On the regularization of the Lagrange principle in a nonlinear optimal control problem for a Goursat–Darboux system with a pointwise state equality-constraint
Abstract:
Based on the conjugation of optimal control methods, nonlinear analysis and the theory of ill-posed problems, the regularization of the Lagrange principle (LP) in a non-differential form, in regular and irregular variants, in a nonlinear (non-convex) optimization problem of a general Goursat–Darboux system with a pointwise state equality-constraint is considered. This constraint is understood as an equality in the Hilbert space of square-summable functions and contains a parameter additively included in it, which makes it possible to use a “nonlinear version” of the perturbation method for studying the problem. The main purpose of both variants of the regularized LP is the stable generation of generalized minimizing sequences (GMS) in the problem under consideration, the existence of a solution to which is not assumed a priori. They can be interpreted as GMS-forming (regularizing) operators, which associate with each set of initial data of the problem a subminimal (minimal) of its regular augmented Lagrangian corresponding to this set, the dual variable in which is generated in accordance with the procedures specified in these variants. The construction of the augmented Lagrangian is completely determined by the form of “nonlinear” subdifferentials of a lower semicontinuous and, generally speaking, nonconvex function of values as a function of the problem parameter. The proximal subgradient and the Frechet subdifferential, well known in nonlinear analysis, are used as the latter. In the special case, when the problem is regular, in the sense of the existence of a generalized Kuhn–Tucker vector in it, and its initial data (the integrand of the quality functional and the right-hand side of the controlled system) depend affinely on the control, the limit passage in the relations of the regularized LP leads to classical optimality conditions in the form of the nondifferential Kuhn-Tucker theorem and the Pontryagin maximum principle.
Key words:nonlinear optimal control, vector hyperbolic equation, Goursat–Darboux problem, pointwise state equality-constraint, regularization, perturbation method, value function, proximal subgradient, frechet subdifferential, augmented Lagrangian, generalized minimizing sequence, regularizing algorithm, Lagrange principle in non-differential form, Pontryagin maximum principle.
