Regularization of classical optimality conditions

V. I. Sumin ; M. I. Sumin

Vestnik rossijskih universitetov. Matematika, Tome 28 (2023) no. 143, pp. 298-325

Résumé

We consider the regularization of classical optimality conditions (COCs) — the Lagrange principle (LP) and the Pontryagin maximum principle (PMP) — in a convex optimal control problem with functional constraints such as equalities and inequalities. The controlled system is given by a linear functional-operator equation of the second kind of general form in the space $L^m_2,$ the main operator on the right side of the equation is assumed to be quasi-nilpotent. The problem functional to be minimized is convex (probably not strongly). The regularization of the COCs in the non-iterative and iterative forms is based on the use of the methods of dual regularization and iterative dual regularization, respectively. Obtaining non-iterative regularized COCs uses two regularization parameters, one of which is “responsible” for the regularization of the dual problem, the other is contained in a strongly convex regularizing Tikhonov addition to the objective functional of the original problem, thereby ensuring the correctness of the problem of minimizing the Lagrange function. The main purpose of regularized LP and PMP is the stable generation of minimizing approximate solutions (MASs) in the sense of J. Warga. Regularized COCs: 1) are formulated as existence theorems for MASs in the original problem with simultaneous constructive representation of specific MASs; 2) are sequential generalizations of classical analogues — their limiting variants and preserve the general structure of the latter; 3) “overcome” the ill-posedness properties of the COCs and give regularizing algorithms for solving optimization problems. Illustrating examples are considered: the problem of optimal control for the equation with delay, the problem of optimal control for the integrodifferential equation of the type of transport equation.

Détail

Keywords: convex optimal control, functional constraints, Volterra-type functional-operator equation, ill-posedness, regularization, iterative regularization, duality, minimizing approximate solution, regularizing operator, Lagrange principle, Pontryagin maximum principle.

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