Necessary stability conditions for neutral-type systems with multiple commensurate delays

Vladimir Kharitonov: Achievements in research

“…[127][128][129][130][131][132] Another application is, of course, the stability analysis of individual systems based on Theorem 3. Some effective necessary stability conditions can be obtained directly, 100,[133][134][135][136][137][138][139][140][141][142] but sufficiency requires substitution of every point of an infinite-dimensional space to (24). With some special techniques, the validation can be reduced first to a compact set of the infinite-dimensional space and then to a finite-dimensional space leading to necessary and sufficient stability conditions.…”

Lyapunov–Krasovskii functionals for homogeneous time delay systems of neutral type

“…Notice that Assumption 1 is a necessary condition for the asymptotic stability of the trivial solution of system (6). It is also known 13 that under Assumption 2 there exists a positive definite, homogeneous of degree 𝛾 > 2 and twice continuously differentiable Lyapunov function V(x) for delay free system (7):…”

“…Theorem 2. If the matrix D is Schur stable and delay free system (7) is asymptotically stable, then the trivial solution of system ( 6) is asymptotically stable for all delays 𝜏 ≥ 0, h j ≥ 0.…”

“…It was shown recently that these functionals allow analyzing the stability directly as well. 6,7 A lot of frequency domain techniques 8,9 are also available, and some peculiarities of linear neutral type systems are pointed out by Gu. 10 For time-varying systems, diverse constructions of Lyapunov-Krasovskii functionals are widely used to obtain sufficient stability conditions in the form of linear matrix inequalities.…”

Exponential stability of linear periodic difference-delay equations

“…In the time-invariant case, stability of functional differential equations like (1.3) has been studied from the fifties on. By and large, works on the subject fall into two classes: those relying on Laplace transform and the spectral theory of semi-groups to reduce issues of stability to the location of zeros of almost periodic holomorphic functions (see for instance [13,4,2,1]), and those seeking to construct Lyapunov-Krasovskii functionals whose existence is sufficient for exponential stability to hold (for example as in [1,14,15]). For time invariant difference-delay systems of the form (1.1) (that is: with constant D j ), a well-known necessary and sufficient condition for exponential stability (either C 0 or L q for any q ∈ [1, ∞]) can be given in terms of the matrixvalued function H(p) = I − N j=1 e −p τ j D j of the complex variable p; namely, exponential stability in this case is equivalent to the existence of α < 0 such that H(p) is invertible for ℜ(p) > α.…”