t),\varphi (t)\}\in H\times H$ is the phase state of system (1) at the time t. The following theorem takes place

Theorem 1

[6, p. 25, Assertion 5] Let conditions (A1)–(A3) be fulfilled. Then for any $u(\cdot )\in L_{\infty }(T;U)$ and $v(\cdot )\in L_{\infty }(T;V)$ there exists a unique solution of system (1).

The paper is devoted to the investigation of the game control problem. Let us give the informal formulation of this problem. Let a uniform net $\varDelta = \{ \tau _i \}_{i=0}^m,\quad \tau _i=\tau _{i-1}+\delta ,\quad \tau _0=t_0,\quad \tau _m=\vartheta$ with a diameter $\delta =\tau _i-\tau _{i-1}$ be fixed on a given time interval T. Let a solution of system (1) be unknown. At the times $\tau _i\in \varDelta$, a part of the phase states $x(\tau _i)$ (namely $\phi (\tau _i)$) is inaccurately measured. The results of measurements $\xi _i^h\in H$, $i\in [1:m-1]$, satisfy the inequalities

$$\begin{aligned} |\xi _i^h-\phi (\tau _i)|_H\le h. \end{aligned}$$

(2)

Here, $h\in (0,1)$ is a level of informational noise. Let the following quality criterion be given: $I(x(\cdot ),u(\cdot ))=\sigma (x(\vartheta ))+ \int \limits _{t_0}^{\vartheta } \chi (t,x(t),u(t))\,dt$, where $\sigma :H\times H\rightarrow \mathbb {R}$ and $\chi :T\times H \times H\times U\rightarrow \mathbb {R}$ are given functions satisfying the local Lipschitz conditions. Let also a prescribed value of the criterion, number $I_*$, be fixed. The control problem under consideration consists in the following. There are two players-antagonists controlling system (1) by means of u and v, respectively. One of them is called a partner; another, an opponent. Let $P\subset U$ and $E \subset V$ be given convex bounded and closed sets. The problem undertaken by the partner is as follows. It is necessary to construct a law (a strategy) for forming the control u (with values from P) by the feedback principle (on the base of measurements of $\varphi (\tau _i)$) in order to provide the prescribed value of the quality criterion for any (unknown) realization $v = v(\cdot )$.

To form the control u providing the solution of the problem, along with the information on the “part” of coordinates of the solution of system (1) (namely, on the values $\xi _i^h$ satisfying inequalities (2)), it is necessary to obtain some additional information on the coordinate $\psi (\cdot )$, which is missing. To get such a piece of information during the control process, it is reasonable, following the approach developed in [7,8,9], to introduce an auxiliary controlled system. This system is described by a parabolic equation (the form is specified below). The equation has an output $w_*(t)$, $t\in T$, and an input $p^h(t)$, $t\in T$. The input $p^h(\cdot )$ is some new auxiliary control; it should be formed by the feedback principle in such a way that $p^h(\cdot )$ “approximates” the unknown coordinate $\psi (\cdot )$ in the mean square metric. Thus, along with the block of forming the control in the real system (it is called an controller), we need to incorporate into the control contour one more block (it is called an identifier) allowing to reconstruct the missing coordinate $\psi (\cdot )$ in real time. Note that, in essence, the identifier block solves a dynamical inverse problem, namely, the problem of (approximate) reconstruction of the unknown coordinate $\psi (\cdot )$. In he recent time, the theory of inverse problems for distributed systems has been intensively developed. Among the latest investigations, it is possible to mark out the research [10].

2 Problem statement

Before passing to the problem formulation, we give some definitions. Furthermore, we denote by $u_{a,b}(\cdot )$ the function u(t), $t \in [a,b]$, considered as a whole. The symbol $P_{a,b}(\cdot )$ stands for the restriction of the set $P_T(\cdot )$ onto the segment $[a,b] \subset T$. Any strongly measurable functions $u(\cdot ): T\rightarrow P$ and $v(\cdot ): T\rightarrow E$ are called program controls of the partner and opponent, respectively. The sets of all program controls of the partner and opponent are denoted by the symbols $P_T(\cdot )$ and $E_T(\cdot )$ : $P_T(\cdot )=\{u(\cdot )\in L_2(T;U):u(t)\in P\ \text{ a.e. }\ t\in T\},$ $E_T(\cdot )=\{v(\cdot )\in L_2(T;V):v(t)\in E\ \text{ a.e. }\ t\in T\}$. Elements of the product $T\times \mathcal{H}$ are called positions, $\mathcal{H}=H\times H\times \mathbb {R} \times H\times H\times \mathbb {R}$. Any function (perhaps, multifunction) $\mathcal{U}: T\times \mathcal{H}\rightarrow P$ is said to be a positional strategy of the partner. The positional strategy corrects the controls at discrete times given by some partition of the interval T. Any function $\mathcal{V}:T\times H\times H\rightarrow H$ is said to be a strategy of reconstruction. The strategy $\mathcal{V}$ is formed in order to reconstruct the unknown component $\psi (\cdot )$.

Consider the following ordinary differential equation

$$\begin{aligned} \dot{q}(t)=\chi (t,x(t),u(t)),\quad q(t_0)=0. \end{aligned}$$

(3)

Introducing this new variable q, we reduce the control problem of Bolza type to a control problem with a terminal quality criterion of the form $I=\sigma (x(\vartheta )) + q(\vartheta )$. In this case, the controlled system consists of phase field Eq. (1) and ordinary differential Eq. (3).

