On the Global Asymptotic Stability of Switched Linear Time-Varying Systems with Constant Point Delays

Abstract

This paper investigates the asymptotic stability of switched linear time-varying systems with constant point delays under not very stringent conditions on the matrix functions of parameters. Such conditions are their boundedness, the existence of bounded time derivatives almost everywhere, and small amplitudes of the appearing Dirac impulses where such derivatives do not exist. It is also assumed that the system matrix for zero delay is stable with some prescribed stability abscissa for all time in order to obtain sufficiency-type conditions of asymptotic stability dependent on the delay sizes. Alternatively, it is assumed that the auxiliary system matrix defined for all the delayed system matrices being zero is stable with prescribed stability abscissa for all time to obtain results for global asymptotic stability independent of the delays. A particular subset of the switching instants is the so-called set of reset instants where switching leads to the parameterization to reset to a value within a prescribed set.

1. Introduction

Time-delay systems are receiving important attention in the last years. The reason is that they offer a very significant modeling tool for dynamic systems since a wide variety of physical systems possess delays either in the state (internal delays) or in the input or output (external delays). Examples of time-delay systems are war/peace models, biological systems, like, for instance, the sunflower equation, Minorsky's effect in tank ships, transmission systems, teleoperated systems, some kinds of neural networks, and so forth (see, e.g., [1–9]). Time-delay models are useful for modeling both linear systems (see, e.g., [1–4, 10]) and certain nonlinear physical systems, (see, e.g., [4, 7, 8, 9, 11]). A subject of major interest in time-delay systems, as it is in other areas of control theory, is the investigation of the stability as well as the closed-loop stabilization of unstable systems, [2, 3, 4, 6–13] either with delay-free controllers or by using delayed controllers. Dynamic systems subject to internal delays are infinite dimensional by nature so that they have infinitely many characteristic zeros. Therefore, the differential equations describing their dynamics are functional rather than ordinary. Recent research on time delay systems is devoted to numerical stability tests, to stochastic time-delay systems, diffusive time-delayed systems, medical and biological applications [14–17], and characterization of minimal state-space realizations [18]. Another research field of recent growing interest is the investigation in switched systems including their stability and stabilization properties. A general insight in this problem is given in [19–21]. Switched systems consist of a number of different parameterizations (or distinct active systems) subject to a certain switching rule which chooses one of them being active during a certain time. The problem is relevant in applications since the corresponding models are useful to describe changing operating points or to synthesize different controllers which can adjust to operate on a given plant according to situations of changing parameters, dynamics, and so forth. Specific problems related to switched systems are the following.

(a) The nominal order of the dynamics changes according to the frequency content of the control signal since fast modes are excited with fast input while they are not excited under slow controls. This can imply the need to use different controllers through time.

(b) The system parameters are changing so that the operation points change. Thus, a switched model which adjusts to several operation points may be useful [19–21].

(c) The adaptation transient has a bad performance due to a poor estimates initialization due to very imprecise knowledge of the true parameters. In this case, a multiparameterized adaptive controller, whose parameterization varies through time governed by a parallel multiestimation scheme, can improve the whole system performance. For this purpose, the parallel multiestimation scheme selects trough time, via a judicious supervision rule, the particular estimator associated with either the best identification objective, or the best tracking objective or the best mixed identification and tracking objectives. Such strategies can improve the switched system performance compared to the use of a single estimator/controller pair [5, 22].

This paper is devoted to the investigation of the global asymptotic stability properties of switched systems subject to internal constant point delays, while the matrices defining the delay-free and delayed dynamics are allowed to be time varying while fulfilling some standard additional regularity conditions like boundedness, eventual time differentiability, and being subject to sufficiently slow growing rates [23]. The various obtained asymptotic stability results are either dependent on or independent of the delay size and they are obtained by proving the existence of “ad-hoc” Krasovsky-Lyapunov functionals. It is assumed that either the current system matrix or that describing the system under null delay is stability matrices for results independent of and dependent on the delays sizes, respectively. This idea relies on the well-known fact that both of those matrices have to be stable for any linear time-invariant configuration in order that the corresponding time-delay system may be asymptotically stable, [1, 4, 10], provided that a minimum residence time at each configuration is respected before the next switching to another configuration. The formalism is derived by assuming two classes of mutually excluding switching instants. The so-called reset-free switching instants are defined as those where some parametrical function is subject to a finite jump (equivalently, a Dirac impulse at its time derivative) which is not constrained to a finite set. The so-called reset switching instants are defined as those registering bounded jumps to values within some prescribed set of resetting parameterizations. The distinction between reset-free and reset switching instants is irrelevant for stability analysis since in both cases at least one parameter is subject to a bounded jump, or equivalently, to a Dirac impulse in its time derivative. Impulsive systems are of growing interest in a number of applications related, for instance, to very large forces applied during very small intervals of times, population dynamics, chemostat models, pest, and epidemic models, and so forth (see, e.g., [24–28] and references therein). However, it may be relevant in practical situations to distinguish a switch to prescribed time-invariant parameterizations (see the above situations (a)–(c)) from an undriven switching action. The paper is organized as follows. Section 2 is devoted to obtain asymptotic stability results dependent on the delay sizes. Section 3 gives some extension for global asymptotic stability independent of the delays. Numerical examples are presented in Section 4, where switching through time in between distinct parameterizations is discussed. Finally, conclusions end the paper. Some mathematical derivations concerned with the results of Sections 2 and 3 are derived in Appendix .

2. Asymptotic Stability Dependent on the Delays

Consider the 𝑛th order linear time-varying dynamic system with 𝑞 internal (in general, incommensurate) known point delays: 𝑥̇𝑡=^𝑞_𝑗=0𝐴_𝑗𝑡𝑥𝑡−ℎ_𝑗(2.1) for any given bounded piecewise absolutely continuous function 𝜑∶[−ℎ,0]→𝐑^𝑛 of initial conditions, where ℎ∶=max_{1≤𝑗≤𝑞}(ℎ_𝑗) with ℎ₀=0, for some delays ℎ_𝑗∈[0,ℎ_𝑗], of finite or infinite maximum allowable delays sizes ℎ_𝑗∈𝐑₀₊, forall𝑗∈𝑞∶={1,2,…,𝑞}, where 𝐑₀₊ is the nonnegative real axis 𝐑₀₊∶=𝐑₊∪{0}={0≤𝑧∈𝐑}; and 𝐴_𝑗∶𝐑₀₊→𝐑^𝑛×𝑛, for all 𝑗∈𝑞∪{0}. The following assumptions are made.

2.1. Assumptions on the Time-Delay Dynamic System (2.1)

One or more of the following assumptions are used to derive the various stability results obtained in this paper.

Assumption 2.1. All the entries of the matrix functions 𝐴_𝑗∶𝐑₀₊→𝐑^𝑛×𝑛 are piecewise continuous and uniformly bounded for all 𝑗∈𝑞∪{0}.

Assumption 2.2. All the eigenvalues 𝜆_𝑖(∑^𝑞𝑗=0𝐴_𝑗(𝑡)) of the matrix function ∑^𝑞𝑗=0𝐴_𝑗(𝑡) satisfy Re𝜆_𝑖(∑^𝑞𝑗=0𝐴_𝑗(𝑡))≤−𝜌₀<0 for all 𝑡∈𝐑₀₊, forall𝑖∈𝜎(𝜎≤𝑛) for some 𝜌₀∈𝐑₊∶={0<𝑧∈𝐑}, that is, ∑^𝑞𝑗=0𝐴_𝑗(𝑡) is a stability matrix for all 𝑡∈𝐑₀₊.

Assumption 2.3. The matrix functions 𝐴_𝑗∶𝐑₀₊→𝐑^𝑛×𝑛 are almost everywhere time differentiable with essentially bounded time derivative for all 𝑗∈𝑞∪{0} possessing eventual isolated bounded discontinuities, then esssup_{𝑡∈𝐑𝟎+}(‖∑^𝑞𝑗=0𝐴_𝑗̇(𝑡)‖)≤𝛾_ℓ<∞ with 𝛾_ℓ being a ℓ-norm dependent nonnegative real constant and, furthermore, ^{𝑡𝑡+𝑇}‖‖𝐴_𝑗̇𝜏‖‖𝑑𝜏≤𝜇_𝑗𝑇+𝛼_𝑗≤𝜇𝑇+𝛼∀𝑗∈𝑞∪{0}(2.2) for some 𝛼_𝑗,𝜇_𝑗,𝛼,𝜇∈𝐑₊ for all 𝑡∈𝐑₀₊, and some fixed 𝑇∈𝐑₀₊ independent of 𝑡.
At time instants 𝑡, where the time-derivative of some entry of 𝐴_𝑗(𝑡) does not exist for any 𝑗∈𝑞∪{0}, the time derivative is defined in a distributional Dirac sense as 𝐴̇_𝑗(𝑡)=Γ(𝑡)𝛿(0) what equivalently means the presence of a discontinuity at 𝑡 in 𝐴_𝑗(𝑡) defined as 𝐴_𝑗𝑡⁺=𝐴^𝑗𝑡⁻+lim_𝜀→0+^𝜀−𝜀Γ_𝑗𝜏𝛿𝑡−𝜏𝑑𝜏=𝐴_𝑗𝑡+Γ_𝑗𝑡.(2.3)

Assumption 2.1 is relevant for existence and uniqueness of the solution of (2.1). The differential system (2.1) has a unique state-trajectory solution for 𝑡∈𝐑₊ for any given piecewise absolutely continuous function 𝜑∶[−ℎ,0]→𝐑^𝑛 of initial conditions. This follows from Picard-Lindelöff existence and uniqueness theorem. Assumption 2.2 establishes that ∑^𝑞𝑗=0𝐴_𝑗(𝑡) is a stability matrix for all time what is known to be a necessary condition for the global asymptotic stability of the system (2.1) for a set of prescribed maximum delays in the time-invariant case. It is well known that even if 𝐴_𝑗≡0 for all 𝑗∈𝑞, then the resulting linear time-varying delay-free system cannot be proved to be stable without some additional assumptions, like for instance, Assumptions 2.3. The latest assumption is related to the smallness of the time-derivative of the delay-free system matrix everywhere it exists or generating sufficiently small bounded discontinuities in ∑^𝑞𝑗=0𝐴_𝑗(𝑡) at times, where it is impulsive. An alternative assumption to Assumptions 2.3 which avoids the assumption of almost everywhere existence of a bounded 𝐴̇_𝑗(𝑡), for all 𝑗∈𝑞∪{0} (see second part of Assumption 2.1) might be stated in terms of sufficiently smallness of ‖Δ𝐴_𝑗(𝑡)‖ for Δ𝐴_𝑗(𝑡)∶=𝐴_𝑗(𝑡)−𝐴^∗𝑗 for all 𝑡∈𝐑₀₊ for all 𝑗∈𝑞∪{0} for some constant stability matrix ∑^𝑞𝑗=0𝐴^∗𝑗 whose eigenvalues satisfy Re𝜆_𝑖(∑^𝑞𝑗=0𝐴^∗𝑗)≤−𝜌₀<0. Such an alternative assumption guarantees also the global existence and uniqueness of the state-trajectory solution of (2.1) and it allows obtaining very close stability results to those being obtainable from the given assumptions. For global asymptotic stability dependent of the delay sizes on the first delay interval, the stability of the values taken by the matrix function (∑^𝑞𝑗=0𝐴_𝑗(𝑡)) is required within some real interval of infinite measure. Such an interval possesses a connected component being of infinite measure which is a necessary condition for global asymptotic stability for zero delays (see Theorem 2.12(i)).

