Suboptimal Solutions of Control Problems for Distributed Parameter Systems

Abstract

We consider a parabolic PDE in general form together with initial and first boundary conditions where the controls appear nonlinear in the lower order coefficients and the right hand side of the differential equation. It is very difficult to prove existence theorems for optimal solutions without convexity assumptions, but there are necessary optimality conditions in terms of a suitable defined Hamiltonian function. We prove the existence of minimizing sequences of admissible controls each of which satisfies a necessary condition with a slightly disturbed Hamiltonian. In the proof, we use Ekelands variational principle and the techniques for proving necessary optimality conditions.