On Equivalence of Lyapunov–Razumikhin Conditions and ISS for a Class of Time-Delay Systems

For a retarded nonlinear system with a Lipschitz nonlinearity on bounded sets outside the origin, this article proposes necessary conditions for application of the Lyapunov–Razumikhin method. The presented conditions are illustrated on the classes of linear and nonlinear homogeneous systems.


I. INTRODUCTION
There exist two generic frameworks for stability investigation in time-delay systems, which are based on analysis of properties of a Lyapunov-Razumikhin function (LRF) or a Lyapunov-Krasovskii functional [11], [13] (see also a recent survey [2]).The latter method has been proven to be equivalent to the asymptotic and input-to-state stability (ISS) properties for many classes of the time-delay systems [7], [17], [18], and it can also be used to establish finite-time stability [9], [15].The former approach is only sufficient for the asymptotic stability or ISS [11], [13], [21], but it can also be used to evaluate the rates of convergence in the retarded dynamics [6].Both methods have extensions to distinguish delay-dependent and independent stability scenarios (i.e., separating the cases when the stable compartment is possible for restricted values of delays, or the stability conditions are uniform in the delay values) and can be used for small-gain analysis [4].An advantage of Lyapunov-Razumikhin approach with respect to Lyapunov-Krasovskii one is that in many nonlinear cases it is more simple to find LRF than a Lyapunov-Krasovskii functional [8], [10], [21] [e.g., a Lyapunov function (LF) for the delay-free case can be tested looking for delay-dependent conditions].
The aim of this work is to complement the LRF approach by proposing necessary conditions of its applicability, and relating it with the existence of an LF for the delay-free counterpart.The results are formulated in the ISS framework.The obtained outcomes will be illustrated on homogeneous time-delay models [8].
The rest of this article is organized as follows.Preliminaries are given in Section II.The main results are formulated in Section III.The linear and homogeneous models are considered in Section IV.Finally, Section V concludes this article.

II. PRELIMINARIES
The real numbers are denoted by R, R + = {s ∈ R : s ≥ 0}.Euclidean norm for a vector x ∈ R n is defined as |x|, for a matrix | • | denotes the induced norm.We denote by . For a locally Lipschitz continuous function V : R n → R + , the upper directional Dini derivative is defined as follows: and β(r, •) is strictly decreasing to zero for any fixed r > 0.

A. ISS for Delay-Free Systems
Consider a dynamical system where x(t) ∈ R n is the state and d(t) ∈ R m is the input, d ∈ L m ∞ ; g : R n+m → R n is a locally Lipschitz continuous function (that ensures forward existence and uniqueness of solutions of the system at least locally in time), g(0, 0) = 0.For any x 0 ∈ R n and d ∈ L m ∞ , the respective solution is denoted by x(t, x 0 , d) with x(0, x 0 , d) = x 0 .
The definitions and results given in this section are as follows [5], [19].
Definition 1: System (1) is called practically ISS, if there exist β ∈ KL, γ ∈ K, and q ≥ 0 such that If d = 0, then from ISS we recover the conventional global asymptotic stability property.
Definition 2: System (1) is said to possess the practical asymptotic gain (AG) property, if there exist γ ∈ K and q ≥ 0 such that System (1) with a locally Lipschitz continuous g admits (practical) AG if and only if it is (practical) ISS [5].
Definition 3: A locally Lipschitz continuous function V : R n → R + is called practical ISS-LF for system (1) if there exist α 1 , α 2 , η ∈ K ∞ , ∈ K, and r ≥ 0 such that for all x ∈ R n and all Theorem 1 ( [20]): System ( 1) is (practically) ISS if and only if it admits an (practical) ISS-LF.
Remark 1: If an (practical) ISS-LF verifies the properties of Definition 3 only for max{|x|, |d|} ≤ ρ with some ρ > 0, then it is equivalent to local (practical) ISS.
Remark 2: Existence of a practical ISS-LF implies that system (1) admits α −1 1 • as an AG.Indeed, the interval of definition of solutions [0, +∞) can be decomposed on two subsets For any nonempty connected interval [t s , t e ] ⊂ T 2 and t ∈ [t s , t e ], we get which by comparison lemmas [19] implies existence of β ∈ KL such that for all t ∈ [t s , t e ] and β(ζ, 0) = ζ (since V (x(t)) is strictly decreasing on these time intervals).Due to commutations between T 1 and T 2 , V (x(t s )) has two admissible values: V (x(0)) and t s = 0 or V (x(t s )) = max{ ( d ∞ ), r} and t s > 0. Therefore for all t ≥ 0, which by recalling Definition 2 gives the desired result.
In addition, existence of an ISS-LF leads to the ISS property in (1) for simply continuous g or even for a discontinuous one [1].We will need the following result.
Lemma 1: Consider a dynamical system where x(t) ∈ R n is the state, d(t) ∈ R m and u(t) ∈ R p are the inputs, where d ∈ L m ∞ and u ∈ L p ∞ , and : R n × R m × R p → R n is a continuous map.Assume that for u ≡ 0, it admits an (practical) ISS-LF and for any ψ0 > 0 there exist ψ 0 ∈ [0, ψ0 ),ψ 1 ∈ R + , and ψ ∈ K such that and u ∈ R p .Then, there exists a continuous matrix function If is locally Lipschitz continuous, then ψ 0 = 0, while ψ 0 > 0 covers the case with lack of this property at the origin (e.g., a homogeneous system of negative degree).
Proof: According to conditions of the lemma, there exists a locally Lipschitz continuous function V : R n → R + , and In such a case, there exist ρ 0 ∈ R + and ρ ∈ K ∞ such that for all x ∈ R n .Hence, for V (x) ≥ max{ (|d|), r}, we get , r}, we have Construct a continuous function B such that From this estimate, the local practical ISS property can be derived.A variant of this lemma has been proven in [3] for a smooth V and ψ 0 = 0.As in [3], it can be observed that if ∈ K ∞ , then B can be chosen to be independent of d.According to the proof, the constant R can be selected arbitrary large, and R = +∞ for the case ψ 0 = 0; the parameter r can be kept zero if ψ 0 = 0.