2.2. Switching Function, Switching Sequence, and Basic Assumptions on the Switching Matrix Function

Assumption 2.1 admits bounded discontinuities in the entries of 𝐴_𝑗(𝑡) for 𝑗∈𝑞∪{0}. At such times 𝐴_𝑗(𝑡⁺) denote right values of the matrix function while 𝐴_𝑗(𝑡⁻) is simply denoted by 𝐴_𝑗(𝑡). A set of 𝑝 resetting systems of (2.1) is defined by the linear time-invariant systems: 𝑧̇_𝑗𝑡=^𝑞_𝑖=0𝐴_𝑖𝑗𝑧^𝑗𝑡−ℎ^𝑗(2.4) for some given 𝐴_𝑗𝑖𝐑^𝑛×𝑛 for all 𝑗∈𝑝 for all 𝑖∈𝑞∪{0} for some given 𝑝∈𝐍. Those parameterizations are used to reset the system (2.1) at certain reset instants defined later on. Assumption 2.2 extends in a natural fashion to include the resetting systems as follows.

Assumption 2.4. All the eigenvalues 𝜆_𝑘(∑^𝑞𝑖=0𝐴_𝑖𝑗) satisfy Re𝜆_𝑘(∑^𝑞𝑖=0𝐴_𝑖𝑗)≤−𝜌₀<0 for all 𝑡∈𝐑₀₊ for all 𝑘∈𝜎_𝑗0(𝜎_𝑗0≤𝑛) for all 𝑗∈𝑝; that is, ∑^𝑞𝑖=0𝐴_𝑖𝑗 are constant stability matrices with prescribed stability abscissa −𝜌₀<0 for all 𝑗∈𝑝.

The following definitions are then used.

Definition 2.5. The switching matrix function is a mapping 𝜎∶𝐑₀₊→{(𝐴_𝑗(𝑡⁺)−𝐴_𝑗(𝑡)),∀𝑗∈𝑞∪{0}}⊂𝐑^{𝑛×(𝑞+1)𝑛} from the nonnegative real axis to the set of real 𝑛×(𝑞+1)𝑛 matrices.

The trivial switching matrix function is that being identically zero so that no switch occurs. If some switch occurs then the switching matrix function is nonzero. The switching matrix function is colloquially referred to in the following as the switching law.

Definition 2.6 (switching instant). 𝑡∈𝐑0+ is a switching instant if 𝐴_𝑗(𝑡⁺)≠𝐴_𝑗(𝑡) for some 𝑗∈𝑞∪{0}.

The set of switching instants generated by the switching law 𝜎 is denoted by ST(𝜎). Two kinds of switching instants, respectively, reset instants and reset-free switching instants defined in Definitions 2.7 and 2.8 are considered.

Definition 2.7 (reset instant). 𝑡∈𝐑₀₊ is a reset instant generated by the switching law 𝜎 if 𝑡∈ST(𝜎) and 𝐴_𝑖(𝑡⁺)=𝐴_𝑖𝑗 for some 𝑖∈𝑞∪{0} and some 𝑗∈𝑝, provided that 𝐴_𝑖(𝑡)=𝐴_𝑖𝑘≠𝐴_𝑖𝑗 for some 𝑘(≠𝑗)∈𝑝.

The set of reset instants generated by the switching law 𝜎 is denoted by ST_𝑟(𝜎). Note that ST_𝑟(𝜎)⊂ST(𝜎) from Definitions 2.6 and 2.7. Note also that the whole system parameterization is driven to some of the prefixed resetting systems (2.4) when a reset instant happens. Note that at reset instants, 𝐴_𝑖̇(𝑡)=(𝐴_𝑖𝑗−𝐴_𝑖(𝑡))𝛿(0) for some 𝑖∈𝑞∪{0}, 𝑗∈𝑝.

Definition 2.8 (reset-free switching instant). 𝑡∈𝐑₀₊ is a switching reset-free instant generated by the switching law 𝜎 if 𝑡∈ST(𝜎) and 𝐴_𝑖𝑗≠𝐴_𝑖(𝑡⁺)≠𝐴_𝑖(𝑡) for some 𝑖∈𝑞∪{0}, for all 𝑗∈𝑝.

The set of reset-free instants is denoted by 𝑡∈ST_rf(𝜎). Note that at reset-free switching instants some of the switched system parameters suffer an undriven bounded discontinuity. If all the parameters jump to a parameterization (2.4) at the same time, then the corresponding instant is considered a reset time instant. Note that ST_rf(𝜎)⊂ST(𝜎), ST(𝜎)=ST_𝑟(𝜎)∪ST_rf(𝜎), and ST_𝑟(𝜎)∩ST_rf(𝜎)=∅ from Definitions 2.6–2.8, that is, the whole set of switching instants is the disjoint union of the sets of reset and reset-free switching instants.

Definition 2.9. The partial switching sequence ST(𝜎,𝑡), the partial switching sequence ST_𝑟(𝜎,𝑡), and the reset-free partial switching sequence ST_rf(𝜎,𝑡), generated by the switching law 𝜎∶𝐑₀₊→{(𝐴_𝑗(𝑡)−𝐴_𝑗(𝑡⁻)),∀𝑗∈𝑞∪{0}}⊂𝐑^{𝑛×(𝑞+1)𝑛} up till any time 𝑡∈𝐑₀₊, are defined, respectively, by ST(𝜎,𝑡)∶={𝑡_𝑖∈ST(𝜎)∶𝑡_𝑖<𝑡}, ST_𝑟(𝜎,𝑡)∶={𝑡_𝑖∈ST_𝑟(𝜎)∶𝑡_𝑖<𝑡}, and ST_rf(𝜎,𝑡)∶={𝑡_𝑖∈ST_rf(𝜎)∶𝑡_𝑖<𝑡}.

Remark 2.10. An interpretation of Assumptions 2.3 is that the following conditions hold for any given ℓ-matrix norm for some nonnegative norm dependent real constants 𝜇_𝑗, 𝛼_𝑗, 𝜇, and 𝛼 for all 𝑗∈𝑞∪{0}: ‖esssup_{𝑡∈𝐑+0}‖‖𝐴̇_𝑗𝑡‖‖≤𝜇_𝑗≤maxesssup_{𝑡∈𝐑+0}‖‖𝐴̇_𝑗𝑡‖‖∶𝑗∈𝑞∪0≤𝜇,_𝜏∈ST^𝜎^∩^{𝑡,𝑡+𝑇}𝐴^𝑗̇𝜏‖≤_𝜏∈ST^𝜎^∩^{𝑡,𝑡+𝑇}‖‖𝑋𝜏𝛿0‖‖≤𝛼^𝑗≤max(‖_𝜏∈ST^𝜎^∩^{𝑡,𝑡+𝑇}𝐴^𝑗̇𝜏‖∶𝑗∈𝑞∪0)≤𝛼,(2.5) where 𝛿(0) is a Dirac impulse at 𝑡=0. Note that Assumptions 2.3 imply |‖∑^𝑞𝑗=0𝐴_𝑗(𝑡+𝜏)‖−‖∑^𝑞𝑗=0𝐴_𝑗(𝑡)‖|≤𝜇𝑇+𝜈(𝑡+𝜏) for all 𝜏∈[0,𝑇], with the function 𝜈∶𝐑₀₊→𝐑₀₊ satisfying 𝜈(𝑡+𝜏)≤𝛼₀𝑗+𝛼_1𝑗(𝑡+𝜏)≤𝛼 with 𝛼_1𝑗(𝑡+𝜏)=0 if 𝐴̇_𝑗(𝑡+𝜏) exists with ‖𝐴̇_𝑗(𝑡+𝜏)‖≤𝛼_𝑗 or 𝛼_𝑗−𝛼_0𝑗≥𝛼_1𝑗(𝑡+𝜏)≥‖Δ_𝑗(𝑡+𝜏)‖ if 𝐴̇_𝑗(𝑡+𝜏)=Δ_𝑗(𝑡+𝜏)𝛿(0), that is, at least one of its entries is impulsive.

Note that Assumption 2.1 implies that switching does not happen arbitrarily fast neither to reset parameters nor to reset-free ones . The subsequent result is direct.

Assertions 1. The following properties are true irrespective of the switching function:
(i) 𝑡∉ST(𝜎)⇔𝜎(𝑡⁺)=𝜎(𝑡)=0_{𝑛×(𝑞+1)𝑛} (i.e., a zero 𝑛×(𝑞+1)𝑛-matrix);
(ii) 𝑡∈ST(𝜎)⇔𝜎(𝑡⁺)−𝜎(𝑡)≠0_{𝑛×(𝑞+1)𝑛};
(iii) 𝑡∉ST(𝜎)⇔ST(𝜎,𝑡⁺)=ST(𝜎,𝑡);
(iv) 𝑡∈ST(𝜎)⇔ST(𝜎,𝑡⁺)≠ST(𝜎,𝑡).

Proof. (i) 𝜎(𝑡⁺)=𝜎(𝑡)=0_{𝑛×(𝑞+1)𝑛}⇒𝑡∉ST_𝑟(𝜎)⋀𝑡∉ST_rf(𝜎)⇔𝑡∉ST(𝜎),
𝑡∉ST⇔𝜎(𝑡)=0 (since switching is not arbitrarily fast), 𝜎𝑡⁺−𝜎𝑡=0_𝑛×^𝑞+1^𝑛⟺𝜎𝑡⁺=𝜎𝑡=0_𝑛×^𝑞+1^𝑛.(2.6) Property (i) has been proven. Property (ii) is the contrapositive logic proposition to Property (i), and thus equivalent, since switching is not arbitrarily fast.
Properties (iii)-(iv) are also contrapositive logic propositions, then equivalent since ST𝜎,𝑡∶=𝑡_𝑖∈ST𝜎∶𝑡_𝑖<𝑡⟺⟺SSTT𝜎𝜎,,𝑡𝑡⁺₊∶=𝑡^𝑖∈ST𝜎∶𝑡_𝑖<𝑡⁺=𝑡^𝑖∈ST𝜎∶𝑡_𝑖≤𝑡=≠SSTT𝜎𝜎,,𝑡𝑡iiff𝑡𝑡∉∈SSTT𝜎𝜎,(2.7) since switching cannot happen arbitrarily fast. Properties (iii)-(iv) have been proven.

