ON THE UNIFORM EXPONENTIAL STABILITY OF LINEAR TIME-DELAY SYSTEMS

This paper deals with the global uniform exponential stability independent of delay of timedelay linear and time-invariant systems subject to point and distributed delays for the initial conditions being continuous real functions except possibly on a set of zero measure of bounded discontinuities. It is assumed that the delay-free system as well as an auxiliary one are globally uniformly exponentially stable and globally uniform exponential stability independent of delay, respectively. The auxiliary system is, typically, part of the overall dynamics of the delayed system but not necessarily the isolated undelayed dynamics as usually assumed in the literature. Since there is a great freedom in setting such an auxiliary system, the obtained stability conditions are very useful in a wide class of practical applications.


Introduction.
The stability and feedback stabilization of time-delay systems subject to constant point and distributed delays as well as time-varying ones has received important attention in the last years (see, e.g., [1,2,4,5,6,8,10,11,13]). A key point is that a system exhibiting stability in the absence of delays may lose that property for small delays and, in contrast, a stable delayed system may lose the property in the absence of delay (see, e.g., [1,6,8]). This paper deals with the global uniform exponential stability independent of delay (g.u.e.s.i.d.) of a class of homogeneous time-delay systems subject to combined point and distributed delays as well as integrodifferential Volterra-type delayed dynamics. The global stability is investigated for any function of initial conditions being everywhere continuous on its definition domain, a real interval [−h, 0], where h is the maximum delay in the system, except possibly on a set of zero measure where the function possesses bounded discontinuities. Necessary and sufficient global uniform stability conditions independent of delay are obtained if the delay-free system is globally uniformly exponentially stable (g.u.e.s.) and an auxiliary system is g.u.e.s.i.d. The obtained results are then applied to a number of particular cases of interest by setting different auxiliary systems including the standard delay-free one. The mathematical proofs are based on conditions which guarantee that a linear operator in a Banach space is compact within a domain that contains the closed complex right half-plane provided that another one defined for the auxiliary system is also compact within a (non necessarily identical) domain that contains the closed complex right half-plane. The auxiliary system may be a delay-free one or, in general, any particular parametrization of the whole system under study where part of the delayed dynamics is deleted. Some sufficient conditions for the system to be g.u.e.s. dependent on delay are also obtained by using the same mathematical outlines. Extensions are given for the case when the system is forced by impulsive inputs and also by considering the closed-loop stabilization of time-delay systems of the given class. The paper is organized as follows. Section 2 deals with the class of homogeneous delayed systems under study and with the definition of the auxiliary system. Section 3 is devoted to the main uniform stability result and the related ones for some particular auxiliary systems of interest. Some of those systems are defined by considering only delay-free dynamics or either pointdelayed, distributed-delayed, or even Volterra-type delayed dynamics together with a delay-free dynamics. Section 4 extends the above results to the presence of impulsive forcing functions. Section 5 is devoted to the stabilization of closed-loop systems of the given class under linear state or output-feedback controllers which can include delays. Some simple examples are discussed in Section 6 and, finally, conclusions end the paper.  3 ) denotes a triple for sets of indices referred to as the particular subsets of real constants describing point delays (J 1 ), infinitely distributed Volterra-type delays (J 2 ), and finitely distributed delays (J 3 ) of the system which are also present in the auxiliary system. For instance, 1 ∈ J 1 ⇒ h 1 > 0 is a point delay of the time-delay system which is also present in the auxiliary system, and so on. Also, Card(J 1 ) ≤ m, Card(J 2 ) ≤ m + 1, Card(J 3 ) ≤ m . If a pure convolution Volterra-type dynamics t 0 dα 0 (τ)A α 0 x(t −τ) is present, then it is described by a fictitious delay h 0 = 0. If such a term is not present, then Card(J 2 ) ≤ m . The remaining infinitely distributed delays give contributions t 0 dα i (τ)A α i x(t −τ −h i ), with finite real constants h i > 0 with i = 1, 2,...,m , toẋ(t) which are point delays under the integral symbol. It is said that the delays are infinitely distributed because the contribution of the delayed dynamics is made under an integral over is an operator-valued function with domain in C × R m+m +m +1 , whereĥ = (ĥ T 1 ,ĥ T 2 ) T and h 0 = 0 for s = jω, anyĥ with ϕ m+1 = 0, and remaining components ϕ i in [−π,π] whose values depend on h i (i ≤ m) or h i (i ≥ m + 2). Similarly, for the aboveĥ and ϕ. Note thatT ,T , and N −1 are distinct mathematical objects but, however, they take identical values for all pure imaginary s = jω and a corresponding ϕ i ∈ [−π,π] such that e −jω i = e ±jϕ i with ϕ m+1 = h 0 = 0. The same applies to the related objects referred to as the auxiliary system.

