Representations of Relaxations of Linear-Quadratic Optimal Control Problems for Elliptic Systems

摘要

The paper considers optimal control problems of the type {I(h, u) = j [(B(x)Vq.b, Vu) + (g(x, h(x)), Vu,) + F(s, h(x))]dx, where the set M of admissible controls h consists of all measurable vector-functions h(.) with values from a given compact set M subset of R-r. The functions f and g are affine with respect to h, but the matrix B can be negatively definite. It is shown that the relaxation of such problems can be represented as a joint passage from AI to its convex hull and from F to a new function which is lower semicontinuous with respect to h.