The subsequent global stability result is proven in Appendix by guaranteeing that the Krasovsky-Lyapunov functional candidate below is indeed a Krasovsky-Lyapunov functional: 𝑉𝑡,𝑥_𝑡=𝑥^𝑇𝑡𝑃𝑡𝑥𝑡+^𝑞_𝑞𝑖=1_𝑗=0^−ℎ𝑗_{−ℎ𝑖−ℎ𝑗}^{𝑡𝑡+𝜃}𝑥^𝑇𝜏𝑆_𝑖𝑗𝑥𝜏𝑑𝜏𝑑𝜃.(2.8)

Theorem 2.12. The following properties hold.
(i)Assume the following.(i.a)The matrix functions 𝐴_𝑗(𝑡), for all 𝑗∈𝑞∪{0} are subject to Assumption 2.1.(i.b)The switching law 𝜎 is such that 𝑄𝑡∶=⎡⎢⎢⎢⎢⎢⎢⎣𝐻1ℎ¹ℎ₁𝑃𝑡𝐴₁𝑡𝑀𝑡⋯ℎ_𝑞𝑃𝑡𝐴_𝑞𝑡𝑀𝑡𝑀^𝑇𝑡⋮𝐴^𝑇1𝑡𝑃𝑡−0𝑅_1⋮0⋯0ℎ⋱⋮_𝑞𝑀^𝑇𝑡𝐴^𝑇𝑞𝑡𝑃𝑡0⋯−𝑅_𝑞∀𝑡∈⎤⎥⎥⎥⎥⎥⎥⎦𝐑<_𝟎+0,,(2.9)where 𝐻1denotes(∑^𝑞𝑗=0𝐴^𝑇𝑗(𝑡))𝑃(𝑡)+𝑃(𝑡)(∑^𝑞𝑗=0𝐴_𝑗(𝑡))+𝑃̇(𝑡)+∑^𝑞𝑖=1∑^𝑞𝑗=0ℎ_𝑖𝑆_𝑖𝑗, for some time-differentiable real symmetric positive definite matrix function 𝑃∶𝐑₀₊→𝐑^𝑛×𝑛 and some real symmetric positive definite matrices 𝑆_𝑖𝑗∈𝐑^𝑛×𝑛(∀𝑖∈𝑞,∀𝑗∈𝑞∪{0}), where 𝑅_𝑖𝑀𝑡∶=𝐴₀𝑡,…,𝐴_𝑞𝑡∈𝐑^𝑛×^𝑞+1^𝑛,∶=diag𝑆_𝑖0,𝑆_𝑖1,…,𝑆_𝑖𝑞∈𝐑^𝑞+1^𝑛×^𝑞+1^𝑛;∀𝑖∈𝑞,∀𝑡∈𝐑_𝟎+.(2.10)Thus, the system (2.1) is globally asymptotically Lyapunov's stable for all delays ℎ_𝑖∈[0,ℎ_𝑖], forall𝑖∈𝑞. A necessary condition is (∑^𝑞𝑗=0𝐴^𝑇𝑗(𝑡))𝑃(𝑡)+𝑃(𝑡)(∑^𝑞𝑗=0𝐴_𝑗(𝑡))+𝑃̇(𝑡)<0, forall𝑡∈𝐑₀₊ what implies that (∑^𝑞𝑗=0𝐴_𝑗(𝑡)) is a stability matrix of prescribed stability abscissa on 𝐑₀₊ except eventually on a real subinterval of finite measure of 𝐑₀₊.(ii) Assume the following(ii.a)𝐴_𝑖(𝑡)=𝐴_𝑖𝑗, forall𝑖∈𝑞∪{0}, forall𝑡∈𝐑₀₊ for some 𝑗∈𝑝 (eventually being dependent on t) satisfying Assumption 2.4. (ii.b)The switching law 𝜎 is such that ST_rf(𝜎)=∅ (i.e., it generates reset switching instants only) with ST_𝑟(𝜎) being arbitrary, namely, the set of reset times is either any arbitrary strictly increasing sequence of nonnegative real values (i.e., the resetting switching never ends) or any finite set of strictly ordered nonnegative real numbers with a finite maximal (i.e., the resetting switching process ends in finite time).(ii.c)𝑄^∗𝑖∶=⎡⎢⎢⎢⎢⎢⎢⎢⎣(^𝑞_𝑗=0𝐴^𝑇𝑗𝑖)𝑃^∗+𝑃^∗(^𝑞_𝑗=0𝐴_𝑗𝑖)+^𝑞_𝑞𝑖=1_𝑗=0ℎ_𝑖𝑆^∗𝑖𝑗ℎ₁𝑃^∗𝐴_1𝑖𝑀^∗1⋯ℎ_𝑞𝑃^∗𝐴_𝑞𝑖𝑀^∗𝑞ℎ₁𝑀⋮^∗1𝑇𝐴^𝑇1𝑖𝑃^∗−0𝑅^∗1_⋮0⋯0ℎ⋱⋮_𝑞𝑀^∗𝑞𝑇𝐴^𝑇𝑞𝑖𝑃^∗0⋯−𝑅^∗𝑞∀⎤⎥⎥⎥⎥⎥⎥⎥⎦𝑖<∈0;𝑝(2.11)for some 𝐑^𝑛×𝑛∋𝑃^∗=𝑃^∗𝑇>0, 𝐑^𝑛×𝑛∋𝑆^∗𝑖𝑗=𝑆^{∗𝑇𝑖𝑗}>0(∀𝑖∈𝑞,∀𝑗∈𝑞∪{0}), where 𝑅_∗𝑖𝑀_𝑖∶=𝐴_0𝑖,…,𝐴_𝑞𝑖∈𝐑^𝑛×^𝑞+1^𝑛,∶=diag𝑆^∗𝑖0,𝑆^∗𝑖1,…,∈𝐑^{(𝑞+1)𝑛×(𝑞+1)𝑛};∀𝑖∈𝑞(2.12)Thus, the switched system (2.1), obtained from switches among resetting systems (2.4), is globally asymptotically Lyapunov's stable and also globally exponentially stable for all delays ℎ_𝑖∈[0,ℎ_𝑖], forall𝑖∈𝑞. If (2.9) is replaced with 𝑄^∗𝑖≤−2𝜀𝐼_(𝑞+1)𝑛<0, forall𝑖∈𝑞, and some 𝜀∈𝐑₊ then the state trajectory decays exponentially with rate(−𝜀)<0.(iii) There is a sufficiently small ℎ∶=max_𝑖∈𝑞ℎ_𝑖 such that Property (i) holds for any ℎ_𝑖∈[0,ℎ_𝑖], forall𝑖∈𝑞 provided that all the delay-free resetting systems (2.4)𝑧̇_𝑗(𝑡)=(∑^𝑞𝑖=0𝐴_𝑖𝑗)𝑧_𝑗(𝑡) fulfil Assumption 2.4, that is, they are globally exponentially stable.

It is of interest to discuss particular cases easy to test, guaranteeing Theorem 2.12 (i).

2.3. Sufficiency Type Asymptotic Stability Conditions Obtained for Constant Symmetric Matrices 𝑃 and 𝑆_𝑖𝑗

Assume real constant symmetric matrices 𝑃(𝑡)=𝑃 and 𝑆_𝑖𝑗, forall𝑖∈𝑞,forall𝑗∈𝑞∪{0}, forall𝑡∈𝐑₀₊ so that 𝑄𝑡=⎡⎢⎢⎢⎢⎢⎢⎣𝐻2ℎ¹ℎ₁𝑃𝐴₁𝑡𝑀𝑡⋯ℎ_𝑞𝑃𝐴_𝑞𝑡𝑀𝑡𝑀^𝑇𝑡⋮𝐴^𝑇1𝑡𝑃−0𝑅_1⋮0⋯0ℎ⋱⋮_𝑞𝑀^𝑇𝑡𝐴^𝑇𝑞𝑡𝑃0⋯−𝑅_𝑞⎤⎥⎥⎥⎥⎥⎥⎦<0,(2.13) where 𝐻2denotes(∑^𝑞𝑗=0𝐴^𝑇𝑗(𝑡))𝑃+𝑃(∑^𝑞𝑗=0𝐴_𝑗(𝑡))+∑^𝑞𝑖=1∑^𝑞𝑗=0ℎ_𝑖𝑆_𝑖𝑗. In this case, the Krasovsky-Lyapunov functional used in the proof of Theorem 2.12(i) holds defined with constant matrices for all time irrespective of being a switching-free instant or any switching instant (independently of its nature: reset time or reset-free switching instant). A practical test for (2.13) to hold follows. Consider 𝐴^∗𝑖(𝑖∈𝑞∪{0}) such that the time invariant system (2.1) defined with 𝐴_𝑖(𝑡)→𝐴^∗𝑖 is globally asymptotically Lyapunov's stable and define a stability real 𝑛-matrix 𝐴^∗∶=∑^𝑞𝑖=0𝐴^∗𝑖. Decompose 𝑄(𝑡)=𝑄^∗+𝑄(𝑡), where 𝑄^∗∶=⎡⎢⎢⎢⎢⎢⎢⎢⎣𝐴^∗𝑇𝑃+𝑃𝐴^∗+^𝑞_𝑞𝑖=1_𝑗=0ℎ_𝑖𝑆_𝑖𝑗ℎ₁𝑃𝐴^∗1𝑀^∗⋯ℎ_𝑞𝑃𝐴^∗𝑞𝑀^∗ℎ₁𝑀⋮^∗𝑇𝐴^∗1𝑇𝑃−0𝑅^∗1_⋮0⋯0ℎ⋱⋮_𝑞𝑀^∗𝑇𝐴^∗𝑞𝑇𝑃0⋯−𝑅^∗𝑞𝑄𝑡∶=⎡⎢⎢⎢⎢⎢⎢⎣𝐴_𝑇⎤⎥⎥⎥⎥⎥⎥⎥⎦,ℎ𝑡₁𝑃+𝑃𝐴𝑡ℎ₁𝑃Δ₁𝑡⋯ℎ_𝑞𝑃Δ_𝑞𝑡Δ^𝑇1⋮𝑡𝑃00_⋮0⋯0ℎ⋱⋮_𝑞Δ^𝑇𝑞𝑡𝑃0⋯0⎤⎥⎥⎥⎥⎥⎥⎦,(2.14)where 𝑀^∗∶=diag𝐴^∗0,𝐴^∗1,𝐴…_𝑖,𝐴^∗𝑞,𝑅^∗𝑖∶=diag𝑆^∗𝑖0,𝑆^∗𝑖1,…,𝑆^∗𝑖𝑞,𝑀𝑡∶=𝑀𝑡−𝑀^∗,Δ_𝑖𝑡∶=𝐴_𝑖𝑡−𝐴^∗,𝐴𝑡∶=(∑^𝑞_𝑗=0𝐴_𝑗𝑖𝑡)−𝐴^∗,𝑡∶=𝐴_𝑖𝑡𝑀^∗+𝐴^∗𝑖𝑀𝑡+𝐴_𝑖𝑡𝑀^↔𝑡.(2.15) If 𝑡∈ST(𝜎), then 𝛿𝑄𝑡=∶=⎡⎢⎢⎢⎢⎢⎢⎢⎣𝑄𝑡⁺−𝑄𝐻𝑡3ℎ₁ℎ₁𝑃Δ₁𝑡⁺−Δ¹𝑡⋯ℎ_𝑞𝑃Δ₁𝑡⁺−Δ¹𝑡Δ¹𝑡⁺⋮−Δ¹𝑡^𝑇𝑃00_⋮0⋯0ℎ⋱⋮_𝑞‖‖𝛿𝑄𝑡‖‖Δ¹𝑡⁺−Δ¹𝑡^𝑇𝑃0⋯0₂⎤⎥⎥⎥⎥⎥⎥⎥⎦,≤2𝜆_max𝑃[‖‖𝐴^𝑇𝑡⁺−𝐴^𝑇𝑡‖‖₂+^𝑞_𝑖=1ℎ_𝑖‖‖Δ_𝑖=2𝜆_max𝑡⁺−Δ^𝑖𝑡‖‖₂]𝑃[‖^𝑞_𝑖=0𝐴^𝑇𝑖𝑡⁺−𝐴^𝑇𝑖𝑡‖₂+^𝑞_𝑖=1ℎ_𝑖‖‖Δ_𝑖𝑡⁺−Δ^𝑖𝑡‖‖₂],(2.16) where 𝐻3denotes(𝐴^𝑇(𝑡⁺)−𝐴^𝑇(𝑡))𝑃+𝑃((𝐴(𝑡⁺)−𝐴(𝑡))), what leads to ‖‖𝛿𝑄𝑡‖‖₂≤2𝜆_max𝑃[𝑞+1𝑎⁺+ℎ𝑞+(𝑎⁺𝑞(+∑^𝑞_𝑗=01‖‖𝐴^∗𝑗2‖‖𝑎₂++𝑎₊𝑞+1‖‖𝐴^∗𝑖‖‖₂+𝑞+1𝑎+𝑎⁺)𝑎)],(2.17) where 𝑎∶=max_{𝑖∈𝑞∪{0}}(‖𝐴_𝑖‖₂) and 𝑎⁺∶=sup_{𝑡∈ST(𝜎)}max_{𝑖∈𝑞∪{0}}(‖𝐴_𝑖(𝑡⁺)−𝐴_𝑖‖₂) (see A.9 in Appendix ). Direct results from Theorem 2.12 which follow from (2.13) to (2.17) are given below.

