Theorem 1. Consider the following differential system:

subject to initial conditions x(t₀) = x₀ where F : [t₀, ∞) × Rⁿ⟶Rⁿ is given by where A : [t₀, ∞)⟶R^n×n has bounded piecewise continuous entries and f_i, g_j : [t₀, ∞) × Rⁿ⟶Rⁿ; i = 1,2, …, p; j = 1,2, …, q are locally integrable functions of t on [t₀, ∞] in the closed ball B_r = {x ∈ Rⁿ : ‖x‖ ≤ r}, where ‖x‖ denotes some vector norm, and suppose that the following two assumptions hold:

•  

  (A1) f_i(t, 0) = 0 and $$; i = 1,2, …, p for all x₁, x₂ ∈ B_r

•  

  (A2) ‖g_i(t, x₁)‖ ≤ γ_irⁱ⁺¹ and $$; i = 1,2, …, q for all x₁, x₂ ∈ B_r

•  

  (A3) The auxiliary unforced differential system

  $$ ()

•  

  is exponentially stable on [t₀, ∞) of stability abscissa −α < 0; that is, there exist real constants α > 0 and K ≥ 1 such that the fundamental matrix Ψ : [t₀, ∞]⟶Rⁿ, which is the mild solution of $$ for t ≥ t₀ with ψ(t₀, t₀) = I_n (I_n being the n-th identity matrix) satisfies $$ for any t₁ and t₂ such that t₂ ≥ t₁ ≥ t₀.

•  

  (A4) The constants β_i and γ_j, i = 1,2, …, p and j = 1,2, …, q, satisfy

  $$ ()

•  

  for some given r > 0.

•  

  Then, the following properties hold:

  • (i)

    $$ if ‖x₀‖ ≤ (r/K) so that differential systems (1) and (2) are globally Lyapunov stable in the closed ball of radius r centred at the origin.

    $$ ()

  •  

    for any given finite ‖x₀‖.

  • (iii)

    Assume that A4 is restricted to a strict inequality, that is,

    $$ ()

  •  

    for some given r > 0. Then, there is a unique globally Lyapunov stable solution of (1) and (2) in the closed ball of radius r centred at the origin for t ≥ t₀ under Assumptions A1, A2, A3, and A5 satisfying

    $$ ()

Proof. The solution of (1) and (2) for t ≥ t₀ is given by

and, by using Assumptions A1 to A3, one gets provided that A4 holds subject to which is got from (9) for t = t₀. Then, Property (i) is proved. On the other hand, since α > 0, one gets from A4 and (9) that (5) holds and Property (ii) is proved. Now, let Cⁿ[t₀, ∞) be the Banach space of continuous, bounded n-vector real functions y : [t₀, ∞)⟶Rⁿ equipped with the norm $$. Let T an operator on Cⁿ[t₀, ∞) represent solution (1) which is defined pointwise via for t ≥ t₀. Then, if ‖x₀‖ ≤ (r/K), one has from (9) that and one has then from (8) for any solutions x₁, x₂ ∈ B_r on [t₀, ∞) under the same initial condition x₀ that which leads to so that T : Cⁿ[t₀, ∞)⟶Cⁿ[t₀, ∞) is a contraction on the closed subset B_r = {y ∈ Cⁿ[t₀, ∞) : ‖y‖ ≤ r} of Cⁿ[t₀, ∞) if A5 holds. This implies from the contraction mapping theorem that solution (8) is unique. From (11) and Assumptions A–A3, one gets

Also, one has from (15) that e^αt‖x(t)‖ ≤ ξ(t), t ≥ t₀ from Bellman–Gronwall Lemma, where ξ : [t₀, ∞)⟶R is the solution of

for t ≥ t₀, which is which implies (7) and proves Proposition 3.

Some direct corollaries are as follows.

