Exponential estimates for neutral time delay systems with multiple delays

Abstract

SUMMARYExponential estimates and sufficient conditions for the exponential stability of linear neutral time delay for systems with multiple delays are given. The case of systems with uncertainties, including uncertainties in the difference operator, is considered. The proofs follows from new results on non-homogeneous difference equations evolving in continuous time combined with the Lyapunov-Krasovskii functionals approach. The conditions are expressed in terms of linear matrix inequalities. The particular case of neutral time delay systems with commensurate delays, which leads to less restrictive exponential estimates, is also addressed.

Exponential Stability With Decay Rate Estimation for Linear Difference Equations

“…The converse of this assertion is false, as shown in the example below. These matrices satisfy N k=1 A k = 1.4435, therefore the exponential stability of (43) cannot be analyzed using the conditions in [12]. However, this system is exponentially stable.…”

“…Substitute (4) into these expressions, and using the decomposition blocks (11) in (12), we arrive at the Dini upper right-hand derivative of vµ(xt(ϕ)) along the trajectory of (4), that is…”

“…Since (2) requires iterative optimization methods to be tested and is not constructive, in general, for state-feedback synthesis, some numerically tractable sufficient stability conditions were proposed. In [12] (see also the references therein), it was shown that the sufficient exponential stability (delay-independent) condition N k=1…”

“…, N . Difference equations in the form (1) appear in systems of conservation laws modeled by first order hyperbolic partial differential equations, where the delay operator describes some transport phenomenon with finite velocity [5], [3], in coupled differential-difference equations [9], [16], [15], neutral timedelay systems [10], [12], [7], [13], integral equations or difference equations with distributed delays [2], [11], [14], and in references therein, covering a wide class of applications, as electric transmission lines, road traffic, water channels or gas flow in pipeline networks.…”

Mean square stability of linear stochastic neutral‐type time‐delay systems with multiple delays

“…Let the associated D operator be defined as D(x t ) = x(t) − 0 (X(t)). To the best of our knowledge, when m is a general integer and 0 contains more derivatives of retarded state variable, unless for corresponding deterministic systems (see other works [30][31][32][33] ), stability analysis of this class of stochastic systems was not well discussed in the literature. Only recently, the stability analysis of stochastic neutral-type time-delay systems with two different delays in both the state variables and derivatives of delayed state variables was considered in our recent work.…”