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On regularization of the classical optimality conditions in the convex optimization problems for Volterra-type systems with operator constraints
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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 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 L2m, the main operator on the right side of the equation is assumed to be quasinilpotent.The objective functional of the problem is only convex (perhaps not strongly convex). Obtaining regularized COCs 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, the other is contained in the strongly convex regularizing Tikhonov addition to the target 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 in the sense of J. Warga. Regularized COCs: 1) are formulated as existence theorems for minimizing approximate solutions in the original problem with a simultaneous constructive representation of these solutions; 2) expressed in terms of regular classical functions of Lagrange and Hamilton–Pontryagin; 3) “overcome” the properties of the ill-posedness of the COCs and provide regularizing algorithms for solving optimization problems. Based on the perturbation method, an important property of the regularized COCs obtained in the work is discussed in sufficient detail, namely that “in the limit” they lead to their classical analogues. 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 final observation problem.
Title: On regularization of the classical optimality conditions in the convex optimization problems for Volterra-type systems with operator constraints
Description:
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 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 L2m, the main operator on the right side of the equation is assumed to be quasinilpotent.
The objective functional of the problem is only convex (perhaps not strongly convex).
Obtaining regularized COCs 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, the other is contained in the strongly convex regularizing Tikhonov addition to the target 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 in the sense of J.
Warga.
Regularized COCs: 1) are formulated as existence theorems for minimizing approximate solutions in the original problem with a simultaneous constructive representation of these solutions; 2) expressed in terms of regular classical functions of Lagrange and Hamilton–Pontryagin; 3) “overcome” the properties of the ill-posedness of the COCs and provide regularizing algorithms for solving optimization problems.
Based on the perturbation method, an important property of the regularized COCs obtained in the work is discussed in sufficient detail, namely that “in the limit” they lead to their classical analogues.
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 final observation problem.
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