The scheme of an algorithm for solving the problem undertaken by the partner is as follows. In the beginning, auxiliary systems $M_1$ and $M_2$ (models) are introduced. The system $M_1$ has an input $u^*(\cdot )$ and an output $w(\cdot )$; the system $M_2$, an input $p^h(\cdot )$ and an output $w_*(\cdot )$, respectively. The model $M_2$ with its control law $\mathcal{V}$ forms the identifier, whereas the model $M_1$ and system (1) (with their control laws) form the controller. The process of synchronous feedback control of systems (1), (3), $M_1$, and $M_2$ is organized on the interval T. This process is decomposed into $m-1$ identical steps. At the ith step carried out during the time interval $[\tau _i,\tau _{i+1})$, the following actions are fulfilled. First, at the time $\tau _i$, according to some chosen rules $\mathcal{V}$ and $\mathcal{U}$, the elements $p_i^h\in \mathcal{V}(\tau _i,\xi _i^h,w_*(\tau _i)),\quad u_i^h\in \mathcal{U}(\tau _i,\xi _i^h,p_i^h,\psi ^h_i,w(\tau _i))$ are calculated. Here, $\psi ^h_i$ is the result of measuring $q(\tau _i)$. Then (till the moment $\tau _{i+1}$), the control $p^h(t)=p^h_i$, $\tau _i\le t < \tau _{i+1}$, is fed onto the input of the system $M_2$; the control $u^h(t)=u_i^h$, $\tau _i\le t<\tau _{i+1}$, onto the input of system (1), (3). Under the action of these controls, as well as of the given control $u^*(t)$, $\tau _i\le t < \tau _{i+1}$, and the unknown control of the opponent v(t), $\tau _i\le t < \tau _{i+1}$, the states $x(\tau _{i+1})$, $q(\tau _i)$, $w(\tau _{i+1})$, and $w_*(\tau _{i+1})$ are realized at the time $\tau _{i+1}$. The procedure stops at the time $\vartheta$.

Let models $M_1$ and $M_2$ with phase trajectories $w(\cdot )$ and $w_*(\cdot )$ be fixed. A solution $x(\cdot )$ of system (1) starting from an initial state $(t_*,x_*)$ and corresponding to piecewise constant controls $u^h(\cdot )$ and $p^h(\cdot )$ (formed by the feedback principle) and to a control $v_{t_*,\vartheta }(\cdot )\in E_{t_*,\vartheta }(\cdot )$ is called an $(h,\varDelta )$-motion $x_{\varDelta ,w}^h(\cdot )=x_{\varDelta ,w}^h(\cdot ;t_*,x_*,\mathcal{U},\mathcal{V},v_{t_*,\vartheta }(\cdot ))$. This motion is generated by the positional strategies $\mathcal{U}$ and $\mathcal{V}$. Thus, the motions $x_{\varDelta ,w}^h(\cdot )$, $q^h_{\varDelta }(\cdot )$,$w(\cdot )$, and $w_*(\cdot )$ are formed simultaneously. So, for $t\in [\tau _i,\tau _{i+1})$, we define $x_{\varDelta ,w}^h(t)=x(t;\tau _i,x_{\varDelta ,w}^h(\tau _i),u_{\tau _i,\tau _{i+1}}^h(\cdot ), v_{\tau _i,\tau _{i+1}}(\cdot )),$ $\quad q^h_{\varDelta }(t) = q(t;\tau _i,q^h_{\varDelta }(\tau _i),u_{\tau _i,\tau _{i+1}}^h(\cdot )),$ $w(t)=w(t;\tau _i,w(\tau _i),u^*_{\tau _i,\tau _{i+1}}(\cdot )), w_*(t)=w_*(t;\tau _i,w_*(\tau _i),p_{\tau _i,\tau _{i+1}}^h(\cdot )),$ where

$$\begin{aligned} u^h(t)=u_i^h\in \mathcal{U}(\tau _i,\xi ^h_i,p_i^h,q^h(\tau _i),w(\tau _i)), \ \ p^h(t)=p_i^h\in \mathcal{V}(\tau _i,\xi ^h_i,w_*(\tau _i))\quad \end{aligned}$$

(4)

$$ \text{ for }\ t\in [\tau _i,\tau _{i+1}),\ $$

$$\begin{aligned} i\in [i(t_*):m-1],\quad |\xi _i^h-\phi ^h_{\varDelta ,w}(\tau _i)|_H\le h,\quad |\psi _i^h-g^h_{\varDelta }(\tau _i)| \le h, \end{aligned}$$

(5)

$$ i(t_*)=\min \{i:\tau _i>t_*\},\ \ u^h(t)=u_*^h\in P,\ \ p^h(t)=p_*^h\in H \ \ \text{ for }\ t\in [t_*,\tau _{i(t_*)}). $$

The set of all $(h,\varDelta )$-motions is denoted by $X_h(t_*,x_*,\mathcal{U},\mathcal{V},\varDelta ,w)$.

Problem. It is necessary to construct a positional strategy $\mathcal{U}:T\times \mathcal{H}\rightarrow P$ of the partner and a positional strategy $\mathcal{V}: T \times H\times H \rightarrow H$ of reconstruction with the following properties: whatever a value $\varepsilon >0$ and a disturbance $v_T(\cdot )\in E_T(\cdot )$, one can find (explicitly) numbers $h_*>0$ and $\delta _*>0$ such that the inequalities

$$\begin{aligned} |I(x_{\varDelta , w}^h(\cdot ),u^h_T(\cdot ))-I_*|\le \varepsilon \quad \forall x_{\varDelta ,w}^h(\cdot )\in X_h(t_0,x_0,\mathcal{U},\mathcal{V},\varDelta ,w) \end{aligned}$$

(6)

are fulfilled uniformly with respect to all measurements $\xi _i^h$ and $\psi _i^h$ with properties (5), if $h \le h_*$ and $\delta =\delta (\varDelta ) \le \delta _*$.

