2014/06/13 by Gengsheng Wang, Can Zhang, Wang, Gengsheng +1 · 1 citation
Computer Science · Engineering · Mathematics · #93B07 #93C25 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Nonlinear Differential Equations Analysis #Optimization and Control (math.OC) #Stability and Controllability of Differential Equations #math.OC #msc:93B07 #msc:93C25
paper · pdf · doi:10.48550/arxiv.1406.3422
29 pages
arxiv created 2014/06/13 · openalex publication_date 2014/06/13 · arxiv updated 2014/06/16 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
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. \citeTG), and the pair (A,B) verifies some observability inequality from time intervals. With the help of the propagation estimate of analytic functions (cf. \citeV) 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 C0 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 \citeAEWZ and \citePW2 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.