On regularization of the Lagrange principle in the optimization problems for linear distributed Volterra type systems with operator constraints

V. I. Sumin ; M. I. Sumin

Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 59 (2022), pp. 85-113

Résumé

Regularization of the classical optimality conditions — the Lagrange principle and the Pontryagin maximum principle — in a convex optimal control problem subject to functional equality and inequality constraints is considered. The controlled system is described by a linear functional-operator equation of second kind of the general form in the space $L_2^m$. The main operator on the right-hand side of the equation is assumed to be quasi-nilpotent. The objective functional to be minimized is strongly convex. The derivation of the regularized classical optimality conditions is based on the use of the dual regularization method. The main purpose of the regularized Lagrange principle and regularized Pontryagin maximum principle is to stably generate minimizing approximate solutions in the sense of J. Warga. As an application of the results obtained for the general linear functional-operator equation of second kind, two examples of concrete optimal control problems related to a system of delay equations and to an integro-differential transport equation are discussed.

Détail

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