Problem statement.
Consider the following linear and time-invariant system subject to point and distributed dynamics and to an impulsive function: where A 0 and A i , A α k (i = 1, 2,...,m; k = 0,1,...,m + m ) belong to the spaces of unbounded and bounded operators, respectively, on a Banach space of n-vector real functions x ∈ X endowed with the supremum norm where the vectors of point and distributed constant delays areĥ = (0,h 1 ,h 2 ,...,h m ) andĥ = (ĥ T : h m +1 ,h m +2 ,...,h m +m ) T , respectively, with h i ≥ 0 and h k ≥ 0 (i = 1, 2,...,m + m ) and h 0 = h 0 = 0, A 0 ≡ A, A α 0 ≡ A α , and α 0 (·) ≡ α(·). The functions α i : [0, ∞) → R and α k : [0,h k ] → R are continuously differentiable real functions within their definition domains except possibly on sets of zero measure where the time-derivatives have bounded discontinuities. All or some of the α i (·) and α k (·) may be alternatively matrix functions, α i : [0,t] → R n×n for t ∈ R + and α i : [0,h k ] → R n×n . We will not make any explicit difference between both possibilities in the notation for the sake of simplicity. The impulsive input v(t) = i∈I b i δ(t − t i ) is built with the finite or infinite sequence of Dirac impulses δ(t − t i ) at the sequence of time instants {t i ; i ∈ I} with t i+1 > t i for some totally ordered proper or improper numerable subset I ⊆ N. If Card(I) = p < ∞, then v(t) := p i=1 b i δ(t −t i ) and I := {i ∈ N : i ≤ p}. Note that system (2.1) is very general since it includes point-delayed dynamics like, for instance, in typical war/peace models or the so-called Minorski's problem appearing when controlling the lateral dynamics of a ship. It also includes real constants h i (i = 0, 1,...,m ), with h 0 = 0, associated with infinitely distributed delayed contributions to the dynamics through integrals, related to the α i (·), i = 0, 1,...,m . Such delays are relevant, for instance, in viscoelastic fluids, electrodynamics, and population growth [1,4,6]. In particular, an integrodifferential Volterra-type term is also included through h 0 = 0. Apart from those delays, the action of finite distributed delays characterized by real constants h i (i = 0, 1,...,m + m ) is also included in (2.1). That kind of delays is well known, for instance, in econometric models related to production rate [4]. Finally, the impulsive input v(t) = i∈I b i δ(t −t i ) generates bounded discontinuities of the solution trajectory x(t) at t = t i (i ∈ I), see, for instance, [10,11,12]. The following technical hypothesis are made.