Corollary 1. Consider differential system (1) with

with $$. Then, Theorem 1 holds under the same Assumptions A1–A5 with the replacement where the vector-induced matrix norm ‖X‖ = sup_‖z‖=1(‖Xz‖/‖z‖) is used.

Corollary 2. Consider the following controlled differential system:

subject to initial conditions x(t₀) = x₀, where u : [t₀, ∞)⟶R^m is a linear feedback control defined by u(t, x) = K(t)x(t), where K : [t₀, ∞) × Rⁿ⟶R^m is a bounded piecewise continuous matrix control gain and F(t, x) is defined by (18) such that $$ is exponentially stable with stability abscissa −α < 0. Then, Theorem 1 holds with A(t) = B(t) + K(t).

Corollary 3. Assume that in Corollary 2, $$, where $$ is the set of essentially bounded real n-vector functions in t on [t₀, ∞) for all x ∈ Rⁿ and let us define $$. Then, Theorem 1 still holds by replacing (4) by

and (5) by its corresponding strict inequality.

Proof. It follows directly by replacing (8) and (9) by

and following similar arguments to those used to prove Theorem 1.

The following uniform-type result within a bounded ball of the solution of (1) and (2) follows from Theorem 1:

Corollary 4. Assume that A4 is modified by replacing (4) with

where

Then, the following properties hold under Assumptions A1–A3 and A4:

• (i)

  If ‖x₀‖ ≤ (r/K) then $$ for any given r ∈ B_R = {z ∈ R₊ : ‖z‖ ≤ R} so that differential system (1) and (2) is globally Lyapunov stable in B_R.

• (ii)

  Assume that A5 is modified by replacing (5) with

  $$ ()

Then, there is a unique solution of (1) and (2) for t ≥ t₀ under Assumptions A1, A2, A3, and A5 satisfying

Proposition 1. The following properties hold:

• (i)

  Let A₀ be a stability matrix such that $$ for all t ≥ 0 and some real constants α > 0 and K ≥ 1 (with K being norm-dependent) and fix $$. If

  $$ ()

•  

  then ‖x(t)‖ ≤ r; ∀t ≥ 0, and thus, the system is globally Lyapunov stable.

• (ii)

  Let $$ be a stability matrix such that $$ for all t ≥ 0 and some real constants $$ and $$ (with $$ being norm-dependent) and fix $$. If

  $$ ()

•  

  then ‖x(t)‖ ≤ r; ∀t ≥ 0, and this the system is globally Lyapunov stable.

Proof. Note that λ ≤ 1 so that sup_{−h≤τ≤0}‖x(τ)‖ = sup_{−h≤τ≤0}‖φ(τ)‖ ≤ r and there exist some t₀ > 0 such that $$ by continuity of the solution. Now, proceed by complete induction by using (32). Assume that there is a first time instant t₁(>0) ∈ [t − (i + 1)h, t − ih) for some i(∈Z) ∈ [0, maxint(z : z ≤ (t₁/h) − 1)] such that ‖x(t₁)‖ > r. However, this gives a contradiction since then $$ and then there is some t₂ ∈ (0, t₁ − h) which is the first time instant such that ‖x(t₂)‖ > r. As a result, $$ implies that sup_t≥−h‖x(t)‖ ≤ r which proves Property (i). The proof of Property (ii) is similar to that of Property (i) by using (35) instead of (32).

Remark 1. Note that λ_M and $$ are strictly decreasing functions of any delay on the positive real axis. This is seen easily by considering only one delay h = h₁ and define $$ leading to λ_M = λ_M(v) = (αK⁻¹ + v)/(α + v). Assume that h^′ > h, so that v(h^′) > v(h) then λ_M(v^′) < λ_M(v) since, equivalently, 0 < (K⁻¹ + 1)α(v^′(h) − v(h)). A similar conclusion follows for $$. This concern is that we can expect from intuition; that is, the increase in the delay sizes translates into a more strict sufficiently type constraint on the “smallness” constraint on the initial conditions to guarantee that the solution is kept under a certain prescribed closed ball through time.

