The aim of this paper is to prove the superexponential stabilizability to the ground state solution of a degenerate parabolic equation of the formMore precisely, we provide a control function p that steers the solution of the equation, u, to the ground state solution in small time with doubly-exponential rate of convergence. The parameter α describes the degeneracy magnitude. In particular, for α ∈ [0, 1) the problem is called weakly degenerate, while for α ∈ [1, 2) strong degeneracy occurs. We are able to prove the aforementioned stabilization property for α ∈ [0, 3/2). The proof relies on the application of an abstract result on rapid stabilizability of parabolic evolution equations by the action of bilinear control. A crucial role is also played by Bessel's functions.
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.
In a separable Hilbert space X, we study the controlled evolution equation $$\begin{aligned} u'(t)+Au(t)+p(t)Bu(t)=0, \end{aligned}$$ u ′ ( t ) + A u ( t ) + p ( t ) B u ( t ) = 0 , where $$A\ge -\sigma I$$ A ≥ - σ I ($$\sigma \ge 0$$ σ ≥ 0 ) is a self-adjoint linear operator, B is a bounded linear operator on X, and $$p\in L^2_{loc}(0,+\infty )$$ p ∈ L loc 2 ( 0 , + ∞ ) is a bilinear control. We give sufficient conditions in order for the above nonlinear control system to be locally controllable to the jth eigensolution for any $$j\ge 1$$ j ≥ 1 . We also derive semi-global controllability results in large time and discuss applications to parabolic equations in low space dimension. Our method is constructive and all the constants involved in the main results can be explicitly computed.
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