(H1) All the operators the set of linear operators on X of dual X * , and h k and h (k = 1, 2,...,m; = 0, 1,...,m + m ) are nonnegative constants with h 0 = h 0 = 0 and h = Max(Max 1≤i≤n (h i ), Max 1≤i≤m +m (h i )). (H2) The initial conditions of (2.1) are real n-vector . In the following, the supremum norms on L(X) are also denoted with |·|. , v ≡ 0) for each initial condition φ ∈ C e (h). Take Laplace transforms in (2.1) by using the convolution theorem and the relations dα(τ) =α(τ)dτ. It follows that dα i (s) = sα i (s) − α i (0), wheref (s) denotes the Laplace transform of f (t). Thus, one gets from (2.1) Note that (2.1) is guaranteed to be g.u.e.s.i.d. if and only ifT (s) exists within some region including properly the right complex plane, in other words, if it is compact for Re s > −α 0 , for some constant α 0 ∈ R + , since then all the entries of its Laplace transform T (t) decay with exponential rate on [0, ∞) for φ ∈ C e (h), and then |x(t)| decays with exponential rate on R + . The unique solution of the homogeneous system (2.1) for each φ ∈ C e (h) may be equivalently written in infinitely many cases by first rewriting (2.1) by considering different "auxiliary" reference homogeneous systems plus additional terms considered as forcing actions. The next arrangements lead to conditions guaranteeing that the homogeneous system (2.1) is g.u.e.s.i.d. if it is g.u.e.s. in the absence of delay (i.e., for h = 0). Through this arrangement, it is not necessarily requested forż(t) = Az(t), which is in fact one of the possible auxiliary homogeneous systems for (2.1), to be g.u.e.s.i.d. for any φ ∈ C e (h). Thus, note that (2.1) may be compactly written aṡ where L = L JM +L JM is a linear operator in L(X) defined by Lx t equalized by the unforced right-hand side of (2.1), where x t denotes the string x : [t − h, t] → X of the solution to (2.1) for φ ∈ C e (h) for all t ≥ 0, and L JM andL JM are also linear operators in L(X) which define a nonunique additive decomposition of L that depends on M, an n-square arbitrary real matrix, and J, a triple J = (J 1 , The M-matrix and the J-triple define the subsequent g.u.e.s.i.d. auxiliary system. That property is the starting point to derive conditions for the current delayed system (2.1) to be g.u.e.s.i.d. as well. The auxiliary system iṡ , some given matrix M ∈ R n×n , and are respective proper or improper subsets of In view of (2.7), the unique solution of (2.1) for any φ ∈ C e (h) is At is an analytic semigroup if J 1 and J 3 are empty, and Remark 2.1. Note that the compactness of the operator-valued functionsT (s) and T JM (s) for all Re s > −γ and Re s > −γ JM , some γ ∈ R + and γ JM ∈ R + , respectively, ifẋ(t) = Lx t ,ż(t) = Lz t , respectively, are g.u.e.s.i.d for all φ ∈ C e (h), holds directly if they are bounded provided that X is considered as a Hilbert space endowed with the usual inner product norm. The stability properties of the operator-valued function T : [0, ∞) → L(X) are independent of the use of any of both alternative formal characterizations. Thus, if X is a Hilbert space, then there exist dense injective mappings X → X * (dual of X)→ X * * (dual of X * ) ≡ X, instead of the generic result which may include in some cases proper inclusion X * * ⊃ X ≠ X * * so that X is a reflexive linear space and any operator in L(X * * ,X) (≡ L(X, X) = L(X) in this case) is compact if and only if it is completely continuous (i.e., if it maps any weakly convergent sequence into a strongly convergent one with respect to the norm topology). Thus,T (s) is compact (or completely continuous) where it exists since (T ) * ·T is bounded for Re s > −γ. The same property holds for anyT JM for Re s > −γ JM .
in the definition domain ofT JM for any auxiliary system defined from some given J-triple. The following special cases are of interest.

Special cases.
(1) The auxiliary system is delay-free: . This is the case usually treated in the literature (see, e.g., [4,6]). Thus,J i = N i (i = 1, 2, 3) and T JM (t) = e Mt is an analytic semigroup.