3 Algorithm for Solving the Problem

To solve the Problem, we use ideas from [4], namely, the method of a priori stable sets. In our case, this method consists in the following. Let a trajectory of model $M_1$, $w(\cdot )$, possessing the property $\sigma (w_1(\vartheta )) + w_2(\vartheta ) = I_*$ be known. Then, a feedback strategy $\mathcal{U}$ providing tracking the prescribed trajectory of $M_1$ by the trajectory of real system (1) is constructed. This means that the $(h,\varDelta )$-motion $x_{\varDelta }^h(\cdot )$ formed by the feedback principle (see (4)) remains at a “small” neighborhood of the trajectory $w(\cdot )$ during the whole interval T. This property of the $(h,\varDelta )$-motion allows us to conclude that the chosen strategy solves the considered control problem.

Let us pass to the realization of this scheme. Define $\varPhi (t,x,u,v)=\{Bu-Cv,\chi (t,x,u)\} \in H\times \mathbb {R}$, $\varPhi _{u}(t,x,v)=\bigcup \limits _{u\in P} \varPhi (t,x,u,v)$, $H_*(t;x)=\bigcap \limits _{v\in E} \varPhi _{u}(t,x,v)$, $H_*(\cdot ;x)=\{ u(\cdot )\in L_2(T;H\times \mathbb {R}): u(t)\in H_*(t;x)$ $for\quad a.~a.\ t\in T\}$. As a model $M_1$, we take the system including two subsystems, i.e., the phase field equation

$$\begin{aligned} \displaystyle \frac{\partial }{\partial t}w^{(1)}+ l\displaystyle \frac{\partial }{\partial t}w^{(2)}=\varDelta _L w^{(1)}+u_1\quad \text{ in }\quad \varOmega \times (t_0,\vartheta ], \end{aligned}$$

(7)

$$ \displaystyle \frac{\partial }{\partial t}w^{(2)}=\varDelta _L w^{(2)}+ g(w^{(2)})+w^{(1)} $$

with the boundary condition $\frac{\partial }{\partial n} w^{(1)}=\frac{\partial }{\partial n}w^{(2)}=0\quad \text{ on }\quad \partial \varOmega \times (t_0,\vartheta ]$ and the initial condition $w^{(1)}(t_0)=\psi _0,\quad w^{(2)}(t_0)=\varphi _0\quad \text{ in }\quad \varOmega$ , as well as the ordinary differential equation

$$\begin{aligned} \dot{w}^{(3)}(t) = u_2(t),\quad \text{ w }^{(3)} \in \mathbb {R}, \quad \text{ w }^{(3)}(0)=0. \end{aligned}$$

(8)

By the symbol $w(\cdot )$, we denote the solution of system (7), (8). Then, the model $M_1$ has the control $u(\cdot ) = \{u_1(\cdot ),u_2(\cdot )\}$. As a model $M_2$, we use the equation

$$\begin{aligned} \frac{\partial }{w}_*{\partial t} = \varDelta _L w_*+ p^h + g(w_*)\quad \text{ in }\quad \varOmega \times (t_0,\vartheta ] \end{aligned}$$

(9)

with the boundary condition $\frac{\partial w_*}{\partial n} = 0 \quad \text{ on }\quad \partial \varOmega \times (t_0,\vartheta ]$ and the initial condition $w_*(t_0)=\varphi _0\quad \text{ in }\quad \varOmega$.

Condition 1

There exists a control $u(\cdot )= u^*(\cdot ) = \{u^*_1(\cdot ),u^*_2(\cdot )\} \in H(t; w^{(1)}(t),w^{(2)}(t))$ for a.a. $t \in T$ such that $I_* = \sigma (w^{(1)}(\vartheta ))+ w^{(2)}(\vartheta )$.

The strategies $\mathcal{U}$ and $\mathcal{V}$ (see (4)) are defined in such a way:

$$\begin{aligned} \mathcal{U}(t,\xi ,p,\psi ,w)=\arg \min \{L(u,y) +(\psi -w^{(3)})\chi (t,\xi ,p,u) :u\in P\}, \end{aligned}$$

(10)

$$\begin{aligned} \mathcal{V}(t,\xi ,w_*)=\arg \min \{l(t,\alpha ,u,s):u\in U_d\}, \end{aligned}$$

(11)

where $w=\{w^{(1)},w^{(2)},w^{(3)}\}$, $L(u,y)=(-y,Bu)_H$, $y=w^{(1)}-p+l(w^{(2)}-\xi )$, $l(t,\alpha ,u,s)=\exp (-2b_*t)(s,u)_H+\alpha |u|^2_H$, $s=w_*-\xi$, $b_*= |a+1/3b|_{L_{\infty }(Q)}$, $\alpha = \alpha (h)$, $U_d =\{p(\cdot ) \in L_2(T;H) : |p(t)|_H \le d\$ for a.a. $t \in T\}$, $d\ge \sup _{t \in T}\{|\psi (t;t_0,x_0,u(\cdot ),v(\cdot ))|_H: u(\cdot ) \in P_T(\cdot ), v(\cdot ) \in E_T(\cdot )\}$.

