On piecewise constant approximation in distributed optimization problems

The paper is devoted to optimal control problems for distributed parameter systems representable by functional operator equations of Hammerstein type in a Banach space compactly embedded in a Lebesgue space. The problem of minimizing an integral functional on a set of “state-control” pairs satisfying a control equation of the mentioned type is considered. We prove that this problem is equivalent to an optimization problem obtained from the original one by passing to a description of the control system in terms of V.I. Sumin's functional operator equation in a Lebesgue space. The equivalent optimization problem is called S-dual. For an S-dual optimization problem, we investigate a piecewise constant approximation for the “state-control” pair. For this approximation method, we state the following results: (1) convergence of piecewise constant approximations with respect to the functional and the equation for the S-dual optimization problem; (2) existence of a global solution of an approximating finite-dimensional mathematical programming problem; (3) convergence with respect to the functional of solutions of an approximating optimization problem to a solution of the original problem. As an auxiliary result of independent interest, we prove a theorem on the total (over the whole set of admissible controls) preservation of solvability for a control equation of Hammerstein type. The Dirichlet problem for a semilinear elliptic equation of diffusion-reaction type is considered as an example of reducing a distributed parameter control system to such an equation.