(2) The auxiliary system is subject to delay-free and all point delays: The auxiliary system is subject to delay-free dynamics and Volterra-integraltype dynamics: In particular, T JM (t) is ensured to be a transition operator with |T JM (t)| ≤ Ke −ρt , for some positive real constants K and ρ and all t ≥ 0 (see, e.g., [1,9] is compact for Re s > −ρ, any real constant ρ < γ JM , and |d i ( (4) The auxiliary system has delay-free dynamics and all the infinitely distributed delays: under initial conditions φ ∈ C e (h). Thus, one getṡ for t > 0 with T JM (0) = I, T JM (t) = 0 for t < 0, whose unique solution for all t > 0 is The auxiliary system has delay-free dynamics and all the finitely distributed delays: Under the same initial conditions as in the above case, one getsż(t) = , which is also satisfied by the transition operator of the auxiliary system whose unique solution under the same initial conditions as in case (4) is (2.14)

)) if and only if the operator-valued functioñ
Proof. First note that the argument ω = 0 for the above operator-valued function is excluded from the conditions since (2.1) is g.u.e.s. forĥ = 0. System (2.1) is g.u.e.s.i.d. if and only if N −1 (s,ĥ) exists for Re s > −γ (some γ ∈ R + ) for any sets of delays. Since and N JM (s,ĥ) has an inverse Re s > −γ JM (some γ JM ∈ R + ) for all the sets of delays explicit in the auxiliary system (2.5), N −1 (s,ĥ) exists for a pair (s,ĥ) if and only ifÑ −1 (s,ĥ) exists for (s,ĥ), wherẽ Proof of necessity. The rank condition cannot fail for ω = 0 since system (2.1) is g.u.e.s. and then N(j0,ĥ) is full rank for any set of delays. Assume that rank[T JM (jω, ϕ)] < n for some ω ≠ 0, then rank[Ñ JM (jω,ĥ)] < n and the set of delays h i = ϕ i /ω is the ith component of ϕ. This is a contradiction and necessity follows. N −1 (s,ĥ)) are compact wherever they exist, any possible singularities ofT (s) andT JM (s) are poles [6,9] 1, 2,...,m + m + 1) with h 0,m+1 = 0. But then, from the definition ofT (jω, ϕ 0 ), [−π,π] always exists such that the ranks of T (jω, ϕ 0 ) andT JM (jω, ϕ 0 ) are less than n and the result has been proved. Note that the test for negative ω is unnecessary since eventual complex poles appear in conjugate pairs. Theorem 3.1 may be used in particular for the special cases of Section 2 as follows.
The global uniform exponential stability of (2.1) may be investigated provided that each group of delayed dynamics (like, for instance, all point delays, infinitely distributed delays, or finitely distributed ones) is successively introduced in the system as addressed as follows. Note, for instance, that the system with combined delay-free and point-delayed

the auxiliary system with both undelayed and point delayed dynamics is g.u.e.s. so that Corollary 3.3 holds with M = A).
We might proceed in that way by giving conditions that ensure that each added group of delays maintains the uniform stability independent of delay provided that it was g.u.e.s.i.d. before adding those delays. It is also interesting to derive conditions for losing or ensuring uniform stability dependent on delay as follows.
Proof (outline). The proof follows directly since for such sets of delays, the proof of Theorem 3.1 fails since there is some pole ofT (s), so that it is not holomorphic, on Re s ≥ 0 sinceT (jω, ϕ) does not have an inverse for some ω ∈ R + and ϕ = Remark 3.9. Note that in Theorem 3.8 the rank of an the operator-valued function (3.1) has to be tested in order to ensure the existence of its inverse within an appropriate stability domain. If the auxiliary system is not g.u.e.s.i.d., the test directly fails. On the other hand, since the eigenvalues of the operator-valued function are continuous functions of the arguments and since such a function is continuously differentiable with respect to its arguments, the implicit function theorem ensures that if the test does not fail at a set of delays (or constants h (·) characterizing distributed delays), it does not fail either within open neighborhoods of such delays (or constants). Thus, the system is g.u.e.s for delays in some open neighborhoods of the h (·) -and h (·) -constants where the system is g.u.e.s.