Condition 2

Let $h, \delta (h)$, and $\alpha (h)$ satisfy the conditions: $\alpha ( h) \rightarrow 0,\delta (h) \rightarrow 0, (h+\delta (h))\alpha ^{-1}(h) \rightarrow 0$ as $h \rightarrow 0$.

Let us pass to the description of the algorithm for solving the Problem. Namely, we describe the procedure of forming the $(h,\varDelta )$-motion $x_{\varDelta ,w}^h(\cdot )=\{\psi _{\varDelta ,w}^h(\cdot ), \varphi _{\varDelta ,w}^h(\cdot )\}$ and trajectory $g^h_{\varDelta }(\cdot )$ corresponding to some fixed partition $\varDelta$ and the strategies $\mathcal{U}$ and $\mathcal{V}$, see (10) and (11). Before the algorithm starts, we fix a value $h \in (0,1)$, a function $\alpha =\alpha (h)$: $(0,1)\rightarrow (0,1)$, and a partition $\varDelta _h= \{\tau _{h,i}\}_{i=0}^{m_h}$ with diameter $\delta = \delta (h)= \tau _{i+1}-\tau _i$, $\tau _i = \tau _{h,i}$, $\tau _{h,0} = t_0, \tau _{h,m_h}= \vartheta$. The work of the algorithm is decomposed into $m_h-1$ identical steps. We assume that

$$ u^h(t)=u_0^h\in \mathcal{U}(t_0,\xi _0^h,p_0^h,0,w(t_0)),\quad p^h(t)=p_0^h\in \mathcal{V}(t_0,\xi _0^h,\varphi _0) $$

($|\xi _0^h-\varphi _0|_H\le h$) on the interval $[t_0,\tau _1)$. Under the action of these piecewise-constant controls as well as of an unknown disturbance $v_{t_0,\tau _1}(\cdot )$, the $(h,\varDelta )$-motion $\{x_{\varDelta ,w}^h(\cdot )\}_{t_0,\tau _1}=\{\psi _{\varDelta ,w}^h (\cdot ;t_0,\psi _0,u^h_{t_0,\tau _1}(\cdot ),v_{t_0,\tau _1}(\cdot )), \varphi _{\varDelta ,w}^h(\cdot ,t_0,\varphi _0,u^h_{t_0,\tau _1}(\cdot ), v_{t_0,\tau _t}(\cdot )) \}_{t_0,\tau _1}$ of system (1), the trajectory $\{q^h_{\varDelta }(\cdot )\}_{t_0,\tau }=\{q(\cdot ;t_0,q(t_0)), u^h_{t_0,\tau _1}(\cdot ))\}_{t_0,\tau _1}$ of Eq. (3), the trajectory $\{w_*(\cdot )\}_{t_0,\tau _1}=\{w_*(\cdot ;t_0,\phi _0, p^h_{t_0,\tau _1}(\cdot ))\}_{t_0,\tau _1}$ of the model $M_2$, and the trajectory $\{w(\cdot )\}_{t_0,\tau _1}=\{w(\cdot ;t_0,x_0, u^*_{t_0,\tau _1}(\cdot ))\}_{t_0,\tau _1}$ of the model $M_1$ are realized. At the time $t=\tau _1$, we determine $u_1^h$ and $p_1^h$ from the conditions

$$ u_1^h \in \mathcal{U}(\tau _1,\xi _1^h,p_1^h, \psi ^h_1,w(\tau _1)), \quad p_1^h\in \mathcal{V}(\tau _1,\xi _1^h,w_*(\tau _1)) $$

($|\xi _1^h -\varphi _{\varDelta ,w}^h(\tau _1)|_H \le h$, $|\psi _1^h -g_{\varDelta }^h(\tau _1)| \le h$), i.e., we assume that $u^h(t)=u_1^h$ and $p^h(t)=p_1^h$ for $t\in [\tau _1,\tau _2).$ Then, we calculate the realization of the $(h,\varDelta )$-motion

$$ \begin{array}{ll} \{x_{\varDelta ,w}^h(\cdot )\}_{\tau _1,\tau _2}=&{}\{\psi _{\varDelta ,w}^h(\cdot ;\tau _1,\psi _{\varDelta ,w}^h(\tau _1),u^h_{\tau _1,\tau _2}(\cdot ), v_{\tau _1,\tau _2}(\cdot )),\\ &{}\varphi _{\varDelta ,w}^h(\cdot ;\tau _1, \varphi _{\varDelta ,w}^h(\tau _1), u^h_{\tau _1,\tau _2}(\cdot ), v_{\tau _1,\tau _2}(\cdot ))\}_{\tau _1,\tau _2}, \end{array} $$

the trajectory $\{q^h_{\varDelta }(\cdot )\}_{\tau _1,\tau _2}=\{q(\cdot ;\tau _1,q^h_{\varDelta }(\tau _1),p^h_{\tau _1,\tau _2}(\cdot ))\}_{\tau _1,\tau _2}$ of Eq. (3), the trajectory $\{w_*(\cdot )\}_{\tau _1,\tau _2}=\{w_*(\cdot ;\tau _1,w_*(\tau _1),p^h_{\tau _1,\tau _2}(\cdot ))\}_{\tau _1,\tau _2}$ of the model $M_2,$ and the trajectory $\{w(\cdot )\}_{\tau _1,\tau _2}=\{w(\cdot ;\tau _1, w(\tau _1), u^*_{\tau _1,\tau _2}(\cdot )\}_{\tau _1,\tau _2}$ of the model $M_1$.

