Shape turnpike for linear parabolic PDE models

Gontran Lance (1, 2) , Emmanuel Trélat (1, 2) , Enrique Zuazua (3, 4)

1 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
2 CaGE - Control And GEometry
3 Departamento de Matemáticas [Madrid]
4 Department Mathematik [Erlangen]

We introduce and study the turnpike property for time-varying shapes, within the viewpoint of optimal control. We focus here on second-order linear parabolic equations where the shape acts as a source term and we seek the optimal time-varying shape that minimizes a quadratic criterion. We first establish existence of optimal solutions under some appropriate sufficient conditions. We then provide necessary conditions for optimality in terms of adjoint equations and, using the concept of strict dissipativity, we prove that state and adjoint satisfy the measure-turnpike property, meaning that the extremal time-varying solution remains essentially close to the optimal solution of an associated static problem. We show that the optimal shape enjoys the exponential turnpike property in term of Hausdorff distance for a Mayer quadratic cost. We illustrate the turnpike phenomenon in optimal shape design with several numerical simulations.

Mots clés: Strict dissipativity; Direct methods; Parabolic equation; Optimal shape design; Turnpike

Domaines: Optimisation et contrôle [math.OC]; Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]

Gontran Lance, Emmanuel Trélat, Enrique Zuazua. Shape turnpike for linear parabolic PDE models. Systems and Control Letters, 2020, 142, pp.104733. ⟨10.1016/j.sysconle.2020.104733⟩. ⟨hal-02388972v2⟩