Null controllability for degenerate parabolic equations with a nonlocal space term

Abstract

<p style='text-indent:20px;'>We consider two degenerate heat equations with a nonlocal space term, studying, in particular, their null controllability property. To this aim, we first consider the associated nonhomogeneous degenerate heat equations: we study their well posedness, the Carleman estimates for the associated adjoint problems and, finally, the null controllability. Then, as a consequence, using the Kakutani's fixed point Theorem, we deduce the null controllability property for the initial nonlocal problems.</p>

On a general degenerate/singular parabolic equation with a nonlocal space term

“…In particular, under suitable conditions on all the parameters of the operator, we establish Carleman estimates and, as a consequence, null controllability for the associated generalized heat problem. Clearly, this result generalizes the ones obtained in [4, 16, 20] or in [18].…”

“…Observe that the assumption $$$ \lambda \ne 0 $$$ is not restrictive since the case $$$ \lambda =0 $$$ is considered in [16]. Actually, in [16], it is considered the case $$$ \lambda =0 $$$ and $$$ {x}_0=0 $$$, but with similar computations, one can prove the results contained in [16] also for the case $$$ {x}_0\in \left(0,1\right] $$$.…”

“…In particular, in [15], it is shown that the considered system is null controllable provided the function $$$ K $$$ enjoys only an exponential decay at the final time $$$ t=T $$$. This assumption is also used to prove in the very recent paper [16] that the problem associated to () is null controllable when $$$ \lambda =0 $$$.…”

Stackelberg exact controllability of a class of nonlocal parabolic equations

“…Such systems like (1.1) and (1.2) can appear for instance in population dynamics where the state y(t, x) represents the density of the species at time t and position x, while the nonlocal term Ω K(t, x, θ)y(t, θ) dθ is considered as the rate of reproduction. This integral term is a way to express that the evolution of the species in a point of space depends on the total amount of the species [1].…”

“…The authors studied both the linear and the semilinear case, by using a Carleman approach. Recently, in [1], the authors have proved the null controllability property for the two degenerate nonlocal systems using the Kakutani's fixed point Theorem. The authors in [17] have proved the internal null-controllability of a heat equation with Dirichlet boundary conditions and perturbed by a semilinear nonlocal term.…”