Corollary 2.13. Consider in (2.9) replacements with constant real matrices 𝑄(𝑡)→𝑄^∗=𝑄^∗𝑇, 𝑃(𝑡)→𝑃=𝑃^𝑇>0, 𝐴_𝑖(𝑡)→𝐴^∗𝑖, 𝐴(𝑡)→𝐴^∗∶=∑^𝑞𝑖=0𝐴^∗𝑖; forall𝑖∈𝑞,forall𝑗∈𝑞∪{0}, forall𝑡∈𝐑₀₊ such that 𝐴^∗ is a stability matrix. Then, Theorem 2.12(i) holds if 𝑄^∗<0 for any switching law 𝜎 such that (1)𝜆_min(−𝑄^∗)=−𝜆_max(𝑄^∗)>‖𝑄(𝑡)‖₂∶=𝜆^1/2max(𝑄^𝑇(𝑡)𝑄(𝑡)), forall𝑡∈𝐑₀₊⧵ST(𝜎),(2)𝑎∶=max_{𝑖∈𝑞∪{0}}(‖𝐴_𝑖‖₂) and 𝑎⁺∶=sup_{𝑡∈ST(𝜎)}max_{𝑖∈𝑞∪{0}}(‖𝐴_𝑖(𝑡⁺)−𝐴_𝑖‖₂) are sufficiently small such that 𝜆_min−𝑄^∗−‖‖𝑄𝑡‖‖₂>2𝜆_max𝑃[𝑞+1𝑎⁺+ℎ𝑞(𝑎⁺(^𝑞_𝑗=0‖‖𝐴^∗𝑗‖‖₂+𝑞+1max_{𝑖∈𝑞∪}⁰‖‖𝐴^∗𝑖+𝑞+12𝑎+𝑎₊‖‖₂+𝑞+1𝑎+𝑎⁺)𝑎)],∀𝑡∈ST𝜎.(2.18)

Corollary 2.14. Consider in (2.9) replacements with constant real matrices 𝑄(𝑡)→𝑄^∗𝑗=𝑄^∗𝑗𝑇, 𝑃(𝑡)→𝑃=𝑃^𝑇>0, 𝐴_𝑖𝑗(𝑡)→𝐴^∗𝑖𝑗, 𝐴(𝑡)→𝐴^∗𝑗∶=∑^𝑞𝑖=0𝐴^∗𝑖𝑗 for all 𝑗∈𝑝,forall𝑘∈𝑞∪{0}, forall𝑡∈𝐑₀₊ such that each 𝐴^∗𝑗(∀𝑗∈𝑝) is a stability matrix with 𝐴^∗𝑖𝑗;forall𝑖∈𝑞,forall𝑗∈𝑝 being the parameterizations defining the resetting systems (2.4). Assume that the system (2.1) is one of the resetting systems (2.4) at 𝑡=0. Then, Theorem 2.12(i) holds with a common Krasovsky-Lyapunov function for all those resetting systems if 𝑄^∗𝑗<0(∀𝑗∈𝑝) for any switching law 𝜎 such that
(1)𝜆_min(−𝑄^∗)=−𝜆_max(𝑄^∗)>‖𝑄(𝑡)‖₂∶=𝜆^1/2max(𝑄^𝑇(𝑡)𝑄(𝑡)),forall𝑡∈𝐑₀₊⧵ST(𝜎),(2)𝑎_𝑗∶=max_{𝑖∈𝑞∪{0}}(‖𝐴_𝑗𝑖‖₂) and 𝑎^+𝑗∶=sup_{𝑡∈ST(𝜎)}max_{𝑖∈𝑞∪{0}}(‖𝐴_𝑖(𝑡⁺)−𝐴_𝑗𝑖‖₂) are sufficiently small such that 𝜆_min−𝑄^∗−‖‖𝑄𝑡‖‖₂>2𝜆_max𝑃[𝑞+1𝑎^+𝑗+ℎ𝑞(𝑎^+𝑗(^𝑞_𝑖=0‖‖𝐴_𝑗𝑖‖‖₂+𝑞+1max_{𝑖∈𝑞∪}⁰‖‖𝐴^𝑗𝑖+𝑞+1𝑎_𝑗‖‖₂)+𝑎^+𝑗+𝑞+12𝑎^𝑗+𝑎^+𝑗𝑎^+𝑗)](2.19)𝑡∈ST_fr(𝜎), provided that at time max(𝑡^<𝑡∶𝑡^∈ST_𝑟(𝜎)), the system (2.1) coincides with at the 𝑗∈𝑝𝑟 setting system (2.4).

The proof of Corollary 2.14 is close to that of Corollary 2.13 from A.9 in Appendix with the replacements 𝑎→𝑎_𝑗,𝑎⁺→𝑎^+𝑗 for all 𝑗∈𝑝. If (2.19) is rewritten with the replacements 𝑎_𝑗→𝑎∶=max_𝑗∈𝑝𝑎_𝑗,𝑎^+𝑗→^→𝑎⁺∶=max_𝑗∈𝑝𝑎^+𝑗 then the reformulated weaker Corollary 2.14 is valid for all 𝑡∈ST_fr(𝜎) irrespective of the preceding reset switching. A result which guarantees Corollary 2.13, and then Theorem 2.12(i), is now obtained by replacing the (1,1) block matrix of 𝑄^∗ by a Lyapunov matrix equality as follows. Consider a real 𝑛-matrix 𝑄^∗0=𝑄^0∗𝑇>0 such that 𝜆_min(𝑄^∗0)>𝜆_max(∑^𝑞𝑖=1∑^𝑞𝑗=0ℎ_𝑖𝑆_𝑖𝑗) and 𝑃∶=∫^∞0𝑒^{𝐴∗𝑇𝜏}𝑄^∗0𝑒^𝐴∗𝜏𝑑𝜏 satisfying the Lyapunov equation 𝐴^∗𝑇𝑃+𝑃𝐴^∗=−𝑄^∗0<0 as its unique solution. Note that 𝜆_max(𝑃)≤𝐾^∗𝜆_max(𝑄^∗0)/2𝜌^∗ for some 𝐾^∗∈𝐑₊, where −𝜌^∗<0 is the stability abscissa of 𝐴^∗ with ‖𝑒^{𝐴∗𝑇𝑡}‖≤√𝐾^∗𝑒^{−𝜌∗𝑡} for all 𝑡∈𝐑. Define the decomposition 𝑄(𝑡)=𝑄^∗+𝑄(𝑡), where 𝑄^∗∶=BlockDiag(−𝑄^∗0+^𝑞_𝑞𝑖=1_𝑗=0ℎ_𝑖𝑆_𝑖𝑗,−𝑅^∗1,…,−𝑅^∗𝑞),𝑄𝑡∶=⎡⎢⎢⎢⎢⎢⎢⎢⎣𝐴^𝑇ℎ₁𝑡𝑃+𝑃𝐴𝑡ℎ₁𝑃𝐴^∗1𝑀^∗+Δ₁𝑡⋯ℎ_𝑞𝑃𝐴^∗𝑞𝑀^∗+Δ_𝑞𝑡𝐴^∗1𝑀^∗⋮+Δ₁𝑡^𝑇𝑃00_⋮0⋯0ℎ⋱⋮_𝑞⇒‖‖𝐴^∗𝑞𝑀^∗+Δ_𝑞𝑡^𝑇𝑃0⋯0⎤⎥⎥⎥⎥⎥⎥⎥⎦𝑄𝑡‖‖₂≤(𝐾^∗𝜆_max𝑄^∗0𝜌^∗)(‖‖𝐴^𝑇𝑡‖‖₂+^𝑞_𝑖=1ℎ_𝑖‖‖𝐴^∗𝑖𝑀^∗+Δ_𝑖𝑡‖‖₂).(2.20) Thus, the subsequent result follows from Corollary 2.13 and (2.20).

Corollary 2.15. Consider the matrices of Corollary 2.13 with 𝐴^∗ being a stability matrix with stability abscissa (−𝜌^∗)<0 which satisfies the Lyapunov equation 𝐴^∗𝑇𝑃+𝑃𝐴^∗=−𝑄^∗0<0. Then, Theorem 2.12(i) holds if 𝑄^∗<0 for any switching law 𝜎 such that 𝜆_min−𝑄^∗=>−𝜆𝐾^max_∗𝑄^∗𝜆_max𝑄^∗0𝜌^∗(‖‖𝐴^𝑇𝑡‖‖₂+^𝑞_𝑖=1∀𝑡∈𝐑₀₊ℎ_𝑖‖‖𝐴^∗𝑖𝑀^∗+Δ_𝑖𝑡‖‖₂),,∀𝑡∈ST(𝜎).(2.21)

2.4. Sufficiency Type Asymptotic Stability Conditions Obtained for Time-Varying Symmetric Matrices 𝑃(𝑡), 𝑆_𝑖𝑗(𝑡)=𝑆_𝑖𝑗

The following result, which is proven in Appendix , holds.

