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

TL;DR

This paper develops a multi-level energy method to establish exact controllability and observability of cascade hyperbolic systems with fewer controls, extending results to heat and Schrödinger equations in multiple dimensions.

Contribution

It introduces a novel multi-level energy approach for cascade PDE systems, enabling control with fewer observations and controls, and extends controllability results to heat and Schrödinger equations.

Findings

Observation of the last component recovers all initial energies.

Controllability achieved for systems with disjoint control and coupling regions.

Results hold for systems up to 5 dimensions in multi-dimensional cases.

Abstract

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. 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.