On Regularization of Classical Optimality Conditions in Convex Optimization Problems for Volterra-Type Systems with Operator Constraints

Abstract

We consider the regularization of classical optimality conditions—the Lagrange principle and the Pontryagin maximum principle—in a convex optimal control problem with an operator equality constraint and functional inequality constraints. The controlled system is specified by a linear functional–operator equation of the second kind of general form in the space \(L^m_2 \), and the main operator on the right-hand side of the equation is assumed to be quasinilpotent. The objective functional of the problem is only convex (perhaps not strongly convex). Obtaining regularized classical optimality conditions is based on the dual regularization method. In this case, two regularization parameters are used, one of which is “responsible” for the regularization of the dual problem, and the other is contained in the strongly convex regularizing Tikhonov addition to the objective functional of the original problem, thereby ensuring the well-posedness of the problem of minimizing the Lagrange function. The main purpose of the regularized Lagrange principle and Pontryagin maximum principle is the stable generation of minimizing approximate solutions in the sense of J. Warga. The regularized classical optimality conditions

1. 1.

   Are formulated as existence theorems for minimizing approximate solutions in the original problem with a simultaneous constructive representation of these solutions.

2. 2.

   Are expressed in terms of regular classical Lagrange and Hamilton–Pontryagin functions.

3. 3.

   “Overcome” the properties of the ill-posedness of the classical optimality conditions and provide regularizing algorithms for solving optimization problems.

Based on the perturbation method, an important property of the regularized classical optimality conditions obtained in the work is discussed in sufficient detail; namely, “in the limit” they lead to their classical counterparts. As an application of the general results obtained in the paper, a specific example of an optimal control problem associated with an integro-differential equation of the transport equation type is considered, a special case of which is a certain inverse final observation problem.