A hierarchic multi-level energy method for the control of bi-diagonal and mixed n-coupled cascade systems of PDE's by a reduced number of controls

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This work is concerned with the exact controllability/observability of abstract cascade hyperbolic systems by a reduced number of controls/observations. We prove that the observation of the last component of the vector state allows to recover the initial energies of all of its components in suitable functional spaces under a necessary and sufficient condition on the coupling operators for cascade bi-diagonal systems. The approach is based on a multi-level energy method which involves n-levels of weakened energies. We establish this result for the case of bounded as well as unbounded dual control operators and under the hypotheses of partial coercivity of the n-1 coupling operators on the sub-diagonal of the system. We further extend our observability result to mixed bi-diagonal and non bi-diagonal n+p-coupled cascade systems by p+1 observations. Applying the HUM method, we derive the corresponding exact controllability results for n-coupled bi-diagonal cascade and n+p-coupled mixed cascade systems. Using the transmutation method for the wave operator, we prove that the corresponding heat (resp. Schrödinger) multi-dimensional cascade systems are null-controllable for control regions and coupling regions which are disjoint from each other and for any positive time for n ≤ 5 for dimensions larger than 2, and for any n ≥ 2 in the one-dimensional case. The controls can be localized on a subdomain or on the boundary and in the one-dimensional case the coupling coefficients can be supported in any non-empty subset of the domain.