[Paper Review] 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

Fatiha Alabau‐Boussouira|Jan 28, 2014

Stability and Controllability of Differential EquationsEngineering18 citations

TL;DR

This paper introduces a hierarchic multi-level energy method to achieve exact controllability of n-coupled bi-diagonal and mixed cascade systems of PDEs using a reduced number of controls. It establishes observability and controllability results for hyperbolic, heat, and Schrödinger systems under partial coercivity conditions, proving null-controllability for n ≤ 5 in dimensions >2 and any n ≥ 2 in 1D, even with disjoint control and coupling regions.

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. 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 $n \le 5$ 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.

Motivation & Objective

• To address the challenge of controlling complex cascade systems of PDEs with a minimal number of controls, motivated by practical and cost-efficient applications in engineering and physics.
• To establish necessary and sufficient conditions for exact observability and controllability of bi-diagonal and mixed n+p-coupled cascade systems using only p+1 observations or controls.
• To extend controllability results to parabolic (heat) and dispersive (Schrödinger) systems via the transmutation method, even when control and coupling regions are disjoint.
• To develop a novel multi-level energy method that systematically weakens energy estimates across n levels to recover full initial energy from observation of only the last component.

Proposed method

• A hierarchic multi-level energy method is constructed, involving n levels of weakened energy estimates to propagate observability from the last component to all others in bi-diagonal systems.
• The method relies on a recursive energy decay argument using the structure of the coupling operators on the sub-diagonal, under partial coercivity assumptions.
• Observability estimates are derived using the HUM method, transforming the control problem into an observability inequality for the dual system.
• For parabolic and Schrödinger systems, the transmutation method is applied to the wave operator to transfer controllability results from the hyperbolic case.
• Induction is used to extend observability estimates from the first n+q equations to the full n+p system, incorporating additional non-bi-diagonal couplings.
• The approach handles both bounded and unbounded control operators, and applies to systems with disjoint control and coupling regions.

Experimental results

Research questions

• RQ1 Can exact controllability be achieved for n-coupled bi-diagonal cascade systems of PDEs using only one control, observing only the last component?
• RQ2 What conditions on the coupling operators ensure that observation of the last component recovers the full initial energy of all components?
• RQ3 Can the observability and controllability results be extended to mixed bi-diagonal and non-bi-diagonal n+p-coupled systems with p+1 observations?
• RQ4 Is null-controllability of heat and Schrödinger systems possible when the control region and coupling region are disjoint?
• RQ5 What is the maximal number of components n for which null-controllability holds in multi-dimensional settings with disjoint control and coupling regions?

Key findings

• For n-coupled bi-diagonal cascade systems, exact controllability is achieved with a single control by observing only the last component, under a necessary and sufficient condition on the coupling operators.
• The method proves that observability of the last component implies recovery of the full initial energy in suitable functional spaces, even with partial coercivity of coupling operators.
• Null-controllability of multi-dimensional heat and Schrödinger systems is established for any n ≥ 2 in 1D and for n ≤ 5 in dimensions >2, with disjoint control and coupling regions.
• The transmutation method successfully transfers controllability results from the wave equation to the heat and Schrödinger equations, preserving null-controllability for any positive time.
• The approach allows controls to be localized on a subdomain or on the boundary, and in 1D, coupling coefficients can be supported in any non-empty subset of the domain.
• A recursive induction argument establishes observability for mixed n+p-coupled systems by extending estimates from the first n+q components to the full system with p+1 observations.