Insensitizing exact controls for the scalar wave equation and exact controllability of $2$-coupled cascade systems of PDE's by a single control

摘要

We study the exact controllability, by a reduced number of controls, ofcoupled cascade systems of PDE's and the existence of exact insensitizingcontrols for the scalar wave equation. We give a necessary and sufficientcondition for the observability of abstract coupled cascade hyperbolic systemsby a single observation, the observation operator being either bounded orunbounded. Our proof extends the two-level energy method introduced in\cite{sicon03, alaleau11} for symmetric coupled systems, to cascade systemswhich are examples of non symmetric coupled systems. In particular, we provethe observability of two coupled wave equations in cascade if the observationand coupling regions both satisfy the Geometric Control Condition (GCC) ofBardos Lebeau and Rauch \cite{blr92}. By duality, this solves the exactcontrollability, by a single control, of $2$-coupled abstract cascadehyperbolic systems. Using transmutation, we give null-controllability resultsfor the multidimensional heat and Schr\"odinger $2$-coupled cascade systemsunder (GCC) and for any positive time. By our method, we can treat cases wherethe control and coupling coefficients have disjoint supports, partially solvingan open question raised by de Teresa \cite{DeT00}. Moreover we answer thequestion of the existence of exact insensitizing locally distributed as well asboundary controls of scalar multidimensional wave equations, raised by J.-L.Lions \cite{lions89} and later on by D\'ager \cite{Dager06} and Tebou\cite{tebou08}.