{"title":"On Regularization of Classical Optimality Conditions in Convex Optimization Problems for Volterra-Type Systems with Operator Constraints","authors":"V. I. Sumin, M. I. Sumin","doi":"10.1134/s0012266124020071","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We consider the regularization of classical optimality conditions—the Lagrange\nprinciple and the Pontryagin maximum principle—in a convex optimal control problem\nwith an operator equality constraint and functional inequality constraints. The controlled system\nis specified by a linear functional–operator equation of the second kind of general form in the\nspace <span>\\(L^m_2 \\)</span>, and the main operator on the right-hand side of\nthe equation is assumed to be quasinilpotent. The objective functional of the problem is only\nconvex (perhaps not strongly convex). Obtaining regularized classical optimality conditions is\nbased on the dual regularization method. In this case, two regularization parameters are used, one\nof which is “responsible” for the regularization of the dual problem, and the other is contained in\nthe strongly convex regularizing Tikhonov addition to the objective functional of the original\nproblem, thereby ensuring the well-posedness of the problem of minimizing the Lagrange function.\nThe main purpose of the regularized Lagrange principle and Pontryagin maximum principle is the\nstable generation of minimizing approximate solutions in the sense of J. Warga. The regularized\nclassical optimality conditions\n</p><ol>\n<li>\n<span>1.</span>\n<p>Are formulated as existence theorems for minimizing approximate solutions in the original\nproblem with a simultaneous constructive representation of these solutions. </p>\n</li>\n<li>\n<span>2.</span>\n<p>Are expressed in terms of regular classical Lagrange and Hamilton–Pontryagin functions.\n</p>\n</li>\n<li>\n<span>3.</span>\n<p>“Overcome” the properties of the ill-posedness of the classical optimality conditions and\nprovide regularizing algorithms for solving optimization problems. </p>\n</li>\n</ol><p>Based on the perturbation method, an important property of the regularized\nclassical optimality conditions obtained in the work is discussed in sufficient detail; namely, “in the\nlimit” they lead to their classical counterparts. As an application of the general results obtained in\nthe paper, a specific example of an optimal control problem associated with an integro-differential\nequation of the transport equation type is considered, a special case of which is a certain inverse\nfinal observation problem.\n</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1134/s0012266124020071","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
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.
Are formulated as existence theorems for minimizing approximate solutions in the original
problem with a simultaneous constructive representation of these solutions.
2.
Are expressed in terms of regular classical Lagrange and Hamilton–Pontryagin functions.
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.
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号
