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

In 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 fit 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. Keyworlds: controllability of abstract linear semi-groups, indirect controllability of systems, Schrödinger and wave equations, fictitious control method, algebraic solvability. MSC: 35F35, 35G35, 47D03, 93B05, 93B07.