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

Abstract

This work is concerned with the exact controllability/obser-vability 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 one 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 bidiagonal 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 bidiagonal and non-bidiagonal $n+p$-coupled cascade systems by $p+1$ observations. Applying the HUM method, we derive the corresponding exact controllability results for $n$-coupled bidiagonal cascade and $n+p$-coupled mixed cascade systems. Using the transmutation method for the wave operator, we prove that the corresponding heat (respectively Schr\"odinger) 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 $n \le 5$ for dimensions larger than $2$, and for any $n \ge 2$ 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.