Regularization of classical optimality conditions \\in optimization problems of linear distributed Volterra-type systems with pointwise state constraints

V. I. Sumin ; M. I. Sumin

Vestnik rossijskih universitetov. Matematika, Tome 29 (2024) no. 148, pp. 455-484

Résumé

The regularization of classical optimality conditions (COCs) — the Lagrange principle (LP) and the Pontryagin maximum principle (PMP) — in a convex optimal control problem with a strongly convex objective functional and with pointwise state constraints of the equality and inequality type is considered. The control system is defined by a linear functional-operator equation of the second kind of general form in the space $L^s_2,$ the main operator of the right-hand side of the equation is assumed to be quasi-nilpotent. Obtaining regularized COCs is based on the dual regularization method. The main purpose of regularized LP and PMP is stable generation of minimizing approximate solutions (MASs) in the sense of J. Warga. Regularized COCs: 1) are formulated as existence theorems of MASs in the original problem with simultaneous constructive representation of these solutions; 2) are expressed in terms of regular classical Lagrange and Hamilton–Pontryagin functions; 3) “overcome” the ill-posedness properties of the COCs and provide regularizing algorithms for solving optimization problems. The article continues a series of works by the authors on the regularization of the COCs for a number of canonical problems of optimal control of linear distributed systems of the Volterra type. As an application of the “abstract results” obtained in the work, the final part considers the regularization of the COCs in a specific optimization problem with pointwise state constraints of the equality and inequality type for a control system with delay.

Détail

Keywords: convex optimal control, pointwise state constraints, distributed system, functional-operator equation of the Volterra type, optimality conditions, regularization, duality