Null-controllability of some systems of parabolic type by one control force

Abstract

Abstract.We study the null controllability by one control force of some linear systems of parabolic type. We give sufficient conditions for the null controllability property to be true and, in an abstract setting, we prove that it is not always possible to control.Mathematics Subject Classification. 93B05, 93C20, 93C25, 35K90.

New phenomena for the null controllability of parabolic systems: Minimal time and geometrical dependence

“…The first results on controllability of coupled parabolic equations (n > 1) have been established in [35,13,3,25]. They concern mainly system (1.4) with n = 2, C = 0 (distributed control) and…”

“…Under this condition, in [7] the authors give a necessary and sufficient condition for the approximate and null controllability at time T > 0 of system (1.1). As a consequence, they also obtain the null controllability property at any positive time T for (1.2) under the same conditions.…”

“…From [35,13,3,25], we know that the null controllability result for system (1.2) is valid for any positiver time T .…”

Controllability results for cascade systems of $m$ coupled parabolic PDEs by one control force

“…In [19] and [5] the authors consider a nonlinear system of two heat equations, one of them being forward and the other one backward in time, and show the null controllability of this system with sublinear nonlinearities ( [19]) or slightly superlinear nonlinearities ( [5]). In [1] and [2], the authors give a null controllability result for a phase-field system and for reactiondiffusion systems (two nonlinear heat equations). The results in [1] and [2] have been generalized in [12] (see also [11]) in two directions: on the one hand, there are not restrictions on the dimension N , and on the other hand, the authors consider nonlinearities which depend on the gradient of the state.…”

“…In [1] and [2], the authors give a null controllability result for a phase-field system and for reactiondiffusion systems (two nonlinear heat equations). The results in [1] and [2] have been generalized in [12] (see also [11]) in two directions: on the one hand, there are not restrictions on the dimension N , and on the other hand, the authors consider nonlinearities which depend on the gradient of the state. Finally, in [8] the authors prove a result of local exact controllability to the trajectories for the Boussinesq system (N + 1 equations) when N (or N − 1) distributed controls are exerted on the system.…”