Constrained control problems with convex cost in Hilbert space

Abstract

In recent papers [3, 41 necessary and sufficient conditions were derived for convex control problems involving linear differential equations in Hilbert spaces. In this paper we show that, under relatively weak assumptions (see Section l), one can derive the above mentioned optimality conditions, for a wide class of control problems for linear evolution equations in Hilbert space. The plan of the paper is as follows. Section 1 is concerned principally with the problem formulation. The necessary and sufficient conditions for optimality are in the “subdifferential” form and do not require the differentiability of cost functional. Thus our result is very much in the spirit of Rockafellar works [15, 16-j. These conditions are spelled out in Theorem 1, whose proof is set forth in Section 3. In Section 4 we formulate a control problem for linear hereditary differential systems with convex criterion. The necessary and sufficient conditions are then specialised for this particular problem (see Theorem 2). These results may be compared with those of R. Datko [7,8], H. T. Banks and M. Q. Jacobs [l], H. T. Banks and G. A. Kent [2], M. C. Delfour and S. K. Mitter [9], A. Halanay [l l] (further references may be found in these papers). However, our results on necessary optimality conditions in problems involving linear evolution equations in Hilbert space setting (see Lions’s book [ 121 for significant results in this field) differ from previous results (even specialised to functional differential equations) which involve stronger regularity assumptions than ours. For the most part, these papers are concerned with quadratic cost criterion and certainly the methods used here are very different. Our approach is much similar to that used by the author in [3] and it relies on the methods of convex analysis. For significant results in this field, relevant to this paper we refer to the books of Rockafellar [13] and BrCzis [6].