Adaptive spatial homogenization and synchronization of structurally perturbed parabolic PDEs via asymptotic embedding methods

摘要

This paper proposes synchronization controllers for networked systems whose dynamics are described by parabolic partial differential equations. The control design objectives are to ensure that each networked system agrees with each other (synchronization) and that each networked system follows the state of a partial differential equation (leader-following). The novelty here is that the leader is governed by time invariant partial differential equation and thus the leader-following controller ensures that each networked system is spatially homogenized. This allows each system to dynamically reach a spatial distribution which is the solution to the time invariance partial differential equation. To achieve both control objectives the control signals are decomposed into two parts; one addressing synchronization via an appropriate consensus protocol with adaptation of the synchronization gains, and the other via a regulation controller of an associate error equation. This state error equation is constructed via asymptotic embedding methods which embed the spatially-dependent and time-invariant dynamics of the leader into the dynamics of each networked system. Both the stability of the proposed controllers and the well-posedness of the resulting aggregate system are summarized and an example of five networked parabolic partial differential equations tasked with synchronization and following a spatially-dependent leader are presented to provide insights on the proposed spatial homogenization.