B. ISS for Time-Delay Systems
Consider a generic functional differential equation of retarded type [14] where x(t) ∈ R n , and x t ∈ C [−τ,0] is the state function, x t (s) = x(t + s), −τ ≤ s ≤ 0, and τ > 0 is a finite delay [the main results will be presented for a particular example of this model with pointwise delay (4)]; and it provides (at least local) existence and uniqueness of solutions in forward time for system (3).Denote such a unique solution satisfying the initial condition Definition 4 ( [12], [16]): System (3) is called practically ISS, if there exist β ∈ KL, γ ∈ K, and q ≥ 0 such that There exist two methods evaluating ISS property of system (3) based on an LRF or a Lyapunov-Krasovskii functional.The former approach can be formulated as follows [21].
Definition 5: A locally Lipschitz continuous function V : R n → R + is called practical ISS-LRF if the following holds: i) for some α 1 , α 2 ∈ K ∞ and all x ∈ R n : for all ϕ ∈ C [−τ,0] and all d ∈ R m .If r = 0, then such a function V is called ISS-LRF.Theorem 2: If for system (3) there exists an (practical) ISS-LRF, then it is (practically) ISS.

III. MAIN RESULT
In this article, we will consider a particular example of system (3) where d) is a continuous function ensuring existence and uniqueness of solutions at least locally in forward time, and the remaining notation stays the same, F (0, 0, 0) = 0. We will need a local boundedness of F in the sequel.Assumption 1: For any R > 0 and any ε > 0, there exist ε ∈ [0, ε) and L R > 0 such that The case with ε > 0 corresponds to a non-Lipschitz F at the origin.Our first simple observation connects the existence of an ISS-LRF for (4) with ISS of the delay-free system ( Proposition 1: Let there exists τ > 0 such that for any τ ∈ [0, τ ], (4) admits an (practical) ISS-LRF V as in Definition 5, then ( 5) is (practically) ISS with (practical) ISS-LF V and AG α −1 1 • γ d .Proof: In such a case, there exists a locally Lipschitz continuous (practical) ISS-LRF V : R n → R + such that for any τ ∈ [0, τ ] for some α 1 , α 2 ∈ K ∞ , and all and for some α, γ x , γ d ∈ K and r ≥ 0, with γ x (s) < s for all s > 0 max γ x max which is always smaller than V (ϕ(0)) due to the properties of γ x .Therefore, in the limit case with τ = 0 (by replacing the norm for the state from C [−τ,0] with the respective one in R n ), we obtain for any x ∈ R n and any d ∈ R m , which due to Theorem 1 implies (practical) ISS of (5).The AG of the system can be estimated from the last relation using standard arguments (see Remark 2) and it is equal to Remark 3: In conditions of Proposition 1, according to Lemma 1, the system ẋ(t) = F (x(t), x(t) + B(x(t), d(t))u(t), d(t)) is locally practically ISS with respect to both inputs u and d with the gain γ u (s) = α 2 • max{s, α −1 (s)} and some continuous matrix function B with continuous inverse.Since F (x(t), , with respect to solely x(t − τ ) the Lyapunov-Razumikhin condition locally can be written as γ u (κ|x(t − τ )|) ≤ V (x(t)) for some κ > 0, which is implied by γ u (κα −1 1 (V (x(t − τ )))) ≤ V (x(t)).Therefore, since for a local ISS-LRF the relation between V (x(t − τ )) and V (x(t)) is formulated through γ x , we obtain that for some ρ > 0 provided that the gain γ u has been calculated in a tight nonrestrictive way.The last property can be equivalently rewritten as follows: which corresponds to a contraction constrain on AG in (5) with respect to an auxiliary input in the second argument of the function F .Surprisingly, a converse result for Proposition 1 can be obtained under mild auxiliary hypotheses.