Uniform stability under impulsive forcing terms.
The stability under impulsive forcing terms in (2.1) may be formulated under a direct extension of the basic results of Section 3 as follows.  ∈ (0,ρ), and K i ∈ R + being bounded constants for all i ∈ I. Thus, the solution of (2.1), x(t, φ), is bounded on R + and x(t, φ) → 0 exponentially as t → ∞ for any φ ∈ C e (h).
Proof. Let x 0 (t, φ) be the unique solution of the homogeneousẋ(t) = Lx t for t ≥ 0 for any given φ ∈ C e (h). Thus, the unique solution x(t, φ) for t ≥ 0 for identical φ ∈ C e (h) of the forcedẋ(t) = Lx t +u(t), with v(t) = i∈I b i e −(t−t i ) , is bounded on R + and satisfies It only remains to consider the case when Card(I) = ∞.
Then, it is also exponentially continuous over I. Since the solution x(t, φ) of (2.1) is continuous over the finite intervals of nonzero measures [t k ,t k+1 ), k ∈ I, it cannot diverge within such intervals. Thus, x(t, φ) is bounded and converges exponentially to zero as t → ∞.
Assume that at discontinuity points the solution trajectory satisfies that is, the variation function at the discontinuity points of the trajectory is bounded and the increments converge asymptotically to zero provided that  for all i ∈ I, and τ ∈ [0,T i ). Thus, from (2.1),

for all φ ∈ C e (h) and any impulsive v(t) = i∈I b i δ(t − t i ) with Card(I) being finite or infinite if any of the subsequent conditions holds for all
Taking Euclidean norms in the above relations, one gets with R : [0, ∞) → R n×n being a matrix function that defines the factored representation (4.5) The recursive use of (4.5) for all i ∈ I while relating x(t 1 ,φ) to initial conditions φ : (4.6) A sufficient condition for global uniform exponential stability independent of delay, after excluding any finite number of consecutive impulses in (2.1), what is irrelevant for stability analysis, is e −iγT min i k=1 ( I +B k+ ) < 1 for any finite integer i ≥ 0 provided that t i+1 −t i ≥ T min for any integer i ≥ . Thus, it follows that (2.1) is g.u.e.s.i.d. under (i) by taking logarithms in the above inequality. It is proved that (2.1) is g.u.e.s.i.d. under (ii) by replacing e −iγT min by e −t i = e − i−1 k=1 T k . The fact that (2.1) is g.u.e.s.i.d. under (iii) is direct since the fulfilment of (iii) guarantees that of (ii).

Simple stability tests and stabilization of the closed-loop system.
In the subsequent study, consider the unforced system (2.1). The discussion is limited to the case of delay-free combined point-delayed dynamics in (2.1); that is, m = m = 0. The extension to the general case is direct. The auxiliary system isż(t) = Mz(t), that is, exists for all ω ∈ R + g.u.e.s. forĥ = 0, that is, for any bounded x(0) = φ(0) ∈ R n . Note that (jωI − M) −1 exists for all ω ∈ R + since M is strictly Hurwitzian. Consider the set H ∞ (X) = {x : C 0 + → X : Sup Re s>0 ( x(s) ) < ∞}, where C 0 + is the complex open righthand, side half-plane. A similar H ∞ space is defined for the set of linear operators on X by replacing X→L(X, X). Note thatT ∈ H ∞ (L(X, X)) where it exists. Simple calculations for H ∞ -norms yield  has an eigenvalue on the imaginary axis where R 0 + := R + ∪{0} and · 2 denotes the l 2 -matrix norm for each ω ∈ R + . Now consider the following feedback system subject to m internal and m external (denoted in the sequel as h i ) point delays:ẋ 1, 2,...,m ), where the control function u : [0, ∞) → R q is continuous and has range U , that is, u ∈ C (0) ([0, ∞); U) while being generated from the control law: with real matrices K, K i ∈ R q×r , C ∈ R r ×n . It is assumed that y is an r -measurable output signal y : [0, ∞) → R r defined by y(t) = Cx(t) for all t ≥ 0. Taking Laplace transforms in (5.4) with zero initial conditions with s = jω, one directly gets the closedloop relations The substitution of (5.7) into (5.6) yieldsŜ c (jω)x(jω) = 0 witĥ The closed-loop system is g.u.e.s.i.d. ifT −1 c (jω) exists provided that S −1 u (jω) exists for all ω ∈ R + . The following result holds.
