# Time-delay systems : Lyapunov functionals and matrices by Vladimir Kharitonov

By Vladimir Kharitonov

Half 1. Retarded kind structures -- basic concept -- unmarried hold up case -- a number of hold up case -- structures with disbursed hold up -- half 2. impartial style platforms -- basic concept -- Linear platforms -- disbursed hold up case

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Proof. 16). Since the matrix U(τ ) satisfies the properties, the functional v0 (ϕ ) satisfies Eq. 5) as well. 5 justifies the following definition. 5. 16). On the one hand, the new definition makes it possible to overcome the first limitation of the original definition of the Lyapunov matrices – the exponential stability assumption. 11)? The following statement provides an affirmative answer to this question. 6. 1) be exponentially stable. 11) is the unique solution of Eq. 16). Proof. 11) satisfies Eq.

1) in terms of their initial functions. 1 Fundamental Matrix The key element needed to derive this expression is the fundamental matrix of the system. 1) as the matrix exponent does for linear delay-free systems. 1 ([3]). 1) if it satisfies the matrix equation d K(t) = K(t)A0 + K(t − h)A1, dt t ≥ 0, and K(t) = 0n×n for t < 0, K(0) = I. Here I is the identity matrix. 1. The fundamental matrix also satisfies the matrix equation d K(t) = A0 K(t) + A1K(t − h), dt t ≥ 0. This does not mean that matrix K(t) commutes individually with the coefficient matrices Ak , k = 0, 1.

Let us define the positive-definite function v1 (x) = α1 x 2 and the positivedefinite functional v2 (ϕ ) = α1 ϕ 2h . 5. 3) is uniformly stable, and for every ε > 0 there exists δ (ε ) > 0 such that the inequality ϕ h < δ (ε ) implies x(t,t0 , ϕ ) < ε for t ≥ t0 . 4. To this end, assume that t0 ≥ 0 and ϕ ∈ PC ([−h, 0], Rn ) , ϕ h < Δ0 . The corresponding solution x(t,t0 , ϕ ) is such that x(t,t0 , ϕ ) < H, for t ≥ t0 . The second condition of the theorem implies the inequality v(t, xt (t0 , ϕ )) ≤ v(t0 , ϕ )e−2σ (t−t0 ) , t ≥ t0 .