Uniqueness of solution to systems of elliptic operators and application to asymptotic synchronization of linear dissipative systems

Abstract

We show that under Kalman's rank condition on the coupling matrices, the uniqueness of solution to a complex system of elliptic operators can be reduced to the observability of a scalar problem. Based on this result, we establish the asymptotic stability and the asymptotic synchronization for a large class of linear dissipative systems.

Uniqueness theorem for partially observed elliptic systems and application to asymptotic synchronization

“…In the previous discussion, we have assumed that the matrix A is symmetric, which is actually essential for the stability of the evolution systems that we will study in the next section. A generalization of Theorem 5 can be found in the complete version [6].…”

“…In other words, the pair (L, γ) defined by ( 46) is observable (see [6] for details). Noting that γ is compact from H 1 Γ 0 (Ω) into (H 1 Γ 0 (Ω)) , by Theorem 12 we have the following Theorem 15.…”

Approximate internal controllability and synchronization of a coupled system of wave equations

“…The first attempt for realizing this idea was carried out in [13,14] for a system of wave equations with Dirichlet boundary condition by incomplete Neumann observations. Later, this idea was used in [11,19] for Neumann and Robin boundary conditions, and further developed in [18,21] for an elliptic system with Neumann boundary conditions observed by incomplete Dirichlet observations. Additionally, the coupling matrix A should be nilpotent (see [2,14,16]), or symmetric and close to a scalar matrix etc.…”