The subsequent cases are of interest to address Theorem 5.1.
Case A (C = I (i.e., linear state feedback) and (A, B) is a completely controllable pair). Thus, the eigenvalues of (A+BKC) and then those of (M −A−BKC) may be prefixed to arbitrary positions in Re s < 0 (see, e.g., [3,7,13]) and then any norm of (M − A − BKC) may be made as small as required. Furthermore, γ M may be as small as suitable to fulfil (5.9) for any sets of controller gains K (·) and K (·) that fulfil | m i=1 K i | 2 < 1. Assume, for instance, that the eigenvalues of (A + BKC) are chosen identical to those of M located at Re s≤ − ρ < 0. Assume also that B and C are full column and row rank, respectively. Thus, γ M ≤ 1/ρ n if K is chosen as follows: where P ⊗ Q = (p ij Q) is the direct Kronecker product of the a × b and c × d matrices P and Q, respectively, and where m v and a v are column vectors formed with the consecutive rows m T (·) of M and a T (·) of A, respectively, written in order with ordered entries. It has been used that (P ⊗ Q) T = P T ⊗ Q T and that B T B and CC T are both nonsingular since B and C are full column and row rank, respectively.
Case B (C = I, (A, B) is stabilizable (but not controllable), and (A i ,B) are completely controllable for i = 1, 2,...,m). Thus, a finite gain γ A+BK = (sI − A − BK) −1 ∞ can be designed for each given A-matrix but it cannot be prefixed. Thus, the controller gain matrix K may be chosen so that the controllable and observable modes of (A+BK) are arbitrarily close to those of M [3,7]. Also, the controller gains K i may be calculated so that (A i + B i K i ) are zero (i = 1, 2,...,m) and K i such that K i are arbitrarily small for i = 1, 2,...,m . Thus, condition (d) of Theorem 5.1 may be fulfilled for any finite γ A+BK (i.e., for any K that stabilizes the stabilizable pair (A, B)).
Case C (C ≠ I (i.e., output feedback is used) and the triple (A,B,C) is controllable and observable with rank(B) = q, rank(C) = r , and max(q, r ) ≥ n). Thus, the eigenvalues of (A + BKC) may be prefixed to positions being arbitrarily close to prescribed ones inside the closed left-half complex plane, and any norm of (M −A−BKC) may be made as small as convenient for design purposes. Also, γ A+BKC may be made arbitrarily small, and the design to accomplish with (see, in particular, (5.7)) may be performed similarly as in Case A.
Case D (C ≠ I (i.e., output feedback is used) and the triple (A,B,C) is stabilizable and detectable with rank(B) = q, rank(C) = r , and max(q, r ) ≥ n). Furthermore, (A i ,B,C) is controllable and observable (i = 1, 2,...,m). The design may be performed as in Case C.

Remark 5.2. Once
The global uniform exponential stability of the inhomogeneous closed-loop system has been achieved, then it may be guaranteed to be g.u.e.s. under impulsive forcing signals by establishing additional conditions as in Theorems 4.1 and 4.3. The above analysis dictates that the absence or presence of controller external delays is irrelevant for design purposes. Note, in particular, that condition (c) of Theorem 5.1 is satisfied directly if the related controller gains are zeroed.
