Constructive Averaging of a Class of Parabolic PDEs With Applications to Vibrational Control

All the existing methods for averaging-based stability of PDEs with rapidly varying coefficients are qualitative, guaranteeing stability only for sufficiently high-frequency periodic inputs, provided the averaged system is stable. Inspired by our recent results for averaging of ODEs, in this paper we suggest the first quantitative bounds on the small periodicity for averaging-based stability of a class of PDEs. We consider a linear 1D reaction-convection-diffusion system under the Dirichlet boundary conditions. The reaction coefficient is rapidly varying and periodic in time, and it is spatially varying. By using a change of state variables, we transform the system to the perturbed averaged system and apply Lyapunov method that lead to LMIs for finding an upper bound on the small parameter that preserves the stability. We further apply the presented method to vibrational control of the reaction-convection-diffusion system. The efficiency of the method is illustrated by a vibrational control example.