Two sufficiently type conditions of global asymptotic Lyapunov stability at exponential rate of (30) are obtained in the next result.

Proposition 2. The following properties hold:

• (i)

  Assume that A₀ is a stability matrix such that $$ for all t ≥ 0 and some real constants α > 0 and K ≥ 1. Then, differential system (30) is exponentially stable (i.e., for any given bounded piecewise continuous function of initial conditions) if $$ which is guaranteed for any delays 0 < h_i < h_i+1; i = 1,2, …, ϑ − 1 satisfying $$.

• (ii)

  Assume that $$ is a stability matrix such that $$ for all t ≥ 0 and some real constants $$ and K ≥ 1 so that the delay-free version of (30) is exponentially stable. Then, differential system (30) is exponentially stable if $$ which is guaranteed for any delays 0 < h_i < h_i+1; i = 1,2, …, ϑ − 1 satisfying $$.

Proof. From the first inequality in (32), it follows that

so that, from Gronwall Lemma [1–3], and then

On the other hand, one obtains from the first inequality in (35) and Gronwall Lemma that

Property (i) follows from (40) if A₀ is stability matrix while Property (ii) follows from (41) if A is a stability matrix.

The extensions of Propositions 1 and 2 to the time-varying linear case are direct as follows in the subsequent result:

Proposition 3. The following properties hold if differential system (30) is time-varying:

• (i)

  Assume that

• (A6)

  A_i : [0, ∞)⟶R^n×n, (i = 0.1, …, ϑ) have piecewise continuous and bounded entries.

• (A7)

  the unforced system $$ is exponentially stable, that is, its fundamental matrix of Ψ₀(t, 0), such that $$ with Ψ₀(0,0) = I_n, satisfies ‖Ψ₀(t, 0)‖ ≤ Ke^−αt for all t ≥ 0 with real constants K ≥ 1 and α > 0.

•  

  Thus, if $$ then ‖x(t)‖ ≤ r; ∀t > 0. Also, if $$, then differential system (30) is globally asymptotically stable, with the solution trajectory being constrained to the ball ‖x(t)‖ ≤ r; ∀t > 0, and x(t)⟶0 at exponential rate $$ as t⟶∞.

• (ii)

  Assume that Assumption A6 holds and, furthermore,

• (A8)

  the unforced system $$ is exponentially stable, that is, its fundamental matrix of Ψ(t, 0), such that $$ with Ψ(0,0) = I_n, satisfies $$ for all t ≥ 0 with real constants $$ and $$.

Thus, if $$, then ‖x(t)‖ ≤ r; ∀t > 0. Also, if $$, then differential system (30) is exponentially stable, with the solution trajectory being constrained to the ball ‖x(t)‖ ≤ r; ∀t > 0, so that x(t)⟶0 at exponential rate $$ as t⟶∞.

The following main total stability result gives sufficiency-type global stability conditions of the delayed system in the presence of nonlinearities which generalize those considered in Theorem 1.

Theorem 2. Consider the following differential system:

with A_i ∈ R^n×n (i = 0,1, …, ϑ) which is subject to any bounded piecewise continuous function of initial conditions φ : [t − h, 0]⟶Rⁿ with φ(0) = x₀ where F : [0, ∞) × Rⁿ⟶Rⁿ is given by where A_i : [t₀, ∞)⟶R^n×n (i = 1,2, …, ϑ) have piecewise continuous entries and f_i, g_j : [t₀, ∞) × R^{(ϑ + 1)n}⟶Rⁿ; i = 1,2, …, p; j = 1,2, …, q are locally integrable functions of t on [0, ∞] in the closed ball B_r = {x ∈ Rⁿ : ‖x‖ ≤ r}. Suppose that Assumption A6 and that furthermore the following assumptions hold for some given r > 0: which is guaranteed if α > α₁ + α₂, where

