Theoretical and numerical local null controllability of a quasi-linear parabolic equation in dimensions 2 and 3. Issue 5 (March 2021)

• Record Type: Journal Article
• Authors: Fernández-Cara, E.; Límaco, J.; Marín-Gayte, I.
• Is Part Of: Journal of the Franklin Institute. Volume 358:Issue 5(2021)
• Page Start: 2846
• Page End: 2871
• Publication Date: 2021-03
• Subjects: Quasi-linear parabolic PDEs -- Null controllability -- Iterative algorithms for control problems; 35A35 -- 35K59 -- 49K20 -- 49M15 -- 65M60; Science -- Periodicals; Technology -- Periodicals; Patents -- United States -- Periodicals; 505
• DOI: 10.1016/j.jfranklin.2021.01.031
• Languages: English
• ISSNs: 0016-0032

Abstract

Highlights: Some known results for the linearized parabolic equation. Proof of the null the null controllability of the linearized parabolic equation. Proof of the null controllability of the quasi-linear parabolic equation. Approximate the solution of the quasi-linear parabolic equation with a numerical algorithm (quasi-Newton algorithm) Proof of the algorithm's convergence. Presentation of some numerical experiments. Abstract: This paper is devoted to the theoretical and numerical analysis of the null controllability of a quasi-linear parabolic equation. First, we establish a local controllability result. The proof relies on an appropriate inverse function argument. Then, we formulate an iterative algorithm for the computation of the null control and we prove a convergence result. Finally, we illustrate the analysis with some numerical experiments.