Taylor expansions of the value function associated with a bilinear optimal control problem
Breiten, Tobias1 ; Kunisch, Karl1, 2 ; Pfeiffer, Laurent1
1 Institute of Mathematics, University of Graz, Austria
2 RICAM Institute, Austrian Academy of Sciences, Linz, Austria
Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 5, pp. 1361-1399.
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A general bilinear optimal control problem subject to an infinite-dimensional state equation is considered. Polynomial approximations of the associated value function are derived around the steady state by repeated formal differentiation 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-hand sides. They form the basis for defining a suboptimal 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.

DOI : 10.1016/j.anihpc.2019.01.001
Classification : 49J20, 49N35, 93D05, 93D15
Mots-clés : Value function, Hamilton–Jacobi–Bellman equation, Bilinear control systems, Riccati equation, Generalized Lyapunov equations, Fokker–Planck equation
Breiten, Tobias 1 ; Kunisch, Karl 1, 2 ; Pfeiffer, Laurent 1

1 Institute of Mathematics, University of Graz, Austria
2 RICAM Institute, Austrian Academy of Sciences, Linz, Austria 
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Breiten, Tobias; Kunisch, Karl; Pfeiffer, Laurent. Taylor expansions of the value function associated with a bilinear optimal control problem. Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 5, pp. 1361-1399. doi : 10.1016/j.anihpc.2019.01.001. http://www.numdam.org/articles/10.1016/j.anihpc.2019.01.001/

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