Optimal control of infinite dimensional bilinear systems: application to the heat and wave equations

• Published in: Mathematical programming Vol. 168; no. 1-2; pp. 717 - 757
• Main Authors: Aronna, M. Soledad, Bonnans, J. Frédéric, Kröner, Axel
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01-03-2018; Springer Nature B.V; Springer Verlag
• Subjects: Calculus of Variations and Optimal Control; Optimization; Combinatorics; Full Length Paper; Mathematical analysis; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Mathematics of Computing; Nonlinear programming; Numerical Analysis; Operators (mathematics); Optimal control; Optimization; Optimization and Control; Theoretical; Wave equations; Wave equation; 49K20; Partial differential equations; Heat equation; Optimal control; 35L05; Second-order optimality conditions; 35K05; Semigroup theory; 90C48; Bilinear control systems; Goh transform; second-order optimality conditions; heat equation; bilinear control systems; wave equation; partial differential equations; semigroup theory
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-016-1093-4

In this paper we consider second order optimality conditions for a bilinear optimal control problem governed by a strongly continuous semigroup operator, the control entering linearly in the cost function. We derive first and second order optimality conditions, taking advantage of the Goh transform. We then apply the results to the heat and wave equations.