On the reachable space of the heat equation in one space dimension: recent results and open questions

We consider a system described by the heat equation some bounded interval of the real axis with square integrable controls at the boundary. The main focus is on the studiy the reachable space at some strictly positive time. The main results assert that this space is generally sandwiched between two Hilbert spaces of holomorphic functions defined on a square in the complex plane and which has the considered segment as one of the diagonals. More precisely, in the case of Dirichlet boundary controls acting at both ends we prove that the reachable space contains the Hardy-Smirnov space and it is contained in the Bergman space associated to the above mentioned square. The methodology, quite different of the one employed in previous literature, is a direct one. We first represent the input-to-state map as an integral operator whose kernel is a sum of Gaussians and then we study the range of this operator by combining the theory of Riesz bases for Hardy-Smirnov spaces in polygons and a result of Aikawa, Hayashi and Saitoh on the range of integral transforms associated with the heat kernel. We next discuss possible extensions (variables coefficients, nonlinear problems, control cost estimates) and we state some open questions.

举办单位：数学与统计学院

发 布 人：吴双 发布时间： 2019-05-31

发 布 人：吴双 发布时间： 2019-05-31

Marius Tucsnak, chair professor at University of Bordeaux France. His main research interests are control of systems governed by partial differential equations (such as the parabolic, hyperbolic, Schrödinger or nonlinear plate equations) and the analysis and control of fluid-structure interactions. He is a coauthor (with George Weiss) of the book Observation and Control for Operator Semigroups (Birkhauser, 2009). He published around 80 papers in internationally referred journals, including Archive for Rational Mechanics and Analysis, Journal of the European Mathematical Society, Transactions of the American Mathematical Society, Journal of Differential Equations, SIAM Journal on Control or Automatica. He is currently corresponding editor for ESAIM COCV and associated editor for SIAM Journal on Control, MCSS, MCRF, JMFM. In 2013 he was elected member of the “Institut Universitaire de France” and in 2018 he received the Spiru Haret Prize of the Romanian Academy.