  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
\cite{sicon03, alaleau11} 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 \cite{blr92}. 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 \cite{DeT00}. 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 \cite{lions89} and later on by D\'ager \cite{Dager06} and Tebou
\cite{tebou08}.