Theorem 2.16. Under Assumptions 2.1–2.3, the following properties hold.
(i)The switched system (2.1) is globally asymptotically Lyapunov's for any delays ℎ_𝑖∈[0,ℎ_𝑖] for all 𝑖∈𝑞 for some ℎ∶=max_𝑖∈𝑞(ℎ_𝑖) and any switching law 𝜎 such that (a)the switching instants are arbitrary;(b)max(esssup‖𝐴̇_𝑗(𝑡)‖∶𝑡∈𝐑₀₊,𝑗∈𝑝) is sufficiently small compared to the absolute value of the prescribed stability abscissa of ∑^𝑞𝑗=0𝐴_𝑗(𝑡);(c)the support testing matrix of distributional derivatives Γ_𝐴𝑑𝑗(𝑡) of the same matrices are semidefinite negative for all time instants, where the conventional derivatives do not exist (i.e., 𝐴̇_𝑗(𝑡)=Γ_𝐴𝑑𝑗(𝑡)(𝑡)𝛿(0)).(ii)The switched system (2.1) is globally exponentially stable for any delays ℎ_𝑖∈[0,ℎ_𝑖] for all 𝑖∈𝑞 for some ℎ∶=max_𝑖∈𝑞(ℎ_𝑖) such that (a)max(esssup‖𝐴̇_𝑗(𝑡)‖∶𝑡∈𝐑₀₊,𝑗∈𝑝) is sufficiently small compared to the absolute value of the prescribed stability abscissa of ∑^𝑞𝑗=0𝐴_𝑗(𝑡);(b)max(‖Γ_𝐴𝑑𝑗(𝑡)(𝑡)‖∶𝐴̇_𝑗(𝑡)=Γ_𝐴𝑑𝑗(𝑡)(𝑡)𝛿(0),∀𝑡∈ST(𝜎),𝑗∈𝑝) is sufficiently small compared to the timeintervals in between any two consecutive switching instants. Furthermore, if Assumptions 2.1–2.4 hold, then (iii)the switched system (2.1) is globally exponentially stable for any delays ℎ_𝑖∈[0,ℎ_𝑖] for all 𝑖∈𝑞, for some ℎ∶=max_𝑖∈𝑞(ℎ_𝑖) such that(a)max(esssup‖𝐴̇_𝑗(𝑡)‖∶𝑡∈𝐑₀₊,𝑗∈𝑝) is sufficiently small compared to the absolute value of the prescribed stability abscissa of ∑^𝑞𝑗=0𝐴_𝑗(𝑡);(b)the switching law 𝜎 is such that(a)max(‖Γ_𝐴𝑑𝑗(𝑡)‖∶𝐴̇_𝑗(𝑡)=Γ_𝐴𝑑𝑗(𝑡)𝛿(0),∀𝑡∈ST_rf(𝜎),𝑗∈𝑝) is sufficiently small compared to the lengths of time intervals between any two consecutive switching instants;(b)it exists a common Krasovsky-Lyapunov functional 𝑉(𝑡,𝑥_𝑡) defined with constant matrices 𝑃=𝑃^𝑇>0 and 𝑆_𝑖𝑗=𝑆^𝑇𝑖𝑗>0, forall(𝑖,𝑗)∈(𝑞∪{0})×𝑞 for all the time-invariant resetting systems (2.4) and some of the subsequent conditions hold forall𝑡∈ST_𝑟(𝜎) under the resetting action 𝑃(𝑡⁺)=𝑃; forall(𝑖,𝑗)∈(𝑞∪{0})×𝑞:(b.1)𝑉(𝑡⁺,𝑥_𝑡)≤𝑉(𝑡,𝑥_𝑡) which is guaranteed, in particular, if 𝑃(𝑡⁺)=𝑃≤𝑃(𝑡),(b.2)the tradeoff (a) is respected between sufficiently small norms of the matrices of distributional derivatives and the length |𝑡−𝑡^|, at any 𝑡∈ST_𝑟(𝜎), if any, where the condition (b.1) is not satisfied, where 𝑡^=max(𝜏∈𝐑₀₊∶ST(𝜎)∋𝜏<𝑡).

The characterization of the “sufficient smallness” of the involved magnitudes in Theorem 2.16 is given explicitly in its proof. The proof considers that when some entry time derivative of the involved matrices does not exist, it equivalently exists a distributional derivative at this time instant which is equivalent to the existence of a bounded jump-type discontinuity in its integral, so that the corresponding time instant is in fact a switching instant. The sufficiently large time intervals required in between any two consecutive switching times compared with the amplitudes of the amplitude (in terms of norm errors) among consecutive parameterizations are related to the need for a minimum residence time at each parameterization for the case when those ones do not possess a common Krasovsky-Lyapunov functional.

3. Asymptotic Stability Independent of the Delays

Some results concerning sufficiency type properties of global asymptotic stability independent of the delays, that is, for any ℎ_𝑖∈𝐑₀₊, forall𝑖∈𝑞 of the switched system (2.1) are obtained under very close guidelines as those involved in the results on stability dependent of the delays given in Section 2. The Krasovsky-Lyapunov functional candidate of Section 2 and Appendix is modified as follows: 𝑉𝑡,𝑥_𝑡=𝑥^𝑇𝑡𝑃𝑡𝑥𝑡+^𝑞_𝑖=1^{𝑡𝑡−ℎ}_𝑖𝑥^𝑇𝜏𝑆_𝑖𝜏𝑥𝜏𝑑𝜏(3.1) whose time derivative along the state-trajectory solution of (2.1) is 𝑉̇𝑡,𝑥_𝑡=𝑥^𝑇𝑡(𝐴^𝑇𝑡𝑃𝑡+𝑃𝑡𝐴𝑡+^𝑞_𝑖=1+2𝑥_𝑇𝑆_𝑖𝑡+𝑃̇𝑡)𝑥𝑡=𝑡^𝑞_𝑖=1𝑃𝐴_𝑖𝑡𝑥𝑡−ℎ_𝑖−^𝑞_𝑖=1𝑥^𝑇𝑡−ℎ^𝑖𝑆^𝑖𝑡−ℎ^𝑖𝑥𝑡−ℎ^𝑖𝑥^𝑇𝑡𝑄^𝑡𝑥𝑡<0(3.2) for all nonzero 𝑥^𝑇(𝑡)=(𝑥^𝑇(𝑡),𝑥^𝑇(𝑡−ℎ₁),…,𝑥^𝑇(𝑡−ℎ_𝑞)) if 𝑄^𝑡∶=⎡⎢⎢⎢⎢⎢⎢⎢⎣𝐴^𝑇𝑡𝑃𝑡+𝑃𝑡𝐴𝐴𝑡+^𝑞_𝑖=1𝑆_𝑖𝑡+𝑃̇𝑡𝑃𝐴₁𝑡⋯𝑃𝐴𝑞_𝑇1𝑡𝐴⋮𝑡𝑃−𝑆₁𝑡−ℎ¹0⋮⋯⋮0_𝑇𝑞𝑡𝑃0−𝑆_𝑞𝑡−ℎ^𝑞⎤⎥⎥⎥⎥⎥⎥⎥⎦<0.(3.3) Assumption 2.1 of Section 2 remains unchanged while Assumptions 2.2 and 2.4 of Section 2 are modified under similar justifications as follows.

Assumption 3.1 b. All the eigenvalues 𝜆_𝑖(𝐴₀(𝑡)) of the matrix function 𝐴₀(𝑡) satisfy Re𝜆_𝑖(𝐴₀(𝑡))≤−𝜌₀₀<0; forall𝑡∈𝐑₀₊, forall𝑖∈𝜎(𝜎≤𝑛) for some 𝜌₀₀∈𝐑₊∶={0<𝑧∈𝐑}; that is, 𝐴₀(𝑡) is a stability matrix, forall𝑡∈𝐑₀₊.

Assumption 3.2 b. 𝐴_𝑗∶𝐑₀₊→𝐑^𝑛×𝑛 are almost everywhere time-differentiable with essentially bounded time derivative, forall𝑗∈𝑞∪{0} possessing eventual isolated bounded discontinuities, then esssup_{𝑡∈𝐑0+}(‖𝐴₀̇(𝑡)‖)≤𝛾_0ℓ<∞ with 𝛾_0ℓ being a ℓ-norm dependent nonnegative real constant and, furthermore, ∫^{𝑡𝑡+𝑇}‖𝐴₀̇(𝜏)‖𝑑𝜏≤𝜇₀𝑇+𝛼₀ for some 𝛼₀,𝜇₀∈𝐑₊, forall𝑡∈𝐑₀₊, and some fixed 𝑇∈𝐑₀₊ independent of 𝑡. If the time derivative does not exist then it is defined in the distributional sense as in Assumptions 2.3.

Assumption 3.3 b (for the resetting systems). All the eigenvalues 𝜆_𝑘(𝐴_0𝑗) satisfy Re𝜆_𝑘(𝐴_𝑗0)≤−𝜌₀₀<0; forall𝑡∈𝐑₀₊, forall𝑘∈𝜎_0𝑗(𝜎_0𝑗≤𝑛), forall𝑗∈𝑝; that is, 𝐴_0𝑗 are constant stability matrices with prescribed stability abscissa.

A parallel result to Theorem 2.12(i)-(ii) is the following.

Theorem 3.4. The subsequent properties hold.
(i)Assume that(i.a)the matrix functions 𝐴_𝑗(𝑡), forall𝑗∈𝑞∪{0} are subject to Assumption 2.1;(i.b)the switching law 𝜎 is such that 𝑄^(𝑡)<0, (3.3), forall𝑡∈𝐑₀₊for some time-differentiable real symmetric positive definite matrix function 𝑃∶𝐑₀₊→𝐑^𝑛×𝑛 and some real symmetric positive definite matrix functions 𝑆_𝑖∶𝐑₀₊→𝐑^𝑛×𝑛(∀𝑖∈𝑞). Thus, the system (2.1) is globally asymptotically Lyapunov's stable independent of the delays (i.e., for all delays ℎ_𝑖∈[0,∞), forall𝑖∈𝑞). A necessary condition is 𝐴^𝑇0(𝑡)𝑃(𝑡)+𝑃(𝑡)𝐴₀(𝑡)+𝑃̇(𝑡)<0, forall𝑡∈𝐑₀₊ what implies that 𝐴₀(𝑡) is a stability matrix of prescribed stability abscissa on 𝐑₀₊ except eventually on a real subinterval of finite measure of 𝐑₀₊.(ii) Assume that(ii.a)𝐴_𝑗(𝑡)=𝐴_𝑗𝑖, forall𝑗∈𝑞∪{0}, forall𝑡∈𝐑₀₊ for some 𝑖∈𝑝 (eventually being dependent on 𝑡) satisfying Assumption 3.3; (ii.b)the switching law 𝜎 is such that ST_rf(𝜎)=∅ (i.e., it generates reset switching instants only) with ST_𝑟(𝜎) being arbitrary, namely, the set of reset times is either any arbitrary strictly increasing sequence of nonnegative real values (i.e., the resetting switching never ends) or any finite set of strictly ordered nonnegative real numbers with a finite maximal (i.e., the resetting switching ends in finite time);(ii.c)𝑄^𝑖∗∶=⎡⎢⎢⎢⎢⎢⎢⎢⎣𝐴^𝑇0𝑖𝑃^∗+𝑃^∗𝐴_0𝑖+^𝑞_𝑖=1𝑆^∗𝑖𝑃^∗𝐴_1𝑖⋯𝑃^∗𝐴_𝑞𝑖𝐴⋮^𝑇1𝑖𝑃^∗−0𝑆^∗1_⋮0⋯0𝐴⋱⋮_𝑇𝑞𝑖𝑃^∗0⋯−𝑆^∗𝑞⎤⎥⎥⎥⎥⎥⎥⎥⎦<0;∀𝑖∈𝑝(3.4)for some 𝐑^𝑛×𝑛∋𝑃^∗=𝑃^∗𝑇>0, 𝐑^𝑛×𝑛∋𝑆^∗𝑖=𝑆^∗𝑖𝑇>0(∀𝑖∈𝑞). Thus, the switched system (2.1), obtained from switches among resetting systems (2.4), is globally asymptotically Lyapunov's stable and also globally exponentially stable independent of the delays forall𝑖∈𝑞. If (3.4) is replaced with 𝑄^𝑖∗≤−2𝜀𝐼_(𝑞+1)𝑛<0, forall𝑖∈𝑞, and some 𝜀∈𝐑₊ then the state trajectory decays exponentially with rate(−𝜀)<0.

