Observability from measurable sets for a parabolic equation involving the Grushin operator and applications

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.

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Observability from measurable sets for a parabolic equation involving the Grushin operator and applications

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References 22 publications

Observability Inequality from Measurable Sets for Degenerate Parabolic Equations and its Applications

Observability Inequality from Measurable Sets for Degenerate Parabolic Equations and its Applications

A spectral inequality for degenerate operators and applications

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