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

We consider the regularization of classical optimality conditions—the Lagrangeprinciple and the Pontryagin maximum principle—in a convex optimal control problemwith an operator equality constraint and functional inequality constraints. The controlled systemis specified by a linear functional–operator equation of the second kind of general form in thespace \(L^m_2\) , and the main operator on the right-hand side ofthe equation is assumed to be quasinilpotent. The objective functional of the problem is onlyconvex (perhaps not strongly convex). Obtaining regularized classical optimality conditions isbased on the dual regularization method. In this case, two regularization parameters are used, oneof which is “responsible” for the regularization of the dual problem, and the other is contained inthe strongly convex regularizing Tikhonov addition to the objective functional of the originalproblem, 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 thestable generation of minimizing approximate solutions in the sense of J. Warga. The regularizedclassical optimality conditions 1.

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

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