Parallel results to Corollaries 2.13–2.15 are direct from Theorem 3.4 with the replacements 𝐴₀(𝑡)→𝐴^∗0 (a constant stability matrix), 𝐴_0𝑗(𝑡)→𝐴^∗0𝑗, forall𝑗∈𝑝 (a set of constant stability matrices with prescribed stability abscissa for the resetting configurations). Also, the subsequent result for global asymptotic stability independent of the delays, which is close to Theorem 2.16, follows by replacing Assumptions 2.2–2.4 by Assumptions 3.1–3.3.

Theorem 3.5. Under Assumptions 2.1 and 3.1–3.2, the following properties hold.
(i)The switched system (2.1) is globally asymptotically Lyapunov's stable independent of the delays, that is, for any delays ℎ_𝑖∈[0,∞), forall𝑖∈𝑝 and any switching law 𝜎 such that (a)the switching instants are arbitrary;(b)max(esssup‖𝐴̇₀(𝑡)‖∶𝑡∈𝐑₀₊) is sufficiently small compared to the absolute value of the prescribed stability abscissa of 𝐴₀(𝑡);(c)the support testing matrix of distributional derivatives Γ_𝐴𝑑0(𝑡) of the same matrices are semidefinite negative for all time instants, where the conventional derivatives do not exist (i.e., if𝐴̇₀(𝑡)=Γ_𝐴𝑑0(𝑡)(𝑡)𝛿(0)) .(ii)The switched system (2.1) is globally exponentially stable independent of the delays if (a)max(esssup‖𝐴̇₀(𝑡)‖∶𝑡∈𝐑₀₊) is sufficiently small compared to the absolute value of the prescribed stability abscissa of 𝐴₀(𝑡);(b)max(‖Γ_𝐴𝑑0(𝑡)(𝑡)‖∶𝐴̇₀(𝑡)=Γ_𝐴𝑑0(𝑡)(𝑡)𝛿(0),∀𝑡∈ST(𝜎)) is sufficiently small compared to the time intervals in between any two consecutive switching instants. If Assumptions 2.1 and 3.1–3.3 hold, then (iii)the switched system (2.1) is globally exponentially stable independent of the delays if max(esssup‖𝐴̇₀(𝑡)‖∶𝑡∈𝐑₀₊) is sufficiently small compared to the absolute value of the prescribed stability abscissa of 𝐴₀(𝑡) and, furthermore, the switching law 𝜎 is such that (a)At reset-free switching instants, max(‖Γ_𝐴𝑑0(𝑡)(𝑡)‖∶𝐴̇₀(𝑡)=Γ_𝐴𝑑0(𝑡)(𝑡)𝛿(0),∀𝑡∈ST_rf(𝜎)) is sufficiently small compared to the time intervals in between any two consecutive reset switching instants;(b)there exists a common Krasovsky-Lyapunov functional 𝑉(𝑡,𝑥_𝑡) defined with constant matrices 𝑃(𝑡)→𝑃=𝑃^𝑇>0 and 𝑆_𝑖(𝑡)→𝑆_𝑖=𝑆^𝑇𝑖>0, forall𝑖∈𝑞∪{0} in (3.1) for all the time-invariant resetting systems (2.4) and some of the conditions (b.1)-(b.2) of Theorem 2.16 hold forall𝑡∈ST_𝑟(𝜎) under the resetting action 𝑃(𝑡⁺)=𝑃; forall𝑖∈𝑞∪{0}.

4. Simulation Examples and Potential Future Research

In this section, some simulation examples showing numerically the application of the results introduced below are carried out. The section contains two examples: one related to the delay-dependent stability property introduced in Section 2 and another concerning the delay-independent one considered in Section 3. The resetting systems and the remaining potential jumps in any parameters are considered without explicit separation of the two phenomena since such a separation is not relevant for stability properties.

4.1. Delay-Dependent Stability

Consider the delay system 𝑥̇(𝑡)=𝐴₀(𝑡)𝑥(𝑡)+𝐴₁(𝑡)𝑥(𝑡−ℎ), where ℎ=0.75 second and each resetting system (2.4) with 𝑝=2 is defined by 𝐴𝐴₀₁(𝑡)=−10.3−co24s.5𝑡,𝐴₁₁(𝑡)=−20.2−𝑎2(𝑡.)1₀₂,(𝑡)=−01s−i4n2𝑡,𝐴₁₂(𝑡)=−10.5𝑏−(𝑡2)(4.1) with 𝑎(𝑡)=1/(⌊𝑡⌋+𝑡+1) and 𝑏(𝑡)=𝑡/(⌈𝑡⌉²+1), where ⌊⋅⌋ denotes the largest integer not larger than (·) and ⌈⋅⌉ denotes the smallest integer not smaller than (·). Note that 𝑎(𝑡) and 𝑏(𝑡) are discontinuous functions at integer values of time. A graphical representation of these functions is shown in Figures 1 and 2.

Figure 1 

Graphical representation of 𝑎(𝑡).

Figure 2 

Graphical representation of 𝑏(𝑡).

Initially, it will be checked that Theorem 2.16(ii) conditions hold. Firstly, the switching instants between resetting systems have been selected arbitrarily and defined by Figure 3.

Figure 3 

Sequence of switching instants between resetting systems.

Secondly, the time derivatives of the resetting systems defined above are given on each real interval (𝑘,𝑘+1)⊂𝑅 with 𝑘∈𝑁 by 𝐴̇₀₁=00−s04in𝑡,𝐴̇₁₁=00𝑎0(̇𝑡),𝐴̇₀₂=00co40s𝑡,𝐴̇₁₂=00𝑏(̇0𝑡)(4.2) with 𝑎(̇𝑡)=−1/(𝑘+𝑡+1); 𝑏(̇𝑡)=1/((1+𝑘)²+1). Thus, the “sufficiently smallness condition” mentioned in Theorem 2.16(i) is fulfilled according to its proof in Appendix if 2(𝜌₀/𝐾₀)²>esssup_{𝑡∈𝑅0+/ST(𝜎)}‖∑^1𝑗=0𝐴_𝑗‖, where 𝜆_max(𝑃(𝑡))≤𝐾₀/2𝜌₀ for the existing unique symmetric positive definite solution 𝑃(𝑡) of (∑^1𝑗=0𝐴^𝑇𝑗)𝑃(𝑡)+𝑃(𝑡)(∑^1𝑗=0𝐴_𝑗)=−𝐼_𝑛. Numerical computations lead to 10.33=2(𝜌₀/𝐾₀)²>esssup_{𝑡∈𝑅0+/ST(𝜎)}‖∑^1𝑗=0𝐴_𝑗‖=1.25 which guarantees the fulfilment the second item of the theorem. Finally, from Figures 1 and 2 above it becomes apparent that the distributional derivative at integer time instants (where the ordinary derivative does not exist) is negative since 𝑎(𝑡)>𝑎(𝑡⁺) and 𝑏(𝑡)>𝑏(𝑡⁺) for all 𝑡∈𝑁. Therefore, from Theorem 2.16(ii), the space-state trajectories of the solution asymptotically converge to zero as time evolves as Figure 4 shows.

Figure 4 

Convergence to zero of the state-trajectories.

The phase plane is shown in Figure 5, where it can be appreciated the convergence of the state evolution to the origin. Note that delay-dependent stability is achieved. As simulations show, the system becomes unstable as delay exceeds a certain threshold. Global asymptotic stability is guaranteed within the delay variation interval [0,1].

Figure 5 

Phase portrait of the evolution of the system.

4.2. Delay-Independent Stability

This example is devoted to the delay independent stability ensured by Theorem 3.5(i). Thus, consider the system 𝑥̇(𝑡)=𝐴₀(𝑡)𝑥(𝑡)+𝐴₁(𝑡)𝑥(𝑡−ℎ) and the resetting systems 𝐴₀₁=−01−123,𝐴₀₂=−20.5−12.6,𝐴₁=−20.5−𝑎1(𝑡.)6(4.3) with 𝑎(𝑡)=2+1/(1+𝑡)^𝑛/10 and 𝑛=⌊𝑡⌋. Again, this function possesses bounded discontinuities at integer values of time as Figure 6 shows.

Figure 6 

Graphical representation of 𝑎(𝑡).

Furthermore, the conditions of Theorem 3.5(i) are especially easy to verify since the resetting matrices 𝐴_0𝑖 are time-invariant and hence its time-derivatives are identically zero. The switching sequence is the same as depicted in Figure 3. Figures 7, 8, 9, and 10 show the convergence of the state trajectories of the system to zero for different values of the delay showing the delay independence property.

Figure 7 

State trajectories evolution for a delay of 1.75 seconds.

Figure 8 

State trajectories evolution for a delay of 7.25 seconds.

Figure 9 

State trajectories evolution for a delay of 15.8 seconds.

Figure 10 

State trajectories evolution for a delay of 35.8 seconds.

4.3. Potential Future Research

It is convenient to point out that the above ideas could be used for a better adjustment in Biology and Ecology mathematical models which have received increasing attention recently concerning epidemic propagation, species evolution, predation, and so forth (see, e.g., [25–35]), which can also include delays to better fix the trajectory solutions. For instance, a control theory point of view is given in [29] for the standard Beverton-Holt equation in Ecology which has two parameterizing sequences, namely, the environment carrying capacity (related to a favorable or not habitat for the population) and the intrinsic growth rate (related to the population ability to grow). The inverse of the environment carrying capacity is the control variable. The objective is that the solution trajectory matches a prescribed reference one. The stability results and the matching properties are revisited in [30] for the generalized Beverton-Holt equation which possesses two extra parameterizing sequences, namely, the harvesting quota (related to human intervention like, fishing/hunting) and the independent consumption (related to perturbations in the population levels). The above two models are discrete with a one-step delay. Other control variables apart from the carrying capacity inverse are taken in [31] and comparative results with the former case are provided. Finally, a modified generalized Beverton-Holt is discussed in [32] which is a more complex model than the former model in [30]. This model has a delay of two sampling periods, the new one introduces a penalty in the dynamics for large levels of populations. The strategy seems to be appropriate for certain populations of insects which have several reproduction cycles per year and whose population tends to blast in very short periods of time what makes it to fall after very much as a result. If a comparative is made between the various (standard, generalized, and modified generalized) models, one sees that the foreseen population evolution might depend significantly on the chosen model. Therefore, a switching model strategy between several kinds of single models each one subject to a set of distinct parameterizations could be useful to better adjust experimental data.

