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We prove rapid stabilizability to the ground state solution for a class of abstract parabolic equations of the form begin{equation*} u(t)+Au(t)+p(t)Bu(t)=0,qquad tgeq0 end{equation*} where the operator $-A$ is a self-adjoint accretive operator on a Hilbert space and $p(cdot)$ is the control function. The proof is based on a linearization argument. We prove that the linearized system is exacly controllable and we apply the moment method to build a control $p(cdot)$ that steers the solution to the ground state in finite time. Finally, we use such a control to bring the solution of the nonlinear equation arbitrarily close to the ground state solution with doubly exponential rate of convergence. We give several applications of our result to different kinds of parabolic equations.
The aim of this paper is to prove the superexponential stabilizability to the ground state solution of a degenerate parabolic equation of the form begin{equation*} u_t(t,x)+(x^{alpha}u_x(t,x))_x+p(t)x^{2-alpha}u(t,x)=0,qquad tgeq0,xin(0,1) end{equati
In a separable Hilbert space $X$, we study the linear evolution equation begin{equation*} u(t)+Au(t)+p(t)Bu(t)=0, end{equation*} where $A$ is an accretive self-adjoint linear operator, $B$ is a bounded linear operator on $X$, and $pin L^2_{loc}(0,+in
In this paper we present a null controllability result for a degenerate semilinear parabolic equation with first order terms. The main result is obtained after the proof of a new Carleman inequality for a degenerate linear parabolic equation with first order terms.
Among the main actors of organism development there are morphogens, which are signaling molecules diffusing in the developing organism and acting on cells to produce local responses. Growth is thus determined by the distribution of such signal. Meanw
A class of optimal control problems of hybrid nature governed by semilinear parabolic equations is considered. These problems involve the optimization of switching times at which the dynamics, the integral cost, and the bounds on the control may chan