Superexponential stabilizability of evolution equations of parabolic type via bilinear control

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 {λ k } k∈N * and {ϕ k } 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.

Exact controllability to eigensolutions of the bilinear heat equation on compact networks

“…This is possible because each element in H is uniquely determined by its first component which belongs to L 2 (e 1 ) ≡ L 2 (0, 1). So, our problem boils down to the controllability of a one-dimensional control system in the interval (0, 1) (see the examples in [2,3,17]). A possible choice of the function µ is µ(x) = x 2 as explained in [2, Section 6.1].…”

“…on a Hilbert space X. When A : D(A) ⊆ X → X is a densely defined, self-adjoint linear operator with compact resolvent, B a bounded linear operator and u is a bilinear control, the controllability of (2) was studied in [2,3,17]. Denoting by {λ k } k∈N * the eigenvalues of the operator A, and by {φ k } k∈N * the corresponding normalized eigenfunctions, it is easy to see that the functions ϕ j (t) = e −λj t φ j are solutions of the free dynamics (u ≡ 0) with initial condition y 0 = φ j .…”

Exact controllability to eigensolutions of the bilinear heat equation on compact networks

“…Let us consider the abstract evolution equation (2) y ′ (t) + Ay(t) + u(t)By(t) = 0, t ∈ (0, T ) y(0) = y 0 on a Hilbert space X. When A : D(A) ⊆ X → X is a densely defined, self-adjoint linear operator with compact resolvent, B a bounded linear operator and u is a bilinear control, the controllability of (2) was studied in [2,3,17]. Denoting by {λ k } k∈N * the eigenvalues of the operator A, and by {φ k } k∈N * the corresponding normalized eigenfunctions, it is easy to see that the functions ϕ j (t) = e −λj t φ j are solutions of the free dynamics (u ≡ 0) with initial condition y 0 = φ j .…”

Nonnegative multiplicative controllability for semilinear multidimensional reaction-diffusion equations

“…Recently, it is also studied the exact controllability for evolution equations via bilinear controls. See, e.g., [1] and [2] by Alabau-Boussouira, Cannarsa and Urbani, [7] by Cannarsa and Urbani, and [8] by Duprez and Lissy.…”