5. Conclusions

This paper has been devoted to the investigation of the stability of switched linear time-varying systems with internal constant point delays. The switching laws are allowed to possess two kinds of switching instants, in general. The reset instants are those related to switching the current system parameterization to some configuration within a prescribed set. At switching time instants which are not reset instants, any bounded jump of any of the system parameter function associated either with the delay-free or with delayed dynamics for any of the delays is allowed. The system delay-free matrix as well as the matrices of delayed dynamics is allowed to be time-varying and eventually time differentiable. Also, either the delay-free system matrix or the system matrix obtained by zeroing the matrices of dynamics of all nonzero delays are assumed to be stability matrices with prescribed stability abscissa for all time. The first assumption is used to obtain results for stability dependent on the sizes of the delays, while the second one is used for results concerning asymptotic stability independent of the delays. The parametrical bounded jumps at switching instants may be interpreted equivalently as Dirac impulses of the corresponding time derivatives. Global asymptotic stability and exponential stability results are obtained dependent on and independent of the sizes of the delays. Stability results are guaranteed based on the existence of a Krasovsky-Lyapunov functional through simple tests of negative definiteness of matrices for sufficiently small norms of the parametrical time derivatives, where such derivatives exist, compared to the above mentioned stability abscissas. In addition, the existence of a minimum residence time at each eventual resetting configuration is required to guarantee global asymptotic stability in the event that the Krasovsky-Lyapunov functional candidate has a positive jump at some reset switching instant.

Appendix

A. Mathematical Proofs

A.1. Proof of Theorem 2.12

(i) Denote by 𝑥_𝑡 the strip of state-trajectory solution 𝑥(𝑡+𝜏) of the system 2.1 for 𝜏∈[−ℎ,0]. Consider the Krasovsky-Lyapunov functional candidate: 𝑉𝑡,𝑥_𝑡=𝑥^𝑇𝑡𝑃𝑡𝑥𝑡+^𝑞_𝑞𝑖=1_𝑗=0^−ℎ𝑗_{−ℎ𝑖−ℎ𝑗}^{𝑡𝑡+𝜃}𝑥^𝑇𝜏𝑆_𝑖𝑗𝑥𝜏𝑑𝜏𝑑𝜃(A.1) which is nonnegative and radially unbounded since 𝑉(𝑡,𝑥_𝑡)→∞ as ‖𝑥_𝑡‖∶=sup_{−ℎ≤𝜏≤0}‖𝑥(𝜏)‖→∞, with ℎ∶=max_{1≤𝑖≤𝑞}(ℎ_𝑖), since all the eigenvalues of 𝑃(𝑡) and 𝑆_𝑖𝑗(𝑡) positive and uniformly bounded from above and below forall𝑡∈𝐑₀₊. This also implies that 𝑃̇(𝑡) cannot be neither positive definite nor negative definite forall𝑡∈𝐑₀₊. Direct calculations via 2.9 yield 𝑉̇(𝑡,𝑥_𝑡)≤−𝑥^𝑇(𝑡)𝑄(𝑡)𝑥(𝑡)<0, forall𝑡∈𝐑₀₊ if and only if 𝑥(𝑡)=(𝑥^𝑇(𝑡),𝑥^𝑇(𝑡−ℎ₁),…,𝑥^𝑇(𝑡−ℎ_𝑞))^𝑇≠0. Then, 𝑉(𝑡,𝑥_𝑡)≤𝑉(0,𝜑)<∞ and 𝑉(𝑡,𝑥_𝑡)→0 as 𝑡→∞ for any given bounded function of initial conditions what imply that ‖𝑥_𝑡‖<∞, forall𝑡∈𝐑₀₊ and ‖𝑥_𝑡‖→0 as 𝑡→∞. The global asymptotic stability has been proven. To prove the last part of Property (i), note that 2.9 implies (^𝑞_𝑗=0𝐴^𝑇𝑗𝑡)𝑃𝑡+𝑃𝑡(^𝑞_𝑗=0𝐴_𝑗𝑡)+𝑃̇𝑡+^𝑞_𝑞𝑖=1_𝑗=0ℎ_𝑖𝑆_𝑖𝑗<0,∀𝑡∈𝐑₀₊⇒(^𝑞_𝑗=0𝐴^𝑇𝑗𝑡)𝑃𝑡+𝑃𝑡(^𝑞_𝑗=0𝐴_𝑗𝑡)+𝑃̇𝑡<0.(A.2) Since 𝑃̇(𝑡)<0, forall𝑡∈𝐑₀₊ is impossible from the preceding part of the proof, it has to exist a nonnecessarily connected subinterval 𝑆_𝑅⊂𝐑₀₊ such that (∑^𝑞𝑗=0𝐴^𝑇𝑗(𝑡))𝑃(𝑡)+𝑃(𝑡)(∑^𝑞𝑗=0𝐴_𝑖(𝑡))<0, forall𝑡∈𝑆_𝑅. Now, proceed by contradiction to prove that 𝑆_𝑅 has infinite measure with a connected component of infinite measure by assuming that the system 2.1 is globally asymptotically stable in the following cases.

(1)𝑆_𝑅 has finite measure so that the complement 𝑆_𝑅 in 𝐑₀₊ is nonconnected with infinite measure with a component being necessarily of infinite measure (otherwise, 𝑆_𝑅 has infinite measure). Thus, (∑^𝑞𝑗=0𝐴^𝑇𝑗(𝑡))𝑃(𝑡)+𝑃(𝑡)(∑^𝑞𝑗=0𝐴_𝑗(𝑡))≥0 or indefinite forall𝑡∈𝑆_𝑅. Since 𝑆_𝑅 has finite measure and 𝑆_𝑅 has a component of infinite measure, it exists a sufficiently large finite 𝑡₀∈𝐑₀₊ such that 𝑆_𝑅∋𝑡≥𝑡₀ so that ‖𝑥_𝑡‖→0 as 𝑡→∞ is impossible what leads to a contradiction.(2)Both intervals 𝑆_𝑅 and 𝑆_𝑅 have infinite measures so that they are nonconnected and have infinite components each of them with finite measure. Thus, asymptotic stability is also impossible.

As conclusion, (∑^𝑞𝑗=0𝐴_𝑗(𝑡)) is a stability matrix forall𝑡∈𝐑₀₊ except possibly within an interval of finite measure.

(ii) Denote by 𝑧_𝑡 the strip of state-trajectory solution 𝑧(𝑡+𝜏), for 𝜏∈[−ℎ,0] and any resetting system. Consider the Krasovsky-Lyapunov functional candidate for all the resetting systems: 𝑉^∗𝑡,𝑧_𝑡=𝑧^𝑇𝑡𝑃^∗𝑧𝑡+^𝑞_𝑞𝑖=1_𝑗=0^−ℎ𝑗_{−ℎ𝑖−ℎ𝑗}^{𝑡𝑡+𝜃}𝑧^𝑇𝜏𝑆^∗𝑖𝑗𝑧𝜏𝑑𝜏𝑑𝜃.(A.3)

The real functional A.1 is a common Krasovsky-Lyapunov functional for all 𝑝 distinct resetting systems for all delays ℎ_𝑖∈[0,ℎ_𝑖](∀𝑖∈𝑞), provided that 𝑧_𝑡≡𝑧(𝑡+𝜏), forall𝜏∈[−ℎ,0] and ℎ∶=max_{1≤𝑖≤𝑞}(ℎ_𝑖) since the 𝑖th resetting system 2.4 satisfies from 2.11𝑉^∗𝑖̇𝑡,𝑧_𝑡≤𝑧^𝑇𝑡𝑄^∗𝑖𝑧𝑡<0;∀𝑖∈𝑞,∀𝑡∈𝐑₀₊(A.4) for all nonzero 𝑧(𝑡) where 𝑧(𝑡)∶=(𝑧^𝑇(𝑡),𝑧^𝑇(𝑡−ℎ₁),…,𝑧^𝑇(𝑡−ℎ_𝑞))^𝑇 and accordingly 𝑉̇^∗𝑡,𝑥_𝑡≤−|||min_𝑖∈𝑞𝑥^𝑇𝑡𝑄^∗𝑖𝑥𝑡|||≤0,∀𝑡∈𝐑₀₊(A.5) for all nonzero 𝑥(𝑡) where 𝑥(𝑡)∶=(𝑥^𝑇(𝑡),𝑥^𝑇(𝑡−ℎ₁),…,𝑥^𝑇(𝑡−ℎ_𝑞))^𝑇 for the switched system 2.1 since ST_rf𝜎=∅⇒⇒S−T(_𝑟m𝜎ax_𝑖∈𝑝=ST_𝑟𝜎≤−(min_𝑖∈𝑝𝜆_max𝑄^∗𝑖)𝑧^𝑇𝑡𝑧𝑡≤𝑉̇^∗𝑡,𝑧_𝑡𝜆_min𝑄^∗𝑖𝑖)𝑧^𝑇𝑡𝑧𝑡<0(A.6) for any 𝑡∈𝐑₀₊ such that 𝑧(𝑡)≠0, where 𝜆_max(⋅) and 𝜆_min(⋅) stand for maximum and minimum eigenvalues of real symmetric matrices. Thus, if 2.9 holds then the candidate A.3 is a common Krasovsky-Lyapunov functional for all the resetting systems, and then for the switched system 2.1 which is then globally asymptotically Lyapunov's stable. Furthermore, the Krasovsky-Lyapunov functional of the switched system 2.1 fulfils from A.3–A.6: 𝑉̇^∗𝑡,𝑥^𝑡𝑉^∗𝑡,𝑥^𝑡≤−𝛿<0⇒𝑉^∗𝑡,𝑥^𝑡≤𝑒^−𝛿𝑇𝑉^∗𝑡−𝑇,𝑥^𝑡−𝑇≤𝐾𝑒^−𝛿𝑡𝑉^∗0,𝜑(A.7)forall𝑡∈𝐑₀₊, for some finite 𝐾∈𝐑₊, where 𝛿∶=|max_𝑖∈𝑝𝜆_max(𝑄^∗𝑖)|/𝜆_min(𝑃^∗)>0. Then, from A.7 and A.3𝜆_min𝑃^∗𝑥^𝑇𝑡≤𝑥𝐾𝑡𝐾≤₁𝑉^∗𝑡,𝑥^𝑡≤𝑒^−𝛿𝑇𝑉^∗𝑡−𝑇,𝑥^𝑡−𝑇≤𝐾𝑒^−𝛿𝑡𝑉^∗0,𝜑𝑒^−𝛿𝑡sup₋ℎ_≤𝜏≤0‖‖𝜑𝜏‖‖²²⇒‖‖𝑥𝑡‖‖₂≤√𝐾𝐾₁𝑒^{−(𝛿/2)𝑡}sup₋ℎ_≤𝜏≤0‖‖𝜑𝜏‖‖₂(A.8)forall𝑡∈𝐑₀₊, and some 𝐾∈𝐑₊, where 𝐾₁∶=𝜆_max(𝑃^∗)+(𝑞+1)max_{𝑖∈𝑝,𝑗∈𝑞∪{0}}𝜆_max(𝑆_𝑖𝑗) and ℎ∶=max_𝑖∈𝑞ℎ_𝑖 and ‖⋅‖₂ denotes the ℓ₂ (or spectral) vector norm or the corresponding induced ones for matrices. Then, the system 2.1 is globally exponentially stable for all delays ℎ_𝑖∈[0,ℎ_𝑖], forall𝑖∈𝑞. The modification of 2.11-2.12 with 𝑄_𝑖≤−2𝜀𝐼_(𝑞+1)𝑛<0, forall𝑖∈𝑞 leads directly to an exponential decay of ‖𝑥(𝑡)‖₂ with rate 𝛿=2𝜀 from a similar slightly extended proof.

