Pointwise-in-space stabilization and synchronization of a class of reaction–diffusion systems with mixed time delays via aperiodically impulsive control

Abstract

This paper investigates the problems of pointwise-in-space stabilization and synchronization of semilinear reaction–diffusion systems with Dirichlet boundary conditions and mixed delays via impulsive control. Combining impulse time-dependent Lyapunov function/functional-based methods, descriptor system approach, the Wirtinger’s inequality for tackling the diffusion terms, with the extended Jensen’s inequality for treating the unbounded distributed delay term, two novel global exponential stability criteria are obtained in terms of linear matrix inequalities. Unlike the previous results which only concerned stability in the sense of \(L_2\)-norm, the obtained stability criteria can not only judge whether the considered systems are pointwise convergence in space or not, but also be applied to the class of impulsive PDEs with unstable continuous and discrete dynamics. Moreover, two novel impulsive synchronization schemes are presented to guarantee the pointwise-in-space synchronization between two identical semilinear diffusion systems. Finally, numerical examples show that our results improve some \(L_2\)-norm stability results.