Abstract:
The regularization of the Lagrange principle (LP) and the Kuhn–Tucker theorem (KTT) in a non-differential form is considered in a nonlinear (non-convex) optimal control problem for a system of ordinary differential equations with a pointwise state equality constraint. The existence of a solution to the problem is not assumed a priori. The equality constraint contains an additive parameter, which makes it possible to use the “nonlinear version” of the perturbation method to study the problem. The main purpose of the regularized LP and KTT is to stably generate generalized minimizing sequences (GMS) in the problem under consideration. They can be interpreted as GMS-forming (regularizing) operators that associate with each set of initial data of the problem a subminimal (minimal) of its regular augmented Lagrangian (AL) functional corresponding to this set, the dual variable in which is generated in accordance with the Tikhonov stabilization procedure of the dual problem. The construction of the AL is completely determined by the form of the “nonlinear” subdifferentials (proximal subgradient, Frechet subdifferential) of the lower semicontinuous function of values as a function of the problem parameter. Regularized LP and KTT “overcome” the properties of ill-posedness of classical analogs, thus giving a theoretical basis for creating stable methods for solving nonlinear optimal control problems. In the particular case when the problem is regular in the sense of the existence of a generalized Kuhn-Tucker vector in it, and its initial data depend affinely on the control, the limit transition in the relations of the regularized KTT leads to optimality conditions in the form of the corresponding non-differential KTT and Pontryagin’s maximum principle.
Keywords:nonlinear optimal control, pointwise state equality constraint, generalized minimizing sequence, perturbation method, subdifferentials of nonlinear analysis, regularization, duality, Lagrange principle
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