Observability Inequalities from Measurable Sets for Some Abstract Evolution Equations

Abstract

In this paper, we build up two observability inequalities from measurable sets in time for some evolution equations in Hilbert spaces from two different settings. The equation reads: u ′ = Au, t > 0, and the observation operator is denoted by B. In the first setting, we assume that A generates an analytic semigroup, B is an admissible observation operator for this semigroup (cf. [36]), and the pair (A, B) verifies some observability inequality from time intervals. With the help of the propagation estimate of analytic functions (cf. [35]) and a telescoping series method provided in the current paper, we establish an observability inequality from measurable sets in time. In the second setting, we suppose that A generates a C 0 semigroup, B is a linear and bounded operator, and the pair (A, B) verifies some spectral-like condition. With the aid of methods developed in [2] and [29] respectively, we first obtain an interpolation inequality at one time, and then derive an observability inequality from measurable sets in time. These two observability inequalities are applied to get the bang-bang property for some time optimal control problems.Key words. Evolution equations in Hilbert spaces, observability inequality in measurable sets, telescoping series method, propagation estimate of analytic functions, bang-bang property of time optimal controls AMS Subject Classifications. 93B07, 93C25

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

“…These together lead to the desired result through utilizing a telescoping series argument. We want to mention here the difference between the present work and the recent work by Wang and Zhang . In , the authors studied the observability from measurable sets in time, for some evolution equations in Hilbert spaces, and the two‐dimensional parabolic equations associated with Grushin operator can be applied to this case; while in the present work, we aim to obtain the observability inequality from measurable subset in time and space variables (space set is not considered in the previous work), for the parabolic equations with Grushin operator, in some multidimension domains.…”

Quantitative unique continuation for the heat equation with Coulomb potentials

“…Secondly, several applications for the above interpolation estimate (1.2) in Control Theory, such as impulse control, observability inequalities from measurable subsets, and bang-bang properties of optimal controls for parabolic equations, have been recently discussed in [1,6,7,23,24,25,29,30,31,32,34].…”

Time optimal sampled-data controls for the heat equation

“…We refer to [15,41] for semi-discrete finite element approximations, and [34,43] for perturbations of equations. About more works on time optimal control problems, we would like to mention [2,10,11,16,17,18,21,22,25,27,30,31,35,37,38,39,40,42,44] and the references therein.…”