Taylor expansions of the value function associated with a bilinear optimal control problem

Tobias Breiten (1) , Karl Kunisch (1, 2) , Laurent Pfeiffer (3, 4)

1 IM - Institut für Mathematik [Graz]
2 RICAM - Johann Radon Institute for Computational and Applied Mathematics
3 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
4 COMMANDS - Controle, Optimisation, modèles, Méthodes et Applications pour les Systèmes Dynamiques non linéaires

A general bilinear optimal control problem subject to an infinitedimensional state equation is considered. Polynomial approximations of the associated value function are derived around the steady state by repeated forma ldifferentiation of the Hamilton–Jacobi-Bellman equation. The terms of the approximations are described by multilinear forms, which can be obtained as solutions to generalized Lyapunov equations with recursively defined right-handsides.They form the basis for defining a sub-optimal feedback law.The approximation properties of this feedback law are investigated. An application to the optimal control of a Fokker-Planck equation is also provided.

Mots clés: Hamilton-Jacobi Bellman equation; Bilinear control systems; Riccati equation; Fokker-Planck equation; Generalized Lyapunov equations; Value function

Domaines: Mathématiques [math]; Optimisation et contrôle [math.OC]

Tobias Breiten, Karl Kunisch, Laurent Pfeiffer. Taylor expansions of the value function associated with a bilinear optimal control problem. Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 2019, 36 (5), pp.1361-1399. ⟨10.1016/j.anihpc.2019.01.001⟩. ⟨hal-02372091⟩