Lipschitz continuity of the value function in mixed-integer optimal control problems

• Published in: Mathematics of control, signals, and systems Vol. 29; no. 1; p. 1
• Main Authors: Gugat, Martin, Hante, Falk M.
• Format: Journal Article
• Language: English
• Published: London Springer London 01.03.2017; Springer Nature B.V
• Subjects: Algorithms; Banach spaces; Communications Engineering; Control; Control theory; Costs; Integer programming; Linear programming; Mathematics; Mathematics and Statistics; Mechatronics; Networks; Optimization; Original Article; Partial differential equations; Robotics; Systems Theory; Mixed-integer optimal control problems; Parametric optimization; Sensitivity; Optimal value function; Parametric optimal control; Parametric switching control; Lipschitz continuity
• DOI: 10.1007/s00498-016-0183-4
• ISSN: 0932-4194

We study the optimal value function for control problems on Banach spaces that involve both continuous and discrete control decisions. For problems involving semilinear dynamics subject to mixed control inequality constraints, one can show that the optimal value depends locally Lipschitz continuously on perturbations of the initial data and the costs under rather natural assumptions. We prove a similar result for perturbations of the initial data, the constraints and the costs for problems involving linear dynamics, convex costs and convex constraints under a Slater-type constraint qualification. We show by an example that these results are in a sense sharp.