Let the $(h,\varDelta )$-motion $x_{\varDelta ,w}^h(\cdot )$, the trajectory $q^h_{\varDelta }(\cdot )$ of Eq. (3), the trajectory $w_*(\cdot )$ of the model $M_2$, and the trajectory $w(\cdot )$ of the model $M_1$ be defined on the interval $[t_0,\tau _i]$. At the time $t=\tau _i$, we assume that

$$\begin{aligned} u_i^h \in \mathcal{U}(\tau _i,\xi _i^h,p_i^h,\psi ^h_i,w(\tau _i)), \quad p_i^h\in \mathcal{V}(\tau _i,\xi _i^h,w_*(\tau _i)) \end{aligned}$$

(12)

($|\xi _i^h - \varphi _{\varDelta ,w}^h(\tau _i)|_H \le h$, $|\psi _i^h - g_{\varDelta }^h(\tau _i)| \le h)$, i.e., we set $u^h(t)=u_i^h \text{ and } p^h(t)=p_i^h \text{ for } t\in [\tau _i,\tau _{i+1})$. As a result of the action of these controls and an unknown disturbance $v_{\tau _i,\tau _{i+1}}(\cdot )$, the $(h,\varDelta )$-motion

$$ \begin{array}{ll} \{x_{\varDelta ,w}^h(\cdot )\}_{\tau _i,\tau _{i+1}}=&{}\{\psi _{\varDelta ,w}^h(\cdot ;\tau _i,\psi _{\varDelta ,w}^h(\tau _i), u^h_{\tau _i,\tau _{i+1}}(\cdot ),v_{\tau _i,\tau _{i+1}}(\cdot ) ),\\ &{}\varphi (\cdot ;\tau _i,\varphi _{\varDelta ,w}^h(\tau _i), u^h_{\tau _i,\tau _{i+1}}(\cdot ),v_{\tau _i,\tau _{i+1}}(\cdot ) )\}_{\tau _i,\tau _{i+1}}, \end{array} $$

the trajectory $\{q^h_{\varDelta }(\cdot )\}_{\tau _i,\tau _{i+1}}=\{q(\cdot ;\tau _i,q^h_{\varDelta }(\tau _i),p^h_{\tau _i,\tau _{i+1}}(\cdot )\}_{\tau _i,\tau _{i+1}}$ of Eq. (3), the trajectory $\{w_*(\cdot )\}_{\tau _i,\tau _{i+1}}\!\!=\!\{w_*(\cdot ;\tau _i,w_*(\tau _i),p^h_{\tau _i,\tau _{1+1}}(\cdot ))\}_{\tau _i,\tau _{i+1}}$ of the model $M_2$, and the trajectory $\{w(\cdot )\}_{\tau _i,\tau _{i+1}}=\{w(\cdot ;\tau _i, w(\tau _i), u^*_{\tau _i,\tau _{i+1}}(\cdot ))\}_{\tau _i,\tau _{i+1}}$ of the model $M_1$ are realized on the interval $[\tau _i,\tau _{i+1}]$. This procedure stops at the time $\vartheta$.

Theorem 2

Let Conditions 1 and 2 be fulfilled. Let also the models $M_1$ and $M_2$ be specified by relations (7), (8), and (9), respectively. Then, the strategies $\mathcal{U}$ and $\mathcal{V}$ of form (10) and (11) solve the Problem.

Proof

To prove the theorem, we estimate the variation of the functional

$$\begin{aligned}&\varLambda (t,x_{\varDelta ,w}^h(\cdot ),q^h_{\varDelta }(\cdot ),w(\cdot ))= \varLambda ^0(x_{\varDelta ,w}^h(t),q^h_{\varDelta }(t),w(t))\\&+0.5\displaystyle {\int \limits _0^t\Big \{\int \limits _\varOmega |\nabla \pi ^h(\varrho ,\eta )|^2\,\mathrm {d}\eta +l^2\int \limits _\varOmega |\nabla \nu ^h(\varrho ,\eta )|^2\,\mathrm {d}\eta \Big \} \mathrm {d}\varrho ,} \end{aligned}$$

where $x^h_{\varDelta ,w}(\cdot ) =\{\psi ^h_{\varDelta ,w}(\cdot ),\phi ^h_{\varDelta ,w}(\cdot )\}$, $w(\cdot )=\{w^{(1)}(\cdot ),w^{(2)}(\cdot ),w^{(3)}(\cdot )\}$, $\pi ^h(t) = w^{(1)}(t) - \psi _{\varDelta ,w}^h(t)$, $\nu ^h(t)=w^{(2)}(t)-\varphi _{\varDelta ,w}^h(t)$, $g^h(t)=\pi ^h(t)+l\nu ^h(t)$, $\varLambda _1(x_{\varDelta ,w}^h(t), w(t))=0.5|g^h(t)|_H^2 + 0.5 l^2|\nu ^h(t)|_H^2$, $\lambda (q^h_{\varDelta }(t),w^{(3)}(t))=0.5|q^h_{\varDelta }(t)-w^{(3)}(t)|^2$, $\varLambda ^0(x_{\varDelta ,w}^h(t),q^h_{\varDelta }(t),w(t))=\varLambda _1(x^h_{\varDelta ,w}(t),w(t))+\lambda (q^h_{\varDelta }(t),w^{(3)}(t)$. It is easily seen that the functions $\pi ^h(\cdot )$ and $\nu ^h(\cdot )$ are solutions of the system

$$\begin{aligned} \frac{\partial \pi ^h(t,\eta )}{\partial t}+ l \frac{\partial \nu ^h(t,\eta )}{\partial t}= \varDelta _L \pi ^h(t,\eta )+u^*_1(t,\eta )-(Bu^h)(t,\eta )+(Cv)(t,\eta ), \end{aligned}$$

