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

摘要

This work is concerned with the exact controllability/observability ofabstract cascade hyperbolic systems by a reduced number ofcontrols/observations. We prove that the observation of the last component ofthe vector state allows to recover the initial energies of all of itscomponents in suitable functional spaces under a necessary and sufficientcondition on the coupling operators for cascade bi-diagonal systems. Theapproach is based on a multi-level energy method which involves $n$-levels ofweakened energies. We establish this result for the case of bounded as well as unbounded dualcontrol 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 nonbi-diagonal $n+p$-coupled cascade systems by $p+1$ observations. Applying theHUM method, we derive the corresponding exact controllability results for$n$-coupled bi-diagonal cascade and $n+p$-coupled mixed cascade systems. Usingthe transmutation method for the wave operator, we prove that the correspondingheat (resp. Schr\"odinger) multi-dimensional cascade systems arenull-controllable for control regions and coupling regions which are disjointfrom each other and for any positive time for $n \le 5$ for dimensions largerthan $2$, and for any $n \ge 2$ in the one-dimensional case. The controls canbe localized on a subdomain or on the boundary and in the one-dimensional casethe coupling coefficients can be supported in any non-empty subset of thedomain.