Computational methods in Hilbert space for optimal control problems with delays

Abstract

The paper consits of two parts. The first part is devoted to basic relations in the abstract theory of optimization and their relevance for computational methods. The concepts of the abstract theory (developed by Hurwicz, Uzawa, Dubovitski, Milyutin, Neustadt and others) linked together with the notion of a projection on a cone result in an unifying approach to computational methods of optimization. Several basic computational concepts, such as penalty functional techniques, problems of normality of optimal solutions, gradient projection and gradient reduction techniques, can be investigated in terms of a projection on a cone.

The second part of the paper presents an application of the gradient reduction technique in Hilbert space for optimal control problems with delays. Such an approach results in a family of computational methods, parallel to the methods known for finite-dimmensional and other problems: conjugate gradient methods, variable operator (variable metric) methods and generalized Newton's (second variation) method can be formulated and applied for optimal control problems with delays. The generalized Newton's method is, as usually, the most efficient; however, the computational difficulties in inverting the hessian operator are limiting strongly the applications of the method. Of other methods, the variable operator technique seems to be the most promissing.