Optimization of systems governed by hyperbolic partial differential equations with equality and inequality constraints (Q1072099)

The author deals with a control process described by the system of nonlinear hyperbolic equations \[ \begin{aligned} z^ i_{ts} &= f_ i(t,s,z,z_ t,z_ s,u), \quad (t,s)\in G, \quad i=1,\ldots,n, \\ z(t,s) &= (z^1,\ldots,z^ n), \quad u(t,s) = (u^1,\ldots,u^ n), \\ G &= \left\{(t,s)\in R^ 2 : t_ 0\leq t\leq t_1,\;s_ 0\leq s\leq s_ 1\right \} \end{aligned} \tag{1} \] with Darboux-Goursat boundary conditions \[ z^ i(t,s_ 0) = a_ i(t), \quad z^ i(t_ 0,s) = b_ i(s) \tag{2} \] and with control constraints \[ u(t,s)\in U(t,s). \tag{3} \] The control problem is formulated as finding a pair (u,z) satisfying (1)- -(3) and minimizing the functional \(J_ 0\) subjected to \(J_ k(u)\leq 0\), \(J_{\ell}(u)=0\), \(k=1,...,\alpha\); \(\ell =\alpha +1,...,\alpha +\beta\), where \[ J_ r(u) = \int^{t_ 1}_{t_ 0} \phi^{(r)} (t,z_ u(t,s_ 1))dt + \int^{s_ 1}_{s_ 0} \psi^{(r)} (s,z_ u(t_ 1,s))ds. \] An optimality condition for the above problem is derived.