Journal Article

Dubovitskii–Milyutin formalism applied to optimal control problems with constraints given by the heat equation with final data

Inmaculada Gayte, Francisco Guillén-González and Marko Rojas-Medar

in IMA Journal of Mathematical Control and Information

Published on behalf of Institute of Mathematics and its Applications

Volume 27, issue 1, pages 57-76
Published in print March 2010 | ISSN: 0265-0754
Published online February 2010 | e-ISSN: 1471-6887 | DOI: http://dx.doi.org/10.1093/imamci/dnq001
Dubovitskii–Milyutin formalism applied to optimal control problems with constraints given by the heat equation with final data

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An optimal control problem with a convex cost functional subject to a (linear) non-well-posed problem (Dirichlet heat equation with a given final data) is considered. The control is distributed and a convex constraint on the control is imposed. For a globally distributed control and a convex constraint on the control with non-empty interior, we deduce first-order necessary (and sufficient) optimality conditions using the so-called Dubovitskii–Milyutin formalism, obtaining, in particular, the existence of the corresponding adjoint problem (which is again a non-well-posed problem). In other cases (either empty interior convex constraint on the control or partially distributed control), we arrive at the optimality conditions but admitting the existence of the adjoint problem. Finally, numerical results are also presented approximating the optimality conditions for 1D domains by finite differences in time and space.

Keywords: optimal control; first-order optimality system; distributed control; dual cones; non-well-posed problem

Journal Article.  0 words. 

Subjects: Mathematics

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