By Gianni Dal Maso

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50 Systems with Delays Application of Lyapunov-Krasovskii functionals allows one to avoid these diﬃculties and investigate stability of DDE without calculating eigenvalues or DDE solutions. Though, using this approach we can obtain, as a rule, only suﬃcient conditions of stability, utilization of diﬀerent types of Lyapunov-Krasovskii functionals enables us to obtain various forms of stability conditions in terms of parameters of systems. In this section we present basic theorems of LyapunovKrasovskii functional method for linear systems with delays.

Let us prove the converse implication. 27) which is continuous and w(r) > 0 for r > 0. The functional V is continuous and positive deﬁnite, and for any r > 0 the sphere Sr is compact, hence lim w(r) > 0, and therefore r→∞ Stability Theory 49 there exists a function a ∈ K such that w(r) > a(r) for r > 0. 6. If V : SLk [−τ, 0] → R is a continuous functional, then there exists b ∈ K such that V [z(·)] ≤ b( z(·) C ) for z(·) ∈ SLk [−τ, 0]. Proof. One can easily check that the function b(r) = max V [z(·)] satisﬁes the terms of the theorem.

Zverkin). 1. 1) tends to zero then the system is asymptotically stable. 1) tends to zero if all eigenvalues have negative real parts. 1). Also now we can note that for linear DDE asymptotic stability and exponential stability are equivalent. 3 Stability via the fundamental matrix At present there are no eﬀective algorithms of computing the eigenvalues for linear systems with distributed delays in order to check stability. In this subsection we discuss another method of practical veriﬁcation of stability of the closed-loop system.