Insensitizing controls for a semilinear parabolic equation: a numerical approach

Franck Boyer (1, 2) , Víctor Hernández-Santamaría (3, 4) , ‡ Luz de Teresa

1 IUF - Institut universitaire de France
2 IMT - Institut de Mathématiques de Toulouse UMR5219
3 CINVESTAV - Centro de Investigacion y de Estudios Avanzados del Instituto Politécnico Nacional
4 DEUSTO - Universidad de Deusto

In this paper, we study the insensitizing control problem in the discrete setting of finite-differences. We prove the existence of a control that insensitizes the norm of the observed solution of a 1-D semi-discrete parabolic equation. We derive a (relaxed) observability estimate that yields a controllability result for the cascade system arising in the insensitizing control formulation. Moreover, we deal with the problem of computing numerical approximations of insensitizing controls for the heat equation by using the Hilbert Uniqueness Method (HUM). We present various numerical illustrations.

Mots clés: HUM.; controllability; observability; semi-discrete Carleman estimates; Insensitizing controls; 93C20; 65M06; HUM MSC2010: 35K15; semi discrete Carleman estimates

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

Franck Boyer, Víctor Hernández-Santamaría, ‡ Luz de Teresa. Insensitizing controls for a semilinear parabolic equation: a numerical approach. Mathematical Control and Related Fields, 2019, 9 (1), pp.117-158. ⟨10.3934/mcrf.2019007⟩. ⟨hal-01521642v2⟩