Superexponential stabilizability of evolution equations of parabolic type via bilinear control
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Journal of Evolution Equations
Superexponential stabilizability of evolution equations of parabolic type via bilinear control
Fatiha Alabau- Boussouira, Piermarco Cannarsa and Cristina Urbani
Abstract. We study the stabilizability of a class of abstract parabolic equations of the form u (t) + Au(t) + p(t)Bu(t) = 0,
t ≥0
where the control p(·) is a scalar function, A is a self-adjoint operator on a Hilbert space X that satisfies A ≥ −σ I , with σ > 0, and B is a bounded linear operator on X . Denoting by {λk }k∈N∗ and {ϕk } j∈N∗ the eigenvalues and the eigenfunctions of A, we show that the above system is locally stabilizable to the eigensolutions 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} to satisfy such a property, namely a gap condition for A and a rank condition for B in the direction ϕ j . We give several applications of our result to different kinds of parabolic equations.
1. Introduction In the field of control theory of dynamical systems, a huge amount of works is devoted to the study of models in which the control enters as an additive term (boundary or internal locally distributed controls); see, for instance, the books [22,23] by Lions. On the other hand, these kinds of control systems are not suitable to describe processes that change their physical characteristics in response to the control. This issue is quite common for the so-called smart materials and in many biomedical, chemical and nuclear chain reactions. Indeed, under the process of catalysis some materials are able to change their principal parameters (see the examples in the monograph [20] by Khapalov for more details.) To deal with these situations, an important role is played by multiplicative controls, that is, controls that appear in the equations as coefficients. Mathematics Subject Classification: 35Q93, 93C25, 93C10, 35K10 Keywords: Bilinear control, Stabilization, Evolution equations, Moment method. This paper was partly supported by the INdAM National Group for Mathematical Analysis, Probability and their Applications. P. Cannarsa: This author acknowledges support from the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006. C. Urbani: This author is grateful to University Italo Francese (Vinci Project 2018).
F. A. Boussouira et al.
J. Evol. Equ.
Due to a weaker control action, exact controllability results are not to be expected with multiplicative controls. Nevertheless, approximate controllability has been obtained for different types of initial/target conditions. For instance, in [18] Khapalov proved a result of nonnegative approximate controllability for a 1D semilinear parabolic equation. In [19], the same author proved approximate and exact null controllability for a bilinear parabolic system with reaction term satisfying Newton’s law. Paper [16], by Floridia, is devoted to the study of global a
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