Perturbations of time optimal control problems for a class of abstract parabolic systems

Marius Tucsnak (1, 2) , Gengsheng Wang (3) , Chi-Ting Wu (1, 2)

1 CORIDA - Robust control of infinite dimensional systems and applications
2 IECL - Institut Élie Cartan de Lorraine
3 School of Mathematics and Statistics [Wuhan]

In this work we study the asymptotic behavior of the solutions of a class of abstract parabolic time optimal control problems when the generators converge, in an appropriate sense, to a given strictly negative operator. Our main application to PDEs systems concerns the behavior of optimal time and of the associated optimal controls for parabolic equations with highly oscillating coefficients, as we encounter in homogenization theory. Our main results assert that, provided that the target is a closed ball centered at the origin and of positive radius, the solutions of the time optimal control problems for the systems with oscillating coefficients converge, in the usual norms, to the solution of the corresponding problem for the homogenized system. In order to prove our main theorem, we provide several new results, which could be of a broader interest, on time and norm optimal control problems.

Mots clés: abstract parabolic equation with perturbed parameter; homogenization AMS subject classifications 93C20; norm optimal control; time optimal control; heat equation with oscillating coefficient

Domaines: Optimisation et contrôle [math.OC]; Mathématiques générales [math.GM]; Recherche d'information [cs.IR]

Marius Tucsnak, Gengsheng Wang, Chi-Ting Wu. Perturbations of time optimal control problems for a class of abstract parabolic systems. 2015. ⟨hal-01076859⟩