Proof: In such a case, according to Definition 3 and Remark 1, for (5) there exists V with some α 1 , α 2 ∈ K ∞ , η, ∈ K and r ≥ 0 such that Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
for all x ∈ R n and all d ∈ R m with max{|x|, |d|} ≤ R for some R > 0. Since all conditions of Lemma 1 are satisfied (the arguments can be repeated for a local practical ISS-LF), there are η ∈ K and a continuous B : R n × R m → R n×n with continuous inverse such that for system (4) with a given τ > 0, we obtain for any φ ∈ C [−2τ,0] (for the sequel we need to consider the state from a lifted space on the interval [−2τ, 0], which is a technical modification) and all d ∈ R m satisfying restriction max{|φ(0)|, |d|} ≤ R (decreasing the value of ε the bounds r and R can be preserved), where Further, consider only behavior of V obtained on the solutions of (4), i.e., when the effect of initial conditions passes through for any φ ≤ R and d ∞ ≤ R, where LR = L R B. Since (a + b) ≤ (2 max{a, b}) for any a, b ∈ R + , the last property follows after: According to (7), there exists γ x ∈ K such that (κs) ≤ γ x • α 1 (s) and γ x (s) < s for all s ∈ (0, ρ] (recall, (7) can be rewritten as −1 (r)}, then, the last implication can be simplified Hence, V is a local practical ISS-LRF, and according to Theorem 2, system (4) is locally practically ISS.
In the case of ISS property of (5) (i.e., r = 0) for ε = 0, the analysis as above can be repeated considering perturbation d t (not as a pointwise input, but as a function on the interval [t − τ, t]), then ISS can be preserved for (4), see the next section for an example.
Remark 4: Note that in the proof, the domain of attraction and the amplitude of admissible inputs are characterized by R and the set of ultimate boundedness is governed by r, which can be chosen arbitrary large or small, respectively, if V is a global ISS-LF.However, the growth of R or decay of r decrease the obtained value of τ .
The condition (7) serves to guarantee that the offset r and the upper bound R stay in (4) the same as in (5), and without (7) the values of r and R cannot be preserved in general (see [21,Th. 3]).The restriction on AG ( 7) is verified if the functions α 1 and are of the same order locally at the origin, then playing with κ it is possible to ensure the required relation.
The result of Proposition 2 can be interpreted as follows: if ( 5) is practically ISS, then (4) is locally practically ISS (there exists a respective ISS-LRF) for any sufficiently small delay, where domain of stability can be arbitrary enlarged at the price of decay of the maximal admissible delay τ , provided that Assumption 1 is satisfied with the restriction (7) imposed on the AG of ( 5) with respect to an additive input in the second argument of F .Proposition 1 provides a kind of converse statement, that existence of practical ISS-LRF for (4) guarantees practical ISS property for (5) [the constraint (7) to the respective AG can be obtained from ( 6) since the functions γ u and are connected].Therefore, existence of an ISS-LRF for (4) can be related with ISS property of the delay-free system (5), which is the main finding of this work.In particular, if F is locally Lipschitz continuous everywhere, for the case of local practical (lp) ISS the following implications are satisfied:

∃lp-ISS-LF
where vertical arrows correspond to existing results given in Theorems 1 and 2.

IV. CASES OF LINEAR AND HOMOGENEOUS SYSTEMS
In this section, two examples are given, when the conditions of Propositions 1 and 2 [together with the constraints on AG (6) and ( 7)] are satisfied.