Then, the following properties hold:

• (i)

  There is a unique globally Lyapunov stable solution of (42) and (43) satisfying

  $$ ()

• (ii)

  Assume that α, α₁, and α₂ and $$ in (45)–(47) are replaced with similar equations under the replacements $$, $$ (obtained by replacing $$), Assumption A1 being modified with the replacement $$ and (47) being modified with the replacements $$, $$, and sup_t≥0‖A_i(t)‖⟶2sup_t≥0‖A_i(t)‖ in the denominator of (47) and to define. Then, there is a unique globally Lyapunov stable solution of (42) and (43) satisfying

  $$ ()

Proof. It follows directly by combining Propositions 2 and 3 for differential system (42) and (43) with a similar technique of proof of Theorem 1.

Remarks 2.

• (1)

  It is convenient now to summarize at a first glance which are the basic ideas behind the above results concerning the studied time-delay system.

• (2)

  Note that Proposition 2 gives two results of global asymptotic Lyapunov stability in the large (i.e., for any bounced piecewise continuous function of initial conditions). In this way, the solution trajectory converges asymptotically to the zero equilibrium point at exponential rate. The result is independent of the delay size type if the delays are small enough provided that either the auxiliary system with delay-free dynamics of that with zero delay is exponentially stable.

• (3)

  Note also that Proposition 1 gives two results of global Lyapunov stability independent of the delay sizes for a certain range of values of the delays with the trajectory solution being constrained within a certain closed ball around the origin for t > 0 sufficiently small initial conditions provided that one of the above auxiliary unforced systems is exponentially stable.

•  

  Proposition 3 states results on global stability and global asymptotic stability with the trajectory being constrained to as ball around the origin if one of the two mentioned auxiliary systems is exponentially stable.

• (4)

  Theorem 2 gives two alternative results of solution uniqueness and global stability within a ball around the origin, based on either the stability of the matrix A₀, corresponding to the system without delayed dynamics, or on the stability of the delay-free system of matrix dynamics A, in the case that systems (42) and (43) become forced under forcing functions with eventual delays which satisfy general nonlinear constraints based on extending those invoked in Theorem 1 for the delay-free system. Both results are supported by the constraints that the stability of the matrices A₀ and A guarantee, respectively, the exponential stability of their respective unforced systems independent of the delay sizes within a range of values with a maximum bound.

In particular, Assumption A11 has a direct interpretation as follows: firstly, α > α₁ guarantees that the exponential stability of the unforced differential system $$, which describes the case of absence of delayed dynamics, under any finite bounded conditions z(0) = z₀, guarantees that of system (30) for any bounded piecewise continuous function of initial conditions involving point delays for a certain maximum allowable size of the maximum delay. Secondly, if furthermore, in (46), α₂ is small enough to satisfy α₂ < α − α₁, for given forcing functions f_(⋅), g_(⋅) subject to Assumptions A9 and A10, i.e., such that Assumption A11, is also satisfied; then, the forcing system remains globally Lyapunov stable within the closed ball of radius r centred at the origin and satisfies (48). The alternative constraints invoked to get (49) are interpreted in a similar way based on the assumption of exponential stability under any finite initial conditions of the unforced delay-free differential system $$, which describes the zero-delay case, that is, the case when h_i = 0 for i = 1,2, …, ϑ.

For the following discussion, recall that (F, G) is a controllable pair with F ∈ R^n×n and G ∈ R^m×n if and only if rank(G, FG, …, Fⁿ⁻¹G) = n, equivalently, if and only if rank[sI_n − F, G] = n for any s ∈ C which is not an eigenvalue of F (the Popov–Belevitch–Hautus controllability test). If (F, G) is controllable, then the spectrum of F + GK can be prefixed arbitrarily through the choice of K ∈ R^m×n [4–6].