The extensions of the above results in this section to the presence of distributed delays are not difficult. Assume, for instance, that the state (or only the output) is available for measurement, that is, C = I (or C ≠ I), and that there are distributed delays in the system. Thus, the control law (5.7) may be generalized to 0, 1,...,m + m ); it follows that |α i (t)| ≤ Ke (ε−β)t , for all t ≥ 0 and any real constant ε > 0, so that Max 0≤i≤m +m (|α i (s)|) ≤ K/|s + β − ε| < K/|s + β| for Re s < −γ. Then, Thus, condition (d) of Theorem 5.1 becomes (after substituting (5.12) into (2.1), via (5.13), and obtaining a relation in Laplace transforms for the closed-loop system description)

Examples
. If a > 0, then Theorem 5.1 yields γ a = (s − a) −1 ∞ = 1/|a| and the system is g.u.e.s.i.d. if 1 > γ a |a 1 | Sup ω∈R 0 + (|e −jhω |) = γ a |a 1 | provided that the auxiliary systeṁ z(t) = −az(t) with z(0) = z 0 is g.u.e.s., that is, a > 0. Thus, the system is g.u.e.s.i.d. if a > |a 1 | > 0. The same conclusion is obtained by applying Gronwall's lemma [12], as follows. Compute the solution to the system of differential equation to obtain ). Thus, exponential stability follows for a > |a 1 | > 0. Assume, for instance, that a < 0 so that the auxiliary system is unstable. Thus, use the delayfree control law u(t) = kx(t) with k > −a. Thus, the above results hold by replacing a→k − |a| so that the closed-loop uniform exponential stability independent of delay is ensured if k > |a| + |a 1 | still from Theorem 5.1. Note that Theorem 3.  − t i ). Thus, the unique solution for any admissible n-vector real function φ of initial conditions is with T (0) = I and T (t) = 0 for t < 0. Several situations are now discussed. (a) Assume that the auxiliary systemż(t) = Az(t) is uniformly exponentially stable for all bounded z(0), that is, A is strictly Hurwitzian with stability abscissa −ϑ = (−ϑ i ) < 0 and the associated dominant eigenvalue µ = µ i . Assume also that there is no impulsive action, that is, all the b i are zero. Thus, from Theorem 5.1, the current delayed system is guaranteed to be g.u.e.s.
If A is not strictly Hurwitzian, then assume that a control u(t) = Kx(t) is applied through the control matrix B with (A, B) being controllable. Thus, the delay-free closedloop dynamics can be defined by the strictly Hurwitzian n-matrix M = A + BK (which may be chosen as the delay-free dynamics of the auxiliary system) of the same stability abscissa and dominant eigenvalue multiplicity as above. Thus, the closed-loop system is g.u.e.s.i.d. if |ϑ| µ > m i=1 A i 2 for a controller gain matrix K being an existing solution of (B ⊗ I) (c) If (A, B) is only stabilizable, then the stability abscissa is Min(−ϑ, −ϑ ) < 0, where (−ϑ) < 0 is obtained from the relocated closed-loop controllable poles through the controller gain matrix K and (−ϑ ) < 0 is the stability abscissa of the uncontrollable openloop stable (since the system is stabilizable) poles which cannot be relocated through feedback. Thus, the delayed system is g.u.e.s.i.d. if Max(|ϑ| µ , |ϑ | µ ) > m i=1 A i 2 . (d) If the state is not available for measurement but (A,B,C) is controllable and observable or (at least) stabilizable and detectable for appropriate control and output matrices B and C, respectively, then the closed-loop stabilization problem may be solved in light of Theorem 5.1.
