Null-Controllability of a Non-Local Heat Equation

2021/11/18 by Walton, Steven
#Analysis of PDEs (math.AP) #FOS: Mathematics #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.2111.09965

Abstract

We consider the null-controllability of a non-local heat equation by interior L²(Ω) controls. We confirm a conjecture of Lissy and Zuazua by showing that it is enough to assume that the kernel k(x,ξ) is symmetric and k(x,ξ)∈ L²(Ω×Ω) in order to obtain the result. The result is obtained by treating the non-local linear operator K:L²(Ω)→ L²(Ω) as a compact and bounded perturbation of the Dirichlet-Laplacian, A, which leads to useful bounds on the semigroup generated by (A+K,D(A)). Using this approach, we are also able to estimate the cost of control.

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