On the other hand, (F, G) is a stabilizable pair if and only if rank[sI_n − F, G] = n for any s ∈ C with Res ≥ 0 which is not an eigenvalue of F (the Popov–Belevitch–Hautus stabilizability test). If (F, G) is stabilizable, then the spectrum of F + GK can be stabilized (but it cannot be either arbitrarily assigned or prefixed subject to a prefixed stability abscissa) through the choice of K ∈ R^m×n so that, for some K ∈ R^m×n, all the eigenvalues of F + GK can be allocated in the open complex left-hand-side plane Res < 0, [4, 5].

It turns out that if (F, G) is controllable, then it is stabilizable, but the converse is not true.

The subsequent results rely on the stabilization of (42) and (43) though a linear feedback control of the form u(t) = K₀x(t).

Corollary 5. Assume that B₀ ∈ R^n×m, with m ≤ n, are constant matrices and that the pair (A₀, B₀) is controllable and let differential system (42) be defined by

under the assumptions on the initial conditions and matrices of dynamics of Theorem 2 and Assumptions A9–A10 for any set of ϑ point delays 0 < h₁ < h₂ < ⋯<h_ϑ ≤ h < +∞ with any associated matrices of dynamics sup_τ≥0max_1≤i≤ϑ‖A_i(τ)‖ ≤ a < +∞ and any nonnegative real constants β_ijk ≤ β < +∞, γ_ℓjk ≤ γ < +∞; i = 1,2, …, p; ℓ = 1,2, …, q; j, k = 1,2, …, ϑ. Then, for any given quadruple (h, a, β, γ) of nonnegative real constants, there exists a matrix K₀ ∈ R^m×n such that differential systems (42) and (43) are globally Lyapunov stable with its unforced s version being exponentially stable.

Proof. Since (A₀, B₀) is controllable, the spectrum (and then the stability abscissa) of any given matrix Λ₀ ∈ R^n×n can be fixed by some existing K₀ ∈ R^m×n which satisfies the matrix equality A₀ + B₀K₀ = Λ₀. So, for any given triple (a, β, γ), it is possible to fix a stability abscissa $$ of some Λ₀ which satisfies Assumption A11, and then Theorem 2 (i) holds.

Under stabilizability conditions, rather than controllability conditions, of the pair (A₀, B₀), a weaker result than Corollary 5 is now given since the stability abscissa of A₀ + B₀K₀ cannot be arbitrarily prefixed.

Corollary 6. Assume the assumptions of Corollary 5 except for that the pair (A₀, B₀) is stabilizable rather than controllable. Then, a quintuple (h, a, β, γ, K₀) can be fixed such that differential systems (42) and (43) are globally Lyapunov stable with its unforced s version being exponentially stable.

The subsequent result relies on the stabilization of a linear time-delay system with a unique delay h₁ though a linear feedback control of the form u(t) = K₀x(t) + K₁x(t − h). It is found that the exponential stability is achievable for large delay ranges if (A₀, B) and (A₁, B) are, respectively, stabilizable and controllable.

Corollary 7. Consider the following delayed unforced system:

with a control law u(t) = K₀x(t) + K₁x(t − h) and assume that B ∈ R^n×m, with m ≤ n and some matrix feedback control gains K₀, K₁ ∈ R^m×n. Assume that the pairs (A₀, B) and (A₁, B) are, respectively, stabilizable and controllable. Then, for any existing stabilizing matrix gain K₀ of the pair (A₀, B), there exists a matrix gain K₁ such that the above system is exponentially stable for $$ and some $$ for each stabilizing control gain K₀ of (A₀, B) and which depends on the control gain K₁. The stabilizing delay maximum bound $$ can be made arbitrarily large depending on the choice of such a control gain K₁.

