Optimal Control of Semilinear Parabolic Equations with Non-smooth Pointwise-Integral Control Constraints in Time-Space

• Published in: Applied mathematics & optimization Vol. 85; no. 1
• Main Authors: Casas, Eduardo, Kunisch, Karl
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.02.2022; Springer Nature B.V
• Subjects: Applied mathematics; Calculus of Variations and Optimal Control; Optimization; Control; Control stability; Mathematical analysis; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Numerical and Computational Physics; Optimal control; Optimization; Simulation; Smoothness; Sparsity; Systems Theory; Theoretical; 49K20; 49J20; 49J52; Optimal control; Non-smooth control constraints; 35K58; Semilinear parabolic equations; First and second order optimality conditions
• ISSN: 0095-4616; 1432-0606
• DOI: 10.1007/s00245-022-09850-7

This work concentrates on a class of optimal control problems for semilinear parabolic equations subject to control constraint of the form ‖ u ( t ) ‖ L 1 ( Ω ) ≤ γ for t ∈ ( 0 , T ) . This limits the total control that can be applied to the system at any instant of time. The L 1 -norm of the constraint leads to sparsity of the control in space, for the time instants when the constraint is active. Due to the non-smoothness of the constraint, the analysis of the control problem requires new techniques. Existence of a solution, first and second order optimality conditions, and regularity of the optimal control are proved. Further, stability of the optimal controls with respect to γ is investigated on the basis of different second order conditions.