Dualization and discretization of linear-quadratic control problems with bang–bang solutions

We consider linear-quadratic (LQ) control problems, where the control variable appears linearly and is box-constrained. It is well-known that these problems exhibit bang–bang and singular solutions. We assume that the solution is of bang–bang type, which is computationally challenging to obtain. We...

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Autores principales:	Walter Alt, C. Yalçın Kaya, Christopher Schneider
Formato:	article
Lenguaje:	EN
Publicado:	Elsevier 2016
Materias:	49N10 49N15 49M25 49J30 49J15 Applied mathematics. Quantitative methods T57-57.97 Electronic computers. Computer science QA75.5-76.95
Acceso en línea:	https://doaj.org/article/9c065113ddd54f059d9bfe129bbf6686
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spelling	oai:doaj.org-article:9c065113ddd54f059d9bfe129bbf66862021-12-02T05:00:48ZDualization and discretization of linear-quadratic control problems with bang–bang solutions2192-440610.1007/s13675-015-0049-4https://doaj.org/article/9c065113ddd54f059d9bfe129bbf66862016-02-01T00:00:00Zhttp://www.sciencedirect.com/science/article/pii/S219244062100054Xhttps://doaj.org/toc/2192-4406We consider linear-quadratic (LQ) control problems, where the control variable appears linearly and is box-constrained. It is well-known that these problems exhibit bang–bang and singular solutions. We assume that the solution is of bang–bang type, which is computationally challenging to obtain. We employ a quadratic regularization of the LQ control problem by embedding the L2-norm of the control variable into the cost functional. First, we find a dual problem guided by the methodology of Fenchel duality. Then we prove strong duality and the saddle point property, which together ensure that the primal solution can be recovered from the dual solution. We propose a discretization scheme for the dual problem, under which a diagram depicting the relations between the primal and dual problems and their discretization commutes. The commuting diagram ensures that, given convergence results for the discrete primal variables, discrete dual variables also converge to a solution of the dual problem with a similar error bound. We demonstrate via a simple but illustrative example that significant computational savings can be achieved by solving the dual, rather than the primal, problem.Walter AltC. Yalçın KayaChristopher SchneiderElsevierarticle49N1049N1549M2549J3049J15Applied mathematics. Quantitative methodsT57-57.97Electronic computers. Computer scienceQA75.5-76.95ENEURO Journal on Computational Optimization, Vol 4, Iss 1, Pp 47-77 (2016)
institution	DOAJ
collection	DOAJ
language	EN
topic	49N10 49N15 49M25 49J30 49J15 Applied mathematics. Quantitative methods T57-57.97 Electronic computers. Computer science QA75.5-76.95
spellingShingle	49N10 49N15 49M25 49J30 49J15 Applied mathematics. Quantitative methods T57-57.97 Electronic computers. Computer science QA75.5-76.95 Walter Alt C. Yalçın Kaya Christopher Schneider Dualization and discretization of linear-quadratic control problems with bang–bang solutions
description	We consider linear-quadratic (LQ) control problems, where the control variable appears linearly and is box-constrained. It is well-known that these problems exhibit bang–bang and singular solutions. We assume that the solution is of bang–bang type, which is computationally challenging to obtain. We employ a quadratic regularization of the LQ control problem by embedding the L2-norm of the control variable into the cost functional. First, we find a dual problem guided by the methodology of Fenchel duality. Then we prove strong duality and the saddle point property, which together ensure that the primal solution can be recovered from the dual solution. We propose a discretization scheme for the dual problem, under which a diagram depicting the relations between the primal and dual problems and their discretization commutes. The commuting diagram ensures that, given convergence results for the discrete primal variables, discrete dual variables also converge to a solution of the dual problem with a similar error bound. We demonstrate via a simple but illustrative example that significant computational savings can be achieved by solving the dual, rather than the primal, problem.
format	article
author	Walter Alt C. Yalçın Kaya Christopher Schneider
author_facet	Walter Alt C. Yalçın Kaya Christopher Schneider
author_sort	Walter Alt
title	Dualization and discretization of linear-quadratic control problems with bang–bang solutions
title_short	Dualization and discretization of linear-quadratic control problems with bang–bang solutions
title_full	Dualization and discretization of linear-quadratic control problems with bang–bang solutions
title_fullStr	Dualization and discretization of linear-quadratic control problems with bang–bang solutions
title_full_unstemmed	Dualization and discretization of linear-quadratic control problems with bang–bang solutions
title_sort	dualization and discretization of linear-quadratic control problems with bang–bang solutions
publisher	Elsevier
publishDate	2016
url	https://doaj.org/article/9c065113ddd54f059d9bfe129bbf6686
work_keys_str_mv	AT walteralt dualizationanddiscretizationoflinearquadraticcontrolproblemswithbangbangsolutions AT cyalcınkaya dualizationanddiscretizationoflinearquadraticcontrolproblemswithbangbangsolutions AT christopherschneider dualizationanddiscretizationoflinearquadraticcontrolproblemswithbangbangsolutions
_version_	1718400843306237952

Dualization and discretization of linear-quadratic control problems with bang–bang solutions

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