The features of gradient methods for distributed optimal-control problems

A wide class of nonlinear distributed control systems is distinguished for which the Frechet differentiability in spaces of the type $L_\infty$ of functionals of fairly general form is proved for arbitrary orders of increase of the “right-hand sides” with respect to the “phase” and controlling variables. Formulas are obtained for the corresponding Frechet derivatives, which can be effectively used when validating methods of descent of the gradient type in optimal-control problems with a limited set of permissible values of the control. Central to this is the theorem of the sufficient conditions of stability (with respect to perturbation of the control) of the existence of global solutions of systems of the distinguished class. Specific examples are given.