First-order necessary optimality conditions for infinite horizon optimal control problems with linear dynamics and convex objective. (3rd June 2018)

• Record Type: Journal Article
• Authors: Ziemann, T.; Pickenhain, S.
• Is Part Of: Optimization. Volume 67:Number 6(2018)
• Page Start: 757
• Page End: 782
• Publication Date: 2018-06-03
• Subjects: Vector-valued optimal control -- infinite horizon -- necessary optimality conditions -- Pontryagin-type maximum principle; Mathematical optimization -- Periodicals; 519.7
• DOI: 10.1080/02331934.2018.1434172
• Languages: English
• ISSNs: 0233-1934

Abstract

Abstract: We consider a class of infinite horizon optimal control problems with vector-valued states and controls involving the Lebesgue integral in the objective and a linear dynamics. The key idea is to show that the solutions of the state-equation belong to a weighted Hilbert space, due to a natural growth condition. This allows to introduce the weighted Sobolev spaceW 2 1, n ( ( 0, ∞ ), μ ) as the state space and the weighted Lebesgue spaceL 2 m ( ( 0, ∞ ), μ ) as the control space in the problem setting. We answer the fundamental question of how one can choose the weight functionμ in dependence of the system dynamics. In the setting of Hilbert spaces, we show a Pontryagin-type maximum principle. Therefore, we use techniques from earlier publications of the authors. In contrast to these publications, we deal with vector-valued states and controls, here. These vector-valued problems arise in many applications. The maximum principle also includes a transversality condition which makes the difference to related publications in the field of necessary optimality conditions. This condition brings advantages for the numerical calculations of the solution by indirect methods, like a pseudospectral method.