Approximate Controllability from the Exterior of Space Fractional Parabolic-Elliptic Coupled Systems

Carole Louis-Rose (1)

1 LAMIA - Laboratoire de Mathématiques Informatique et Applications

We study the approximate controllability of parabolic-elliptic coupled systems involving the fractional Laplacian (−Delta)^s, s ∈ (0, 1). The control is located on a non-empty open subset O of the complementary of the open bounded domainin R^N . For this reason, the approximate controllability is said to be exterior. This concept was recently introduced by Warma M. for the fractional diffusion equation. For this purpose, we first prove the existence and uniqueness of the series solution of the studied systems and their dual. Then we state a unique continuation principle for the dual equation which follows from a unique continuation property for the eigenvalues of (−Delta)^s with the homogeneous exterior Dirichlet condition. Finally, we show that for any s ∈ (0, 1) and any control in D(O × (0,T )), under certain conditions on the coefficients, the approximate controllability at any time T > 0 holds.

Mots clés: Fractional partial differential equations · Approximate controllability · Series solution to PDEs · Unique continuation principle

Domaines: Equations aux dérivées partielles [math.AP]

Carole Louis-Rose. Approximate Controllability from the Exterior of Space Fractional Parabolic-Elliptic Coupled Systems. Acta Applicandae Mathematicae, 2021, 176 (6), pp.31. ⟨10.1007/s10440-021-00452-z⟩. ⟨hal-03420002⟩