(iii) If Assumption 2.4 holds then (∑^𝑞𝑗=0𝐴^𝑇𝑗𝑖)𝑃^∗+𝑃^∗(∑^𝑞𝑗=0𝐴_𝑗𝑖)<0 for any 𝑃^∗=𝑃^∗𝑇>0. Thus, it exists a sufficiently small ℎ∶=max_𝑖∈𝑞ℎ_𝑖 such that 2.9 holds forallℎ_𝑖∈[0,ℎ_𝑖], since 𝑅^∗𝑖>0; forall𝑖∈𝑞.

If Property (i) holds then (∑^𝑞𝑗=0𝐴^𝑇𝑗𝑖)𝑃^∗+𝑃^∗(∑^𝑞𝑗=0𝐴_𝑗𝑖)+∑^𝑞𝑖=1∑^𝑞𝑗=0ℎ_𝑖𝑆_𝑖𝑗<0 what implies (∑^𝑞𝑗=0𝐴^𝑇𝑗𝑖)𝑃^∗+𝑃^∗(∑^𝑞𝑗=0𝐴_𝑗𝑖)<0 since ∑^𝑞𝑖=1∑^𝑞𝑗=0ℎ_𝑖𝑆_𝑖𝑗≥0 for any ℎ_𝑖≥0, forall𝑖∈𝑞. Thus, 𝑄_id∶=BlockDiag((∑^𝑞𝑗=0𝐴^𝑇𝑗𝑖)𝑃^∗+𝑃^∗(∑^𝑞𝑗=0𝐴_𝑗𝑖)+∑^𝑞𝑖=1∑^𝑞𝑗=0ℎ_𝑖𝑆_𝑖𝑗<0,−𝑅^∗1,…,−𝑅^∗𝑞)<0, forall𝑖∈𝑞 and then 𝑄_𝑖<0, forallℎ_𝑖∈[0,ℎ_𝑖] and sufficiently small ℎ∶=max_𝑖∈𝑞ℎ_𝑖 since from 2.11‖𝑄_𝑖−𝑄_id‖ is a monotonically increasing function of the argument (ℎ₁,…,ℎ_𝑞) being zero if ℎ_𝑖=0, forall𝑖∈𝑞. Then, Property (i) holds for a sufficiently small ℎ.

A.2. Derivation of the Inequality 2.17

It follows from the subsequent inequalities: ‖‖𝑀^∗‖‖₂≤^𝑞_𝑖=0‖‖𝐴^∗𝑖‖‖₂,‖‖𝑀(𝑡)‖‖₂≤^𝑞_𝑖=0‖‖𝑀𝑡₊‖‖𝐴_𝑖(𝑡)‖‖₂,−𝑀(𝑡)‖‖₂≤^𝑞_𝑖=0‖‖𝐴_𝑖𝑡⁺−𝐴^𝑖(𝑡)‖‖₂,‖‖𝑀𝑡⁺−𝑀^∗‖‖₂≤^𝑞_𝑖=0‖‖𝐴_𝑖𝑡⁺−𝐴^∗𝑖‖‖Δ_𝑖‖‖₂,𝑡⁺−Δ^𝑖(𝑡)‖‖₂=‖‖+𝐴^𝑖𝐴𝑡⁺−𝐴^𝑖(𝑡)𝑀^∗+𝐴^∗𝑖𝑀𝑡⁺−𝑀(𝑡)_𝑖𝑡⁺𝑀𝑡⁺−𝑀^∗+𝐴^𝑖(𝑡)𝑀(𝑡)−𝑀^∗‖‖₂≤‖‖𝐴_𝑖𝑡⁺−𝐴^𝑖(𝑡)‖‖₂(^𝑞_𝑗=0‖‖𝐴^∗𝑗‖‖₂)+‖‖𝐴^∗𝑖‖‖₂‖(^𝑞_𝑗=0‖‖𝐴_𝑗𝑡⁺−𝐴^𝑗(𝑡)‖‖₂)‖₂+‖‖𝐴_𝑖𝑡⁺−𝐴^∗𝑖‖‖₂(^𝑞_𝑗=0‖‖𝐴_𝑗𝑡⁺−𝐴^∗𝑗+‖‖𝐴_𝑖‖‖₂)(𝑡)−𝐴^∗𝑖‖‖₂(^𝑞_𝑗=0≤𝑎₊‖‖𝐴_𝑗(𝑡)−𝐴^∗𝑗‖‖₂)(^𝑞_𝑗=0‖‖𝐴^∗𝑗‖‖₂+(𝑞+1)‖‖𝐴^∗𝑖‖‖₂+(𝑞+1)𝑎⁺+^𝑞_𝑗=0+‖‖𝐴_𝑖‖‖𝐴_𝑗(𝑡)−𝐴^∗𝑗‖‖₂)(𝑡)−𝐴^∗𝑖‖‖₂[(𝑞+1)𝑎⁺+^𝑞_𝑗=0‖‖𝐴_𝑗(𝑡)−𝐴^∗𝑗‖‖₂+^𝑞_𝑗=0≤𝑎₊‖‖𝐴_𝑗(𝑡)−𝐴^∗𝑗‖‖₂](^𝑞_𝑗=0‖‖𝐴^∗𝑗‖‖₂+(𝑞+1)‖‖𝐴^∗𝑖‖‖₂+(𝑞+1)𝑎+𝑎⁺+(𝑞+1)2𝑎+𝑎₊)𝑎,∀𝑖∈𝑞∪{0}.(A.9)

Proof. (i) It can be considered by the obvious nature of the process that the switching set ST(𝜎) of the switching law 𝜎 is defined by the discrete set of times where the time derivative of some of the entries of some of the delay-free or delayed matrices of dynamics does not exist, or equivalently, is impulsive which translated in a bounded discontinuity of the corresponding matrix function at such a time instant. Conversely, a bounded discontinuity of any of such matrices is equivalent to a distributional time derivative. Thus, if Assumptions 2.1–2.3 hold then (∑^𝑞𝑗=0𝐴_𝑗(𝑡)) is a stability matrix, forall𝑡∈𝐑₀₊, so that a positive definite symmetric 𝑃(𝑡) has to exist as a unique solution to the following matrix Lyapunov equation 𝑄(𝑡) in 2.9 takes the form (^𝑞_𝑗=0𝐴^𝑇𝑗𝑡)𝑃𝑡+𝑃𝑡(^𝑞_𝑗=0𝐴_𝑗𝑡)=−𝐼_𝑛,(A.10) where 𝐼_𝑛 is the 𝑛th identity matrix. Taking time-derivatives in A.9 yields to the subsequent matrix Lyapunov equation: Thus, the unique solutions to the above Lyapunov equations are, respectively, 𝑃̇𝑡=𝑃𝑡=^∞0𝑒^{∑𝑞𝑗=0𝐴𝑇𝑗}^𝑡^𝜏𝑒^{∑𝑞𝑗=0𝐴𝑗}^𝑡^𝜏𝑑𝜏,(A.12)_∞0𝑒^{∑𝑞𝑗=0𝐴𝑇𝑗}^𝑡^𝜏((^𝑞_𝑗=0𝐴̇^𝑇𝑗𝑡)𝑃𝑡+𝑃𝑡(^𝑞_𝑗=0𝐴̇_𝑗𝑡))𝑒^{∑𝑞𝑗=0𝐴𝑗}^𝑡^𝜏𝑑𝜏,(A.13)𝜆_max𝑃𝑡=‖‖𝑃𝑡‖‖₂≤𝐾⁰2𝜌₀||𝜆,(A.14)_max𝑃̇𝑡||=‖‖𝑃̇𝑡‖‖₂≤𝐾⁰𝜆_max𝑃𝑡𝜌₀‖^𝑞_𝑗=0𝐴̇_𝑗𝑡‖₂≤𝐾²⁰2𝜌²⁰‖^𝑞_𝑗=0𝐴̇_𝑗𝑡‖₂≤𝐾²⁰𝛾₂2𝜌²⁰,(A.15) with ∑^𝑞𝑗=0‖𝐴̇_𝑗(𝑡)‖₂≤𝛾₂∈𝐑₀₊𝑃(𝑡) provided that 𝑛², from Assumptions 2.3.
Remark A.1. Note that 𝑃(𝑡), A.12, may be (pointwise) equivalently calculated from the linear algebraic system below of 𝑛²×𝑛² unknowns (the entries of [(^𝑞_𝑗=0𝐴^𝑇𝑗𝑡)⊗𝐼_𝑛+𝐼_𝑛⊗(^𝑞_𝑗=0=vec−𝐼_𝑛𝐴^𝑇𝑗𝑡)]vec𝑃𝑡=−𝑒^𝑇1,𝑒^𝑇1,…,𝑒^𝑇1^𝑇⟺vec𝑃𝑡=[(^𝑞_𝑗=0𝐴^𝑇𝑗𝑡)⊗𝐼_𝑛+𝐼_𝑛⊗(^𝑞_𝑗=0𝐴^𝑇𝑗𝑡)]⁻¹vec−𝐼_𝑛^𝑇,(A.16) at each time) and ⊗ coefficient matrix which is equivalent to the matrix equation A.10: vec(𝑀)∶=(𝑚^𝑇1,𝑚^𝑇2,…,𝑚^𝑇𝑛)^𝑇∈𝐑^𝑛𝑚 where “𝑀=(𝑚₁,𝑚₂,…,𝑚_𝑛)^𝑇” defines the Kronecker (or direct) product of matrices and 𝑛×𝑚 if 𝑚^𝑇𝑖(𝑖∈𝑛) is an 𝑚 real matrix of rows 𝑃(𝑡) each of 𝑄(𝑡)≤𝑄(𝑡) components. Note that the coefficient matrix of the above algebraic system is everywhere nonsingular since forall𝑡∈𝐑₀₊⧵ST(𝜎) exists and it is unique for all time.
Thus, one gets from 2.9 that 𝑄𝑡∶=⎡⎢⎢⎢⎢⎢⎢⎢⎣−1−𝐾²⁰𝛾₂2𝜌²⁰𝐼^𝑛+^𝑞_𝑞𝑖=1_𝑗=0ℎ₁ℎ_𝑖𝑆_𝑖𝑗ℎ₁𝑃𝑡𝐴₁𝑡𝑀𝑡