DISCRETE APPROXIMATION OF NONCONVEX HYPERBOLIC OPTIMAL-CONTROL PROBLEMS WITH STATE CONSTRAINTS

We consider an optimal control problem for systems defined by nonlinea r hyperbolic partial differential equations with state constraints. Si nce no convexity assumptions are made on the data, we also consider th e control problem in relaxed form. We discretize both the classical an d the relaxed problems by using a finite element method in space and a finite difference scheme in time, the controls being approximated by piecewise constant ones. We develop the existence theory and the neces sary conditions for optimality, for the continuous and the discrete pr oblems. Finally, we study the behaviour in the limit of discrete optim ality, admissibility and extremality properties.