(13)

$$ \frac{\partial \nu ^h(t,\eta )}{\partial t}= \varDelta _L\nu ^h(t,\eta )+R^h(t,\eta )\nu ^h(t,\eta )+\pi ^h(t,\eta )\quad \text{ in }\ \varOmega \times (t_0,\vartheta ], $$

with the initial condition $\pi ^h(t_0)=\nu ^h(t_0)=0\ \text{ in }\ \varOmega$ and with the boundary condition $\displaystyle {\frac{\partial \pi ^h}{\partial n}=\frac{\partial \nu ^h}{\partial n}=0\ \text{ on }\ \partial \varOmega \times (t_0,\vartheta ].}$ Here, $R^h(t,\eta )=a(t,\eta )+ b(t,\eta )(w^{(1)}(t,\eta )+ \varphi ^h_{\varDelta ,w}(t,\eta )) {}- (({w^{(1)}}(t,\eta ))^2+w^{(1)}(t,\eta )\varphi ^h_{\varDelta ,w}(t,\eta )+(\varphi ^h_{\varDelta ,w})^2(t,\eta )).$ Multiplying scalarly the first equation of (13) by $g^h(t)$, and the second one, by $\nu ^h(t)$, we obtain

$$ (g^h(t),g_t^h(t))_H+\int \limits _{\varOmega }\{ |\nabla \pi ^h(t,\eta )|^2+ l\nabla \pi ^h(t,\eta )\nabla \nu ^h(t,\eta )\}\,\mathrm {d}\eta $$

$$\begin{aligned} =(g^h(t),u^*_1(t)-Bu^h(t)+Cv(t))_H, \end{aligned}$$

(14)

$$ (\nu ^h(t),\nu _t^h(t))_H+\int \limits _{\varOmega } |\nabla \nu ^h(t,\eta )|^2\,\mathrm {d}\eta \le (\pi ^h(t),\nu ^h(t))_H+b|\nu ^h(t)|_H^2\quad \text{ for } \text{ a.a. }\ t\in T. $$

Here, we use the inequality $\mathop {\mathop {\mathrm {vrai}}\max }\limits _{(t,\eta )\in T\times \varOmega } \{a(t,\eta )+b(t,\eta )(v_1+v_2)- (v_1^2+v_1v_2+v_2^2)\}\le b$, which is valid for any $v_1$, $v_2\in \mathbb {R}$. It is evident that the inequality

$$\begin{aligned} \int \limits _{\varOmega } l(\nabla \pi ^h(t,\eta ),\nabla \nu ^h(t,\eta ))\,\mathrm {d}\eta \ge -0.5\int \limits _{\varOmega } \{ |\nabla \pi ^h(t,\eta )|^2+ l^2|\nabla \nu ^h(t,\eta )|^2\}\,\mathrm {d}\eta \end{aligned}$$

(15)

is fulfilled for a.a. $t \in T$. Let us multiply the first inequality of (14) by $l^2$ and add to the second one. Taking into account (15), we have for a.a. $t\in T$

$$\begin{aligned} (g^h(t),g_t^h(t))_H+l^2(\nu ^h(t),\nu _t^h(t))_H+0.5\int \limits _\varOmega \{|\nabla \pi ^h(t,\eta )|^2+l^2 |\nabla \nu ^h(t,\eta )|^2\}\,\mathrm {d}\eta \end{aligned}$$

(16)

$$ \le (g^h(t), u^*_1(t)-Bu^h(t)+Cv(t))_H{} + l^2(\pi ^h(t),\nu ^h(t))_H+ bl^2|\nu ^h(t)|_H^2. $$

Note that $\pi ^h(t)=g^h(t)-l\nu ^h(t)$. In this case, for a.a. $t\in T$

$$\begin{aligned} (\pi ^h(t),\nu ^h(t))_H+b|\nu ^h(t)|^2_H=(g^h(t)-l\nu ^h(t),\nu ^h(t))_H {}+b|\nu ^h(t)|^2_H \end{aligned}$$

(17)

$$ =(g^h(t),\nu ^h(t))_H+(b-l)|\nu ^h(t)|^2_H \le 0.5 (|g^h(t)|^2_H+ (0.5+|b-l|)|\nu ^h(t)|^2_H. $$

Combining (16) and (17), we obtain

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t}\varLambda ^0(x_{\varDelta ,w}^h(t),q^h_{\varDelta }(t),w(t))+0.5\int \limits _\varOmega \{|\nabla \pi ^h(t,\eta )|^2+ l^2|\nabla \nu ^h(t,\eta )|^2\}\,\mathrm {d}\eta \end{aligned}$$

(18)

$$ \le 2l^2\lambda ^2\varLambda ^0(x_{\varDelta ,w}^h(t),q^h_{\varDelta }(t),w(t))+\gamma _t^{(1)} +\gamma _t^{(2)}, $$

where $\gamma _t^{(1)} = (q^h_{\varDelta }(t)-w^{(3)}(t))(\chi (t,x^h_{\varDelta ,w}(t),u^h(t))-u^*_2(t))$, $\gamma _t^{(2)}=(g^h(t),u^*_1(t)-Bu^h(t)+Cv(t))_H$ for a.a. $t\in T$. For a.a. $t\in \delta _i =[\tau _i,\tau _{i+1}],$ it is easily seen that

$$\begin{aligned} \gamma ^{(1)}_t \le (q^h_{\varDelta }(\tau _i)-w^{(3)}(\tau _i))(\chi (t,x^h_{\varDelta ,w}(t),u^h(t)-u^*_2(t)) + k_0(t-\tau _i)^{1/2}, \end{aligned}$$

