Exponential stability of integral delay systems with a class of analytic kernels

"The exponential stability of a class of integral delay systems with analytic kernels is investigated by using the Lyapunov-Krasovskii functional approach. Sufficient delay-dependent stability conditions and exponential estimates for the solutions are derived. Special attention is paid to the particular cases of polynomial and exponential kernels.

Asymptotic stability of the multidimensional wave equation coupled with classes of positive-real impedance boundary conditions

This paper proves the asymptotic stability of the multidimensional wave equation posed on a bounded open Lipschitz set, coupled with various classes of positive-real impedance boundary conditions, chosen for their physical relevance: time-delayed, standard diffusive (which includes the Riemann-Liouville fractional integral) and extended diffusive (which includes the Caputo fractional derivative). The method of proof consists in formulating an abstract Cauchy problem on an extended state space using a dissipative realization of the impedance operator, be it finite or infinite-dimensional. The asymptotic stability of the corresponding strongly continuous semigroup is then obtained by verifying the sufficient spectral conditions derived by Arendt and Batty (Trans. Amer. Math. Soc., 306 (1988)) as well as Lyubich and V\~u (Studia Math., 88 (1988))

New results on robust exponential stability of integral delay systems

"The robust exponential stability of integral delay systems with exponential kernels is investigated. Sufficient delay-dependent robust conditions expressed in terms of linear matrix inequalities and matrix norms are derived by using the Lyapunov–Krasovskii functional approach. The results are combined with a new result on quadratic stabilisability of the state-feedback synthesis problem in order to derive a new linear matrix inequality methodology of designing a robust non-fragile controller for the finite spectrum assignment of input delay systems that guarantees simultaneously a numerically safe implementation and also the robustness to uncertainty in the system matrices and to perturbation in the feedback gain.

Some remarks on stability for a phase-field model with memory

The phase field system with memory can be viewed as a phenomenological extension of the classical phase equations in which memory effects have been taken into account in both fields. Such memory effects could be important for example during phase transition in polymer melts in the proximity of the glass transition temperature where configurational degrees of freedom in the polymer melt constitute slowly relaxing "internal modes" which are di±cult to model explicitly. They should be relevant in particular to glass-liquid-glass transitions where re-entrance effects have been recently reported [27]. We note that in numerical studies based on sharp interface equations obtained from (PFM), grains have been seen to rotate as they shrink [35, 36]. While further modelling and numerical efforts are now being undertaken, the present manuscript is devoted to strengthening the analytical underpinnings of the model

Robust stability of integral delay systems with exponential kernels

"In this chapter the stability analysis via Lyapunov-Krasovskii method is extended to perturbed integral delay systems with exponential kernels. Several sufficient robust stability conditions given in the form of linear matrix inequalities are derived.

Stability Analysis of Integral Delay Systems with Multiple Delays

This note is concerned with stability analysis of integral delay systems with multiple delays. To study this problem, the well-known Jensen inequality is generalized to the case of multiple terms by introducing an individual slack weighting matrix for each term, which can be optimized to reduce the conservatism. With the help of the multiple Jensen inequalities and by developing a novel linearizing technique, two novel Lyapunov functional based approaches are established to obtain sufficient stability conditions expressed by linear matrix inequalities (LMIs). It is shown that these new conditions are always less conservative than the existing ones. Moreover, by the positive operator theory, a single LMI based condition and a spectral radius based condition are obtained based on an existing sufficient stability condition expressed by coupled LMIs. A numerical example illustrates the effectiveness of the proposed approaches.Comment: 14 page