(e) Now, assume that the impulsive input is nonzero. If there is a finite number of impulses, then the above conditions of uniform stability still remain valid. If there is an infinite number of impulses b i = B i x(t − i ), then the global uniform stability independent of delay is preserved if all the time intervals in-between two consecutive impulses satisfy the lower-bound constraint T min ≥ Sup k∈I (1/iγ ) Then, a sufficient condition for the current system to be g.u.e.s.i.d. when no impulsive input is injected is that 1 > γ aux m i=2 A i 2 , which is guaranteed if 1 > γ A m i=2 A i 2 /(1 − γ A A 1 2 ) provided that A 1 2 < γ −1 A = |ϑ A | µ A . If A + A 1 e −hs has stable eigenvalues but A is not strictly Hurwitzian, that is,ż (t) = Az (t) is not g.u.e.s.i.d., then γ aux is finite but it cannot be calculated from sufficiencytype conditions for stability using (6.4). However, the system is still g.u.e.s.i.d. if 1 > γ aux m i=2 A i 2 .
(h) Now, assume that in case (g) there is an impulsive input as in case (e) consisting of infinitely many impulses. Thus, the current system is g.u.e.s.i.d. if the impulses occur at consecutive times being not less than Sup k∈I (1/iγ aux ) i k=1 I + B k , except possibly on a set of zero measure, with γ −1 → γ −1 , from Theorems 4.3(ii) and 5.1.
Example 6.3. Consider the second-order scalar functional equationẍ(t) = −aẋ(t)+ bx(t − h). The equation is decomposed into two first-order equations as follows: (see [4]), whose time-derivative iṡ (6.7) Thus, the system is globally asymptotically stable, dependent on delay if (−2(a/h) + ξ −b) < 0 for some real constant ξ > 0 if x i (t), for i = 1 or 2, is nonzero for some subinterval of nonzero measure of [t −h, t], any t ≥ t 0 (some finite t 0 ∈ R 0 + ), or, equivalently, Max(0, −b) < ξ < 2(a/h)+b. This holds for all h > 0 if Min(a, b) ≥ 0 and a and b are not simultaneously zero. A general necessary condition for a given h is that (a/h + b) > 0 or, equivalently, h > −a/b. Note also that a necessary condition for exponential stability for h = 0 is that a > 0 and b < 0, which follows from the Routh-Hurwitz criterion. As a result, if a > 0 and b < 0, then the system is globally uniformly asymptotically stable if h ∈ [0,a/|b|). Theorem 4.3 may be applied as follows. Decompose the system of second-order functional differential equations into two first-order differential equations as follows:ẋ where The stability abscissa of A is (−a) if a < 2|b| or −(a − a 2 − 4|b|) otherwise. Thus, the system is g.u.e.s.i.d. if a > 0, b < 0, and a > 2|b|+ a 2 − 4|b| since the stability condition for a ≥ 2|b| is (a − √ a 2 − 4b) > 2|b| (Theorem 5.1), and for a < 2|b|, Theorem 5.1 fails in (6.4).

7.
Conclusions. This paper has dealt with the global uniform exponential stability independent of delay (g.u.e.s.i.d.) of a class of homogeneous time-delay systems being possibly subject to combined point and distributed delays as well as integrodifferential Volterra-type delayed dynamics. The global stability is investigated for any real function of initial conditions being everywhere continuous on its definition domain, a real interval [−h, 0], where h is the maximum delay in the system, except possibly on a set of zero measure where the function of initial conditions possesses bounded discontinuities. Necessary and sufficient global uniform stability independent of delay conditions has been obtained if the delay-free system is globally uniformly exponentially stable (g.u.e.s.) and an auxiliary system is g.u.e.s.i.d. The obtained results have then been applied to a number of particular cases of interest by setting different auxiliary systems including the standard delay-free one. Furthermore, some extensions have been given for the case when the system is forced by impulsive inputs consisting of either a finite number of impulses or infinitely many impulses. It has been assumed either that the impulse amplitudes vanish exponentially or that the time interval between two inputs exceeds a prescribed threshold of sufficiently large length. Some extensions have been given by considering the closed-loop stabilization of time-delay systems of the given class. Finally, some illustrative examples have also been presented.