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Superexponential stabilizability of evolution equations of parabolic type via bilinear control. (English) Zbl 1467.35326

Summary: We study the stabilizability of a class of abstract parabolic equations of the form \[ u'(t)+Au(t)+p(t)Bu(t)=0,\quad t\geq 0 \] where the control \(p(\cdot)\) is a scalar function, \(A\) is a self-adjoint operator on a Hilbert space \(X\) that satisfies \(A\geq-\sigma I\), with \(\sigma>0\), and \(B\) is a bounded linear operator on \(X\). Denoting by \(\{\lambda_k\}_{k\in\mathbb{N}^*}\) and \(\{\varphi_k\}_{j\in\mathbb{N}^*}\) the eigenvalues and the eigenfunctions of \(A\), we show that the above system is locally stabilizable to the eigensolutions \(\psi_j=\text{e}^{-\lambda_jt}\varphi_j\) with doubly exponential rate of convergence, provided that the associated linearized system is null controllable. Moreover, we give sufficient conditions for the pair \(\{A,B\}\) to satisfy such a property, namely a gap condition for \(A\) and a rank condition for \(B\) in the direction \(\varphi_j\). We give several applications of our result to different kinds of parabolic equations.

MSC:

35Q93 PDEs in connection with control and optimization
93C25 Control/observation systems in abstract spaces
93C10 Nonlinear systems in control theory
35K10 Second-order parabolic equations
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References:

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