A. Linear Systems
Consider the linear counterpart of ( 4) where A 0 ∈ R n×n and A 1 ∈ R n×n are constant matrices.The delayfree system (5) can be written as follows: and it is ISS if and only if there exist matrices P, Q ∈ R n×n , such that P = P > 0, Q = Q > 0 and In such a case, an ISS-LF can be chosen as V (x) = x P x whose derivative takes the form: Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
for any x ∈ R n and d ∈ R n , which leads to the estimates given in Definition 3 with α 1 (s) = λ min (P )s 2 , α 2 (s) = λ max (P )s 2 , η(s) = 0.5λ min (Q)s 2 , and , where λ min (P ) and λ max (P ) denote the minimum and maximum eigenvalues of a symmetric matrix P , respectively.Obviously, the condition ( 7) is verified globally 2λmax(P )λmax(P Q −1 P ) .The same V can be considered as an ISS-LRF for the original system [11].First, to formulate delay-independent stability conditions: in such a case if there exist ∈ (0, 1), the symmetric matrices P, R, S ∈ R n×n and a scalar ω d > 0 such that all conditions of Definition 5 can be verified for α and γ x (s) = s.Clearly, the condition ( 6) is satisfied for some κ > 0. Second, to relax these constraints, the delay-dependent stability conditions can be derived by repeating the steps of the proof in Proposition 2: for ∈ (0, 1), find the symmetric matrices P, R ∈ R n×n and scalars ω d , σ > 0 such that then all conditions of Definition 5 are verified with respect to d t for α 1 (s) = λ min (P )s 2 , α 2 (s) = λ max (P )s 2 , γ d (s Again, the condition (6) can be satisfied.Thus, the results of Propositions 1 and 2 can be illustrated on linear systems, where ( 7) is always globally verified, and it is a well-known observation that if a linear delay-free system is globally asymptotically stable (for linear systems it implies ISS), then for a sufficiently small delay, the system is also ISS [11], and above it is shown that in addition it possesses an ISS-LRF.

B. Homogeneous Systems
Linear systems are homogeneous of zero degree [9], let us consider a nonlinear homogeneous system (4) with delays inputs appearing in the same channel In such a case, there exist r i > 0, i = 1, . . ., n called weights (denote r = [r 1 . . .r n ] ) and ν > − min i=1...n r i verifying the equality for any λ > 0 and any x, z ∈ R n , where Λ r (λ) = diag{λ r i } n i=1 is a diagonal matrix having on the main diagonal elements λ r i (it is called the dilation matrix) [8], [10].Following [10, Lemma 2] or [23,Lemma 4], assume that the delay-free counterpart ( 5) is globally asymptotically stable for d = 0, then, there is a continuously differentiable and homogeneous LF V (x) for ( 5) such that for all x ∈ R n , all λ > 0, and any μ > max i=1...n r i (the homogeneity degree of V ), where are all positive constants, and is a homogeneous norm for some χ ≥ max i=1...n r i [9] (it is not a norm in the usual sense since it does not satisfy the triangle inequality, but it gives a distance equivalent to | • |, hence, it can be used for the stability analysis; in addition, by scaling the weights r and the choice of χ it can be made locally Lipschitz continuous [9]).Moreover, V is also an ISS-LF for (5).Indeed, for all x, d ∈ R n we obtain Finally, [10, Lemma 10] or [23, Lemma 4] (see also [22] for the case ν = 0) are the analogues of Proposition 2 in this case, and repeating the steps of its proof we can show that V is an ISS-LRF for a homogeneous system (4) with respect to the input d t .
Proof: Let us just explain the differences with respect to the proof of Proposition 2: here, we replace Assumption 1 by the homogeneity Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
condition.Denote r max = max i=1...n r i and r min = min i=1...n r i , by scaling weights we can always guarantee that r max = 1, then, we can take χ = 1 and the following properties are satisfied for the homogeneous norm: |x + z| r ≤ 2 Proposition 1 in such a case is obviously satisfied together with (6).

V. CONCLUSION
For a nonlinear continuous-time system with pointwise delay, it is shown that how the existence of an LRF can be related with ISS of the delay-free model and delay-dependent ISS of the system itself, which makes the Lyapunov-Razumikhin approach necessary and sufficient for such a robust stability property under mild conditions.The main technical restriction consists in the contraction property of the AG of the system.The obtained results are illustrated on linear and nonlinear homogeneous systems.
Enlarging the class of applicable models and relaxing the technical constraints imposed on AG can be considered as directions of further research works.