Stability and passivity analysis of semilinear diffusion PDEs with time-delays

Abstract

In the present paper, sufficient conditions for the exponential stability and passivity analysis of nonlinear diffusion partial differential equations (PDEs) with infinite distributed and discrete time-varying delays are derived. Such systems arise in many applications, e.g. in population dynamics and in heat flows. The existing Lyapunov-based results on the stability of diffusion nonlinear PDEs treat either systems with infinite delays or the ones with discrete slowly varying delays (with the delay-derivatives upper bounded by d< 1), where the conditions are delay-independent in the discrete delays. In this paper, we introduce the Lyapunov-based analysis of semilinear diffusion PDEs with fast-varying(without any constraints on the delay-derivative) discrete and infinite distributed delays. Two novel methods are suggested leading to conditions in terms of linear matrix inequalities. The first one provides delay-independent with respect to discrete delays stability criterion via combination of Lyapunov–Krasovskii functionals and of the Halanay inequality. Note that the Halanay inequality is not applicable to the passivity analysis. Therefore, the second method develops the direct Lyapunov–Krasovskii method via the descriptor approach that leads to delay-dependent (in discrete delays) conditions for the exponential stability and passivity. Numerical examples illustrate the efficiency of the methods.