A Kalman rank condition for the indirect controllability of coupled systems of linear operator groups

Abstract

International audienceIn this article, we give a necessary and sufficient condition of Kalman type for the indirect controllability of systems of groups of linear operators, under some " regularity and locality " conditions on the control operator that will be made precise later and fits very well the case of distributed controls. Moreover, in the case of first order in time systems, when the Kalman rank condition is not satisfied, we characterize exactly the initial conditions that can be controlled. Some applications to the control of systems of Schrödinger or wave equations are provided. The main tool used here is the fictitious control method coupled with the proof of an algebraic solvability property for some related underdetermined system and some regularity results

On the observability inequality of coupled wave equations: the case without boundary

“…The multispeed case was also studied in [25]. As stated earlier, under Kalman Rank Condition and GCC, exact controllability of systems of wave equations with constant coefficients and general coupling structure of zero order were proved in [36,38]. Note also the recent article of Alabau-Boussouira-Coron-Olive [4] for 1D hyperbolic where appears a condition on ODE problems related to ours and the use of its Gramian.…”

“…Some previous articles (Liard-Lissy [36], Lissy-Zuazua [38]) already obtained some controllability property in this framework under the Kalman rank condition (in a more abstract and general setting). So, an improvement of our Theorem comes from the space where the controllability holds (and not from the assumptions which are more or less the same except that we do not allow boundary here).…”

“…We refer to [40,36] for internal controllability results of systems of Schrödinger equations coupled with constant zero order terms with good algebraic structure.…”

Internal Observability for Coupled Systems of Linear Partial Differential Equations

“…• In Section 2 we present a simple method that can be applied to any first order in time PDE with internal control, giving results of weak observability (in the sense that the observation is made in higher order Sobolev norms) for systems of equations with zero order coupling terms and constant or time-dependent coefficients with a reduced number of observations. Our study is a complement of the results of [38] to the case of all scalar (notably non-conservative) PDE whose solutions verify (1.1), in a different functional setting though.…”

“…• In [38] a Kalman rank condition of controllability was also proved using the fictitious control method, in the abstract setting of groups of operators with bounded control in the case of constant coupling coefficients, with applications to some systems of wave and Schrödinger equations.…”