A spectral inequality for degenerate operators and applications

Abstract

In this paper we establish a Lebeau-Robbiano spectral inequality for a degenerated one dimensional elliptic operator and show how it can be used to impulse control and finite time stabilization for a degenerated parabolic equation.Résumé .-Dans cet article, on s'intérèsseà l'inégalité spectrale de type Lebeau-Robbiano sur la somme de fonctions propres pour une famille d'opérateurs dégénérés. Les applications sont données en théorie du contrôle comme le contrôle impulsionnel et la stabilisation en temps fini.satisfies y(·, T ) = 0. Theorem 1.3 .-There is (t m ) m∈N a increasing sequence of positive real numbers converging to T > 0 and (F m ) m∈N a sequence of linear bounded operators from L 2 (0, 1) into L 2 (0, 1) such that for any z 0 ∈ L 2 (0, 1), the solutionHere δ t=(t m+1 +tm)/2 denotes the Dirac measure at t = (t m+1 + t m ) /2. Note that the above system equivalently readsTheorem 3.1 is new approach to steer the solution to zero at time T and can be seen as a finite time stabilization for the degenerated heat equation by impulse control. This can be compared with [CN]. The standard null-controllability problem is given when E = (0, T ) and has been studied in [CMV]. It is now well-known that the null controllability for higher degeneracies (α ≥ 2) fails to hold (see [CMV2] and the references therein). We also refer to [ABCF], where the null-controllability result has been extended to more general degeneracies at the boundary. When the control is located at the boundary where the degeneracy occurs, we refer to [Gu, CTY, MRR]. We finally refer to the recent book [CMV2] and the references therein for a full description of the field. Note that an estimation of the cost of controllability for small T > 0, as well as for α → 2 − has been recently obtained in [CMV3].The outline of the paper is as follows. In Section 2, we present the key inequalities needed to prove Theorem 1.1 as Hardy inequality and Carleman inequality. Section 3 is devoted to obtaining the applications of the spectral inequality in control theory as observation estimates, impulse approximate controllability, null controllability on measurable set in time (see Theorem 3.4) and finite time stabilization (see Theorem 3.5). Theorem 1.2 and Theorem 1.3 are direct consequence of Theorem 3.4 and Theorem 3.5 respectively. Key inequalitiesThis section is devoted to the statement of the key inequalities: Hardy inequality and Carleman inequality, that will enable us to prove Theorem 1.1. The proof of the Carleman inequality is given at the end of this section.

Impulse null approximate controllability for heat equation with dynamic boundary conditions

“…In this context, the impulse approximate controllability was studied for a linear heat equation with homogeneous Dirichlet and Neumann boundary conditions in [5,13], using a new strategy combining the logarithmic convexity method and the Carleman commutator approach. In [4], the authors have established a Lebeau-Robbiano-type spectral inequality for a degenerate one-dimensional elliptic operator with application to impulse control and finite-time stabilization. It should be pointed out that this method is a new approach to steer the solution to zero using impulse control as a stabilizer in finite time.…”

The fundamental gap of a kind of sub-elliptic operator

“…where f ∈ L 2 (0, 1) are given functions. On the basis of [1,6,25,29,30], we provide a more comprehensive analysis of the weak solution space for this subelliptic operator. We remark that in the next work, the existence and uniqueness of a solution for problem (2.1) follows from the Lax-Milgram Theorem.…”

“…Prior to commencing the study of eigenvalue problem for , we first shall look for the proper weak solution space of the equation: where are given functions. On the basis of [1, 6, 25, 29, 30], we provide a more comprehensive analysis of the weak solution space for this sub-elliptic operator. We remark that in the next work, the existence and uniqueness of a solution for problem follows from the Lax–Milgram Theorem.…”

“…For more aspects relating to degenerate elliptic equations, we can learn from other references [7–9, 26, 34]. Meanwhile, degenerate parabolic equation theory has flourished over the past years, interested readers can refer [1, 6, 27], in particular, [6] established a Lebeau–Robbiano spectral inequality for a degenerate one-dimensional elliptic operator.…”

A Fredholm Transformation for the Rapid Stabilization of a Degenerate Parabolic Equation

“…More recently, these results have been improved in [4], where a Lebeau-Robbiano strategy (see [17,18]) is followed in order to prove the finite-time stabilization of such a system, with distributed controls. Let us mention that, in contrast with the null-controllability property, our result of rapid stabilization of (1) with boundary control cannot be deduced from [4], since the extension method will not give a feedback law in this context. In the case where the degeneracy occurs at both end-points, a result of distributed controllability has been obtained in [21].…”