On exponential observability estimates for the heat semigroup with explicit rates

Abstract

Abstract. This note concerns the final time observability inequality from an interior region for the heat semigroup, which is equivalent to the nullcontrollability of the heat equation by a square integrable source supported in this region. It focuses on exponential estimates in short times of the observability cost, also known as the control cost and the minimum energy function. It proves that this final time observability inequality implies four variants with roughly the same exponential rate everywhere (an integrated inequality with singular weights, an integrated inequality in infinite times, a sharper inequality and a Sobolev inequality) and some control cost estimates with explicit exponential rates concerning null-controllability, null-reachability and approximate controllability. A conjecture and open problems about the optimal rate are stated. This note also contains a brief review of recent or to be published papers related to exponential observability estimates: boundary observability, Schrödinger group, anomalous diffusion, thermoelastic plates, plates with square root damping and other elastic systems with structural damping. IntroductionThe natural setting for the problem to be discussed is on manifolds, but all the statements can be understood, and are already interesting, when the domain M is a smooth bounded open set of R d with the flat metric so that the distance is dist(x, y), always considered with Dirichlet condition on the boundary ∂M . We shall refer to this setting as the Euclidean case.Although it can be skipped, for completeness we now describe the general setting. Let (M, g) be a smooth connected compact d-dimensional Riemannian manifold with metric g and smooth boundary ∂M . When ∂M = ∅, M denotes the interior andUnless mentioned otherwise, the range of the time T is (0, ∞) and the range of the initial state u 0 is L 2 (M ). The corresponding solution of the Cauchy problem for the (forward) heat equation is denoted by u(T, x) = (e T ∆ u 0 )(x), in short: u = e T ∆ u 0 is the (relative) temperature on R + × M .In this note, we make some remarks about the following observability inequality from Ω of the final state at time T : for any T ,

Degenerate parabolic operators of Kolmogorov type with a geometric control condition

“…The null and approximate controllabilities of the heat equation are essentially well understood subjects for both linear and semilinear equations, for bounded or unbounded domains [3,27,30,32,33,34,37,44,46,48,51,52,62,63] and also with discontinuous [28,12,13,57] or singular [61,29] coecients.…”

Null controllability of Kolmogorov-type equations

“…The null and approximate controllability of the heat equation are essentially well understood subjects for both linear and semilinear equations, for bounded or unbounded domains (see, for instance, [16], [19], [21], [22], [23], [26], [30], 3 [31], [33], [36], [37], [42], [43]) and also with discontinuous (see, e.g. [17], [6], [7], [39]) or singular ( [40] and [18]) coecients.…”

On the Reachable Set for the One-Dimensional Heat Equation

“…Let us also emphasize that there are several works related to the cost of controllability of the heat equation in short time. Let us quote in particular the works by [17,18,19,21] studying these questions. It was thought for a while that the understanding of the blow up of the controllability of the heat equation in short time would be more or less equivalent to a good characterization of the reachable set, but this was recently disproved in [13].…”