Steady-state and periodic exponential turnpike property for optimal control problems in Hilbert spaces

Emmanuel Trélat (1, 2, 3) , Can Zhang (4, 2, 3) , Enrique Zuazua (5, 2, 3)

1 CaGE - Control And GEometry
2 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
3 SU - Sorbonne Université
4 School of Mathematics and Statistics [Wuhan University]
5 Departamento de Matemáticas [Madrid]

In this work, we study the steady-state (or periodic) exponential turnpike property of optimal control problems in Hilbert spaces. The turnpike property, which is essentially due to the hyperbolic feature of the Hamiltonian system resulting from the Pontryagin maximum principle, reflects the fact that, in large time, the optimal state, control and adjoint vector remain most of the time close to an optimal steady-state. A similar statement holds true as well when replacing an optimal steady-state by an optimal periodic trajectory. To establish the result, we design an appropriate dichotomy transformation, based on solutions of the algebraic Riccati and Lyapunov equations. We illustrate our results with examples including linear heat and wave equations with periodic tracking terms.

Mots clés: Exponential turnpike property; Periodic tracking; Periodic optimal controls; Stastability analysis; Dichotomy transformation

Domaines: Optimisation et contrôle [math.OC]

Emmanuel Trélat, Can Zhang, Enrique Zuazua. Steady-state and periodic exponential turnpike property for optimal control problems in Hilbert spaces. SIAM Journal on Control and Optimization, 2018, 56 (2), pp.1222-1252. ⟨hal-01377320⟩