Local exact controllability to the steady states of a parabolic system with coupled nonlinear boundary conditions

Kuntal Bhandari (1) , Franck Boyer (2, 3)

1 IM / CAS - Institute of Mathematics of the Czech Academy of Science
2 IMT - Institut de Mathématiques de Toulouse UMR5219
3 IUF - Institut universitaire de France

In this article, we study the boundary local exact controllability to any steady state of a one-dimensional parabolic system with coupled nonlinear boundary conditions by means of only one control. The significant point is that the state components are interacting only at the boundary points in terms of some nonlinear terms. We consider two cases : either the control function is acting through a mixed nonlinear boundary condition on the first component or through a Neumann condition on the second component. The results are slightly different in the two cases. To study this problem, we first consider the associated linearized systems around the given steady state. The method of moments let us to prove its controllability and to obtain a suitable estimate of the control cost of the form $Me^{M(T+1/T)}$. To this end, we need to develop a precise spectral analysis of a non self-adjoint operator. Thanks to those preliminary results, we can use the source term method developed in [Liu-Takahashi-Tucsnak 2013], followed by the Banach fixed point argument, to obtain the small-time local boundary exact controllability to the steady state for the original system.

Mots clés: Parabolic systems; boundary controllability; method of moments; source term method; fixed-point argument

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

Kuntal Bhandari, Franck Boyer. Local exact controllability to the steady states of a parabolic system with coupled nonlinear boundary conditions. Mathematical Control and Related Fields, inPress, ⟨10.3934/mcrf.2023035⟩. ⟨hal-03737472v3⟩