Null controllability for a parabolic equation with nonlocal nonlinearities

Theoretical and numerical local null controllability for a parabolic system with local and nonlocal nonlinearities

“…See for instance [8], where a similar result was established using a fixed point method (that is, rewriting the controllability problem as a fixed point equation in an appropriate space). The main result in this paper can be generalized to other (slightly more general or different) situations.…”

“…There exist positive constants k 0 ; s 0 and C 0 such that, for any s P s 0 and k P k 0 , any F 2 L 2 ðQ Þ and any n T 2 L 2 ðXÞ, the associated solution to (8) satisfies…”

On a nonlinear thermoelastic system with nonlocal coefficients

“…This operator B was also considered in Fernandez-Cara et al [10]. In Límaco et al [17] was derived a model for small transverse vibrations of an elastic membrane assuming that the density of the material is not constant.…”

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

“…The proof of Theorem 1.1 will follow standard arguments (see for instance [21], [10], [25], [19] and [20]), based on Lyusternik's Inverse Mapping Theorem, which can be found in [24] and [26]. To be more specific, we will see that the desired result is equivalent to find a solution to the equation H(u, v, h) = (0, 0, u 0 , v 0 ), (1.4) where H : E → F is a C 1 mapping between two appropriate Hilbert spaces, defined by…”