Sharp observability estimates for heat equations. (English) Zbl 1251.93040

Ervedoza, Sylvain; Zuazua, Enrique

Arch. Ration. Mech. Anal. 202, No. 3, 975-1017 (2011).

Summary: The goal of this article is to derive new estimates for the cost of observability of heat equations. We have developed a new method allowing one to show that when the corresponding wave equation is observable, the heat equation is also observable. This method allows one to describe the explicit dependence of the observability constant on the geometry of the problem (the domain in which the heat process evolves and the observation subdomain). We show that our estimate is sharp in some cases, particularly in one space dimension and in the multi-dimensional radially symmetric case. Our result extends those in H. O. Fattorini and D. L. Russell [”Exact controllability theorems for linear parabolic equations in one space dimension”, Arch. Ration. Mech. Anal. 43, 272-292 (1971; Zbl 0231.93003)] to the multi-dimensional setting and improves those available in the literature, namely those by L. Miller [”Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time”, J. Differ. Equations 204, No. 1, 202-226 (2004; Zbl 1053.93010)], L. Miller [”The control transmutation method and the cost of fast controls”, SIAM J. Control Optim. 45, No. 2, 762-772 (2006; Zbl 1109.93009)], L. Miller [”On exponential observability estimates for the heat semigroup with explicit rates”, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 17, No. 4, 351-366 (2006; Zbl 1150.93006)] and G. Tenenbaum, M. Tucsnak [”New blow-up rates for fast controls of Schrödinger and heat equations”, J. Differ. Equations 243, No. 1, 70-100 (2007; Zbl 1127.93016)]. Our approach is based on an explicit representation formula of some solutions of the wave equation in terms of those of the heat equation, in contrast to the standard application of transmutation methods, which uses a reverse representation of the heat solution in terms of the wave one. We shall also explain how our approach applies and yields some new estimates on the cost of observability in the particular case of the unit square observed from one side. We will also comment on the applications of our techniques to controllability properties of heat-type equations.

MSC:

93B07	Observability
35K20	Initial-boundary value problems for second-order parabolic equations

Citations:

Zbl 0231.93003; Zbl 1053.93010; Zbl 1109.93009; Zbl 1150.93006; Zbl 1127.93016

References:

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