Insensitizing exact controls for the scalar wave equation and exact controllability of 2-coupled cascade systems of PDE’s by a single control
Résumé
We study the exact controllability, by a reduced number of controls, of coupled cascade systems of PDE’s
and the existence of exact insensitizing controls for the scalar wave equation. We give a necessary and sufficient
condition for the observability of abstract coupled cascade hyperbolic systems by a single observation, the
observation operator being either bounded or unbounded. Our proof extends the two-level energy method
introduced in [2, 4] for symmetric coupled systems, to cascade systems which are examples of non symmetric
coupled systems. In particular, we prove the observability of two coupled wave equations in cascade if the
observation and coupling regions both satisfy the Geometric Control Condition (GCC) of Bardos Lebeau and
Rauch [11]. By duality, this solves the exact controllability, by a single control, of 2-coupled abstract cascade
hyperbolic systems. Using transmutation, we give null-controllability results for the multidimensional heat and
Schr¨odinger 2-coupled cascade systems under (GCC) and for any positive time. By our method, we can treat
cases where the control and coupling coefficients have disjoint supports, partially solving an open question raised
by de Teresa [47]. Moreover we answer the question of the existence of exact insensitizing locally distributed
as well as boundary controls of scalar multidimensional wave equations, raised by J.-L. Lions [35] and later on
by D´ager [19] and Tebou [44]
Mots clés
Cascade systems
Geometric conditions
Abstract linear evolution equations
Insensitizing controls
Optimal conditions
Boundary observability
Locally distributed observability
Boundary control
Locally distributed control
HUM
Indirect controllability
Hyperbolic systems
Parabolic systems
Schrödinger equations