Superexponential Stabilizability of Degenerate Parabolic Equations via Bilinear Control

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.

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“…||Bϕ|| (12) and, without loss of generality, we suppose C B ≥ 1. Let A : D(A) ⊂ X → X be a densely defined linear operator such that (6) hold.…”

Section: Resultsmentioning

confidence: 99%

“…The results of this paper have been applied to degenerate parabolic equations in [12]. Moreover, our method can be adapted to obtain exact controllability to eigensolutions (see [1]) under slightly more restrictive assumptions, as well as to treat equations with an unbounded coefficient B such as the Fokker-Planck equations (see [2]).…”

Section: Introductionmentioning

confidence: 99%

Superexponential stabilizability of evolution equations of parabolic type via bilinear control

2020

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“…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 .…”

Section: Introductionmentioning

confidence: 99%

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

Cannarsa¹,

Duca²,

Urbani³

2021

Preprint

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Partial differential equation on networks have been widely investigated in the last decades in view of their application to quantum mechanics (Schrödinger type equations) or to the analysis of flexible structures (wave type equations). Nevertheless, very few results are available for diffusive models despite an increasing demand arising from life sciences such as neurobiology. This paper analyzes the controllability properties of the heat equation on a compact network under the action of a single input bilinear control.By adapting a recent method due to [Alabau-Boussouira, Cannarsa, Urbani, Exact controllability to eigensolutions for evolution equations of parabolic type via bilinear control], an exact controllability result to the eigensolutions of the uncontrolled problem is obtained in this work. A crucial step has been the construction of a suitable biorthogonal family under a non-uniform gap condition of the eigenvalues of the Laplacian on a graph. Application to star graphs and tadpole graphs are included.

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“…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 .…”

mentioning

confidence: 99%

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

Cannarsa

Duca

Urbani

2022

DCDS-S

Self Cite

View full text Add to dashboard Cite

Partial differential equations on networks have been widely investigated in the last decades in view of their application to quantum mechanics (Schrödinger type equations) or to the analysis of flexible structures (wave type equations). Nevertheless, very few results are available for diffusive models despite an increasing demand arising from life sciences such as neurobiology. This paper analyzes the controllability properties of the heat equation on a compact network under the action of a single input bilinear control.By adapting a recent method due to [F. Alabau-Boussouira, P. Cannarsa, C. Urbani, Exact controllability to eigensolutions for evolution equations of parabolic type via bilinear control, arXiv: 1811.08806], an exact controllability result to the eigensolutions of the uncontrolled problem is obtained in this work. A crucial step has been the construction of a suitable biorthogonal family under a non-uniform gap condition of the eigenvalues of the Laplacian on a graph. Application to star graphs and tadpole graphs are included.

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Superexponential Stabilizability of Degenerate Parabolic Equations via Bilinear Control

Cited by 8 publications

References 26 publications

Superexponential stabilizability of evolution equations of parabolic type via bilinear control

Superexponential stabilizability of evolution equations of parabolic type via bilinear control

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

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

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