Superexponential stabilizability of evolution equations of parabolic type via bilinear control

Abstract

AbstractWe study the stabilizability of a class of abstract parabolic equations of the form $$\begin{aligned} u'(t)+Au(t)+p(t)Bu(t)=0,\qquad t\ge 0 \end{aligned}$$ u ′ ( t ) + A u ( t ) + p ( t ) B u ( t ) = 0 , t ≥ 0 where the control $$p(\cdot )$$ p ( · ) is a scalar function, A is a self-adjoint operator on a Hilbert space X that satisfies $$A\ge -\sigma I$$ A ≥ - σ I , with $$\sigma >0$$ σ > 0 , and B is a bounded linear operator on X. Denoting by $$\{\lambda _k\}_{k\in {\mathbb {N}}^*}$$ { λ k } k ∈ N ∗ and $$\{\varphi _k\}_{j\in {\mathbb {N}}^*}$$ { φ k } j ∈ N ∗ the eigenvalues and the eigenfunctions of A, we show that the above system is locally stabilizable to the eigensolutions $$\psi _j={\mathrm{e}}^{-\lambda _j t}\varphi _j$$ ψ j = e - λ j t φ j with doubly exponential rate of convergence, provided that the associated linearized system is null controllable. Moreover, we give sufficient conditions for the pair $$\{A,B\}$$ { A , B } to satisfy such a property, namely a gap condition for A and a rank condition for B in the direction $$\varphi _j$$ φ j . We give several applications of our result to different kinds of parabolic equations.