On the optimal control and relaxation of nonlinear infinite dimensional systems

Summary In this paper we study optimal control problems for infinite dimensional systems governed by a semilinear evolution equation. First under appropriate convexity and growth conditions, we establish the existence of optimal pairs. Then we drop the convexity hypothesis and we pass to a larger system known as the « relaxed system ». We show that this system has a solution and the value of the relaxed optimization problem is equal to the value of the original one. Next we restrict our attention to linear systems and establish two « bang-bang » type theorems. Finally we present some examples from systems governed by partial differential equations.Research supported by N.S.F. Grant-8602313.Work done while on leave at the « University of Thessaloniki, School of Technology, Mathematics Division, Thessaloniki 54006, Greece ».