Superexponential stabilizability of evolution equations of parabolic type via bilinear control

Author:

Alabau-Boussouira Fatiha,Cannarsa Piermarco,Urbani CristinaORCID

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.

Funder

Ministero dell’Istruzione, dell’Università e della Ricerca

Publisher

Springer Science and Business Media LLC

Subject

Mathematics (miscellaneous)

Reference23 articles.

1. F. Alabau-Boussouira, P. Cannarsa, and C. Urbani. Exact controllability to the ground state solution for evolution equations of parabolic type via bilinear control. preprint available onarXiv:1811.08806.

2. F. Alabau-Boussouira, P. Cannarsa, and C. Urbani. Bilinear control for evolution equations of parabolic type with unbounded lower order terms. in preparation.

3. J.M. Ball, J.E. Marsden, and M. Slemrod. Controllability for distributed bilinear systems. SIAM Journal on Control and Optimization, 20(4):575–597, 1982.

4. K. Beauchard. Local controllability and non-controllability for a 1d wave equation with bilinear control. Journal of Differential Equations, 250(4):2064–2098, 2011.

5. K. Beauchard and C. Laurent. Local controllability of 1d linear and nonlinear Schrödinger equations with bilinear control. J. Math. Pures Appl., 94:520–554, 2010.

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