Game problem for a controlled system (Q2742936)

The problem of control, described by the following boundary value problem NEWLINE\[NEWLINEz_{tt}=P(x)z_{xx}+Q(x)z_x+R(x)z,\quad t\in(0,T), x\in(a,b),NEWLINE\]NEWLINE where \(P(x)>0\), \(R(x)\leq 0\), \(z(t,a) =z(t,b)=0\) and \(z(0,x)=u(x)\), \(z_t(0,x)=v(x)\) is considered. The player's I objective, he controls the function \(u(x)\), is to change the process from initial position at \(t=0\) to the given state \(z(T,x)=\varphi(x)\), \(z_t(T,x)=\psi(x)\) for the given time \(T\). The player's II objective, he controls the function \(v(x)\), is to hamper player I. The first player is player II. It is proved that player II wins the game.