(19)

$$\begin{aligned} |\chi (t,x^h_{\varDelta ,w}(t),u^h(t))-\chi (\tau _i,\xi ^h_i,p^h_i,u^h(t))| \le k_1\{h+(t-\tau _i)^{1/2}+|p^h_i-\psi ^h_{\varDelta ,w}(t)|^2_H\}. \end{aligned}$$

(20)

From (19) and (20) we have for a.a. $t \in \delta _i$

$$\begin{aligned} \gamma ^{(1)}_t \le (q^h_{\varDelta }(\tau _i)-w^{(3)}(\tau _i))(\chi (\tau _i,\xi ^h_i,\psi _i^h,u^h(t))-u^*_2(t)) \end{aligned}$$

(21)

$$ +k_2|q^h_{\varDelta }(\tau _i)-w^{(3)}(\tau _i)|\{h+(t-\tau _i)^{1/2}+|p^h_i-\psi ^h_{\varDelta ,w}(t)|_H\}. $$

Estimate the last term in the right-hand side of inequality (18). For a.a. $t\in [\tau _i,\tau _{i+1}{]}$

$$\begin{aligned} |g^h(t)-y_i^h|_H=|\pi ^h(t)+l \nu ^h(t)-y_i^h|_H\le \lambda _{1,i}(t)+\lambda _{2,i}(t), \end{aligned}$$

(22)

where $y_i^h=w^{(1)}(\tau _i)-p_i^h-l(w^{(2)}(\tau _i)-\xi _i^h)$, $\lambda _{1,i}(t)=|w^{(1)}(t)-\psi ^h_{\varDelta ,w}(t)-w^{(1)}(\tau _i)+p_i^h|_H,$ $\lambda _{2,i}(t)=l|w^{(2)}(t)-\varphi ^h_{\varDelta ,w}(t)-w^{(2)}(\tau _i)+\xi _i^h|_H.$ For a.a. $t\in \delta _i= [\tau _i,\tau _{i+1}]$, it is easily seen that $\lambda _{1,i}(t)\le |p_i^h-\psi ^h_{\varDelta ,w}(t)|_H+ \int \limits _{\tau _i}^{t}|\dot{w}^{(1)}(\tau )|_H\,\mathrm {d}\tau$, $\lambda _{2,i}(t)\le lh+ l\int \limits _{\tau _i}^{t}\{|\dot{\varphi }^h_{\varDelta ,w}(\tau )|_H+|\dot{w}^{(2)}(\tau )|_H\}\,\mathrm {d}\tau$. From this equation and (22), for a.a. $t\in \delta _i$, it follows that

$$\begin{aligned} |g^h(t)-y_i^h|_H\le lh+ \int \limits _{\tau _i}^t \{ l|\dot{\varphi }^h_{\varDelta ,w}(\tau )|_H+|\dot{w}^{(1)}(\tau )|_H+|\dot{w}^{(2)}(\tau )|_H\}\,\mathrm {d}\tau + |p_i^h-\psi ^h_{\varDelta ,w}(t)|_H. \end{aligned}$$

(23)

By virtue of (11), taking into account results of [8, 9], from estimate (23) we derive

$$\begin{aligned} \sum _{i=0}^{m-1}\int \limits _{\tau _i}^{\tau _{i+1}} M(t;\tau _i)\,\mathrm {d}t\le k_3(h+\delta )+k_4\int \limits _{t_0}^{\vartheta } |p^h(\tau )-\psi ^h_{\varDelta ,w}(\tau )|_H\,\mathrm {d}\tau \le k_5\nu ^{1/2}(h), \end{aligned}$$

(24)

where $M(t;\tau _i)=|g^h(t)-y_i^h|_H\{ |Bu_i^{h}|_H+|Cv(t)|_H+|u^*_1(t)|_H\}$ for a.a. $t\in \delta _i$, $\nu (h)= (h+\delta (h)+\alpha (h))^{1/2}+ (h+\delta (h))\alpha ^{-1}(h)$. Note that, for a.a. $t \in \delta _i$,

$$\begin{aligned} \gamma _t^{(2)} \le (y_i^h,u^*_1(t)-Bu_i^h+Cv(t))_H +M(t;\tau _i). \end{aligned}$$

(25)

Let the symbol $(\cdot ,\cdot )_{H\times \mathbb {R}}$ denote the scalar product in the space $H\times \mathbb {R}$. Let us define elements $v_{i}^e$ from the conditions

$$\begin{aligned} \inf _{u\in P} (s_i, \varPhi (\tau _i,p^h_i,\xi _i^h,u,v_{i}^e))_{H\times \mathbb {R}}\ge \sup _{v\in E} \inf _{u\in P} (s_i, \varPhi (\tau _i,\xi _i^h,u,v))_{H\times \mathbb {R}}-h, \end{aligned}$$

(26)

where $\varPhi (\tau _i,p^h_i,\xi ^h_i,u^h_i,v(t))= \{Bu^h_i-Cv(t),\chi (\tau _i,\xi ^h_i,p^h_i,u^h_i)\}$, $s_i=\{-y^h_i,\psi ^h_i-w^{(3)}(\tau _i)\}$. It is obvious (see Condition 1) that $u_*(t)\in H(t,w^{(1)}(t),w^{(2)}(t))\subset \bigcup \limits _{u\in P} \varPhi (t,w^{(1)}(t),w^{(2)}(t),u,v^e_i)$ for a. a. $t\in [\tau _i,\tau _{i+1})$. Then, for a.a. $t\in \delta _i$, there exists a control $u^{(1)}(t)\in P$ such that

$$\begin{aligned} \varPhi (t,w^{(1)}(t),w^{(2)}(t),u^{(1)}(t),v_i^e)=u^*(t)\quad \text{ for } \text{ a. } \text{ a. }\quad t\in [\tau _i,\tau _{i+1}]. \end{aligned}$$

(27)

Using the rule of definition of the strategy $\mathcal{U}$, we deduce that

$$\begin{aligned} (s_i,\varPhi (\tau _i,p^h_i,\xi _i^h,u_i^h,v(t)))_{H\times \mathbb {R}} \le \inf _{u\in P}\sup _{v\in E}(s_i,\varPhi (\tau _i,p^h_i,\xi _i^h,u,v))_{H\times \mathbb {R}}+h. \end{aligned}$$

(28)

Here (see (12)), $u_i^h \in \mathcal{U}(\tau _i,\xi _i^h,p_i^h,\psi ^h_i,w(\tau _i))$; $v_{\tau _i,\tau _{i+1}}(\cdot )$ is an unknown realization of the control of the opponent. In turn, from (26) we have

$$\begin{aligned} \sup _{v\in E}\inf _{u\in P}(s_i,\varPhi (\tau _i,p^h_i,\xi _i^h,u,v))_{H\times \mathbb {R}}\le \inf _{u\in P}(s_i,\varPhi (\tau _i,p^h_i,\xi _i^h,u,v_i^e))_{H\times \mathbb {R}}+h. \end{aligned}$$

(29)

Moreover, it is evident that the equality

$$\begin{aligned} \inf _{u\in P}\sup _{v\in E}(s_i,\varPhi (\tau _i,p^h_i,\xi _i^h,u,v))_{H\times \mathbb {R}}= \sup _{v\in E}\inf _{u\in P}(s_i,\varPhi (\tau _i,p^h_i,\xi _i^h,u,v))_{H\times \mathbb {R}} \end{aligned}$$

(30)

is valid. From (28)–(30) we have

$$\begin{aligned} (s_i,\varPhi (\tau _i,p^h_i,\xi _i^h,u_i^h,v(t)))_{H\times \mathbb {R}} \le \inf _{u\in P}(s_i,\varPhi (\tau _i,p^h_i,\xi _i^h,u,v_i^e))_{H\times \mathbb {R}}+2h \end{aligned}$$

(31)

$$ \le (s_i,\varPhi (t,p^h_i,\xi _i^h,u^{(1)}(t),v_i^e))_{H\times \mathbb {R}}+2h+L |\psi ^h_i-w^{(3)}(\tau _i)| (t-\tau _i). $$

Here, L is the Lipschitz constant of the function $\chi (\cdot )$ in the corresponding domain. In this case, for a.a. $t\in \delta _i,$ it follows from (27), (31) that

$$\begin{aligned} (s_i,\varPhi (\tau _i,p^h_i,\xi _i^h,u_i^h,v(t))-u^*(t))_{H\times \mathbb {R}} \end{aligned}$$

(32)

$$ \le 2h+L|\psi ^h_i-w^{(3)}(\tau _i)| \{ t-\tau _i+|\xi _i^h-w^{(1)}(t)|_H + |p^h_i-w^{(2)}(t)|_H\}. $$

By virtue of (32) and (21), for the interval $[\tau _i,\tau _{i+1}],$ we derive

$$\begin{aligned} \gamma _t^{(1)}+\gamma _t^{(2)} \end{aligned}$$

(33)

$$\le \pi ^*_i(t)+ k_6\{h^2+t-\tau _i+\varLambda ^{0}(x^h_{\varDelta ,w}(t),q^h_{\varDelta }(t),w(t))+|p_i^h-\psi _{\varDelta ,w}^h(t)|_H^2\}, $$

where

$$ \pi ^*_i(t)=(s_i,\varPhi (\tau _i,p^h_i,\xi ^h_i,u^h_i,v(t))-u^*(t))_{H\times \mathbb {R}}. $$

We deduce from inequalities (18) and (33) that

$$\begin{aligned} \frac{\mathrm {d}\varLambda (t,x^h_{\varDelta ,w}(\cdot ),q^h_{\varDelta }(\cdot ),w(\cdot ))}{\mathrm {d}t}\le M(t;\tau _i) \end{aligned}$$

(34)

$$ + k_6\varLambda (t,x^h_{\varDelta ,w}(\cdot ),q^h_{\varDelta }(\cdot ),w(\cdot )) +k_7\{h^2+t-\tau _i+|p_i^h-\psi _{\varDelta ,w}^h(t)|_H^2\} \quad \text{ for } \text{ a.a. }\ t\in \delta _i. $$

Using (34) and (24), by virtue of the Gronwall lemma, we obtain for $t\in T$

$$ \varLambda (t,x_{\varDelta ,w}^h(\cdot ),q^h_{\varDelta ,w}(\cdot ),w(\cdot ))\le k_8\sum _{i=0}^{m-1} \Big \{\int \limits _{\tau _i}^{\tau _{i+1}} M(t;\tau _i)\,\mathrm {d}t + \delta (h^2+\delta )\Big \} \le k_9\nu ^{1/2}(h). $$

The statement of the theorem follows from the last inequality.