Proof. Since (A₀, B) is stabilizable, there is K₀ such that (A₀ + B₀K₀) is a stability matrix of some nonprefixed abscissa −α₀ < 0. Now, from Proposition 2, the above system is exponentially stable if h₁ < (1/α₀)ln(α₀/K‖A₁ + B₁K₁‖) with K(≥1) being norm-dependent. However, since (A₁, B) is controllable, the whole spectrum of A₁ + B₁K is freely assignable by the choice of K₁; then, the spectral radius ρ(A₁ + B₁K₁) of A₁ + B₁K₁, which is arbitrarily close to some matrix norm, can be chosen as small as possible so that the exponential stability holds for arbitrarily large delay according to the following formula:

Corollary 8. Assume that A₀ : [0, ∞)⟶R^n×n is bounded piecewise continuous with $$ and that all the remaining assumptions of Corollary 5 hold with the modifications that (A_0∗, B₀) is controllable and $$ and sup_t≥0(max_1≤i≤ϑ‖A_i(t)‖) ≤ a < +∞. Then, for any given quadruple (h, a, β, γ) of nonnegative real constants, there exists a matrix K₀ ∈ R^m×n such that differential systems (42) and (43) are globally Lyapunov stable with its unforced version being exponentially stable.

Proof. It is identical to that of Corollary 5 by adding a new (zero) delay h₀ = 0 with matrix of dynamics $$ and replacing the lower-bound of α₁ in (45) with the corresponding summation from i = 0 to i = ϑ, i.e., by incorporating the extra norm associated with $$.

Remark 3. It turns out that a similar result to Corollary 8 can be directly formulated by considering that (A_0∗, B₀) is stabilizable rather than controllable.

Remark 4. Extensions of Corollaries 5 to 7 and the direct result got as indicated in Remark 3 are direct from Theorem 2 (ii) for the cases of either controllability or stabilizability of the pair (A, B) for a given B ∈ R^m×n and $$ describing the delay-free auxiliary system. If A(t) is time-varying then, the problem can be stated by extending Theorem 2 (ii) with $$, (A_∗, B) being controllable/stabilizable and $$ being a disturbance matrix of zero delay (see Corollary 7).

It can be of interest to give some ideas about how the above results can be applied when switching through time between alternative active configurations of a time-delay system are performed. For this purpose, consider a nonempty set of Ω distinct configurations S of (42) and (43) such that $$, with $$, which admit switching between them through time given by a switching law $$ which is a piecewise continuous function. $$, with t₀ = 0, is either a set or an infinite sequence (i.e., χ ≤ ∞) of switching instants between configurations; that is, $$ and $$; ∀t ∈ [t_i, t_i+1), and it is said that j is the active configuration on [t_i, t_i+1). Assume that $$ such that

•  

  the configurations of S_s satisfy Theorem 2 so that each corresponding unforced version of (42) and (43) is exponentially stable, so that it has a negative stability abscissa

•  

  the configurations of $$ are $$, where S_cs have critically stable unforced versions so that each corresponding unforced version of (42) and (43) has a zero stability abscissa while the configurations of S_us are unstable so that each corresponding unforced version of (42) and (43) has a positive stability abscissa

Theorem 3. Assume that $$ with S_s ≠ ∅ and that there is at least one S^∗ = S_i∗ ∈ S_s with a pair (K^∗, −α^∗) with α^∗ > 0 satisfying (α^∗/K^∗) > R for some given finite real constant R > 0. Then, for any given admissible function of initial conditions satisfying $$, there always exists switching laws $$ with a finite number of switches and switching time instants {t₀, t₁, …, t_k} such that σ(t_k) = i^∗ and ‖x(t)‖ ≤ r_i∗; ∀t ≥ t_k.

Remark 5. Note that, in particular, t₁ = t_i∗ is such a switching law of only one or two switching time instants, the initial one at t₀ = 0, which can be at a distinct active configuration of S_i∗ if S ≠ S_i∗ or directly at S_i∗, necessarily if S = S_i∗, and the second one at t₁ = t_i∗ which is only feasible if S ≠ S_i∗.