Local null controllability for degenerate parabolic equations with nonlocal term

Abstract

We establish a local null controllability result for following the nonlinear parabolic equation:where b(x, r) = (r)a(x) is a function with separated variables that defines an operator which degenerates at x = 0 and has a nonlocal term. Our approach relies on an application of Liusternik's inverse mapping theorem that demands the proof of a suitable Carleman estimate.2010 Mathematics Subject Classification. Primary 35K65, 93B05; Secondary 35K55.

Local null controllability of coupled degenerate systems with nonlocal terms and one control force

“…• In the system (1.1), we can replace each nonlinearity f i (t, x, u, v) by f i (t, x, u, v, u x , v x ), with i ∈ {1, 2}, in order to analyse whether it is possible to prove results about null controllability. • Previously, in [13], we have obtained a local null controlability result for degenarate parabolic equations with nonlocal tems, which implies, throughout standard arguments, a local null boundary controllability result. However, the same fact can not be directly deduced for systems with a reduced number of controls, see [6].…”

“…where a satisfies assumption A.1, c ∈ L ∞ ((0, T )×(0, 1)), F ∈ L 2 ((0, T )×(0, 1)) and ξ T ∈ L 2 (0, 1). The following Carleman estimate, proved in [13], holds for the solution to (2.5):…”

“…Based on this work, in [13], we have considered (1.3), replacing the second-order term (au x ) x by a specific degenerate nonlocal operator. In that new context, we have achieved a local null controllability result.…”

“…For systems of parabolic equations, the main issue is often to reduce the number of control functions acting on the system (see [3,8,11,15], for example), besides that, as it was pointed out in [6], the problem of controlling coupled parabolic equations has a very different behavior with respect to the scalar case, for instance, boundary controllability is not equivalent to distributed controllability, approximate controllability is not equivalent to null controllability, and "the list of open problems is long and there is a lot of work to be done in order to fully understand this challenging subject" [6]. In this direction, the current work may be seen as a natural continuation of [13] and a first step in order to understand parabolic system with nonlocal and degenerate diffusion coefficients of the type µ ·,…”

Carleman inequality for a linear degenerate parabolic problem

“…In order to do that they choose a suitable weight function that change of sign inside the control domain. Following this ideas, in [3,4], the authors extended their Carleman Inequality for a general a = a(x) satisfying hypotheses (3.3), but just for the weak case, and proved a null-controllability result for a degenerate problem with nonlocal nonlinearities. The aim of the present work is extend the Carleman Inequality proved in [3] to the strong case.…”