On the control of strongly nonlinear hyperbolic systems (Q1089904)

The authors consider the following optimal control problem: \[ \text{Minimize} J(v,z)=(1/2)\| f(z)-z_ d\|^ 2_ 2+(N/2)\| v\|^ 2, \] \[ \text{subject to }z_{tt}-\Delta z-f(z)=v,\quad z(0,x)=y_ 0(x),\quad z_ t(0,x)=y_ 1(x),\quad v\in U_{ad}. \] They prove existence of an optimal solution and necessary optimality conditions if f and f' are bounded, or if \(U_{ad}=L^ 2\) or \(n=1\) (one space dimension) and f and f' satisfy some growth condition including polynomials and exponentials. The proof uses a sequence of approximating problems of the form: \[ \text{Minimize} J_{\epsilon}=J+((1/2)\epsilon)\| z_{tt}-\Delta t-f(z)-v\|^ 2+(1/2)\| z-y\|^ 2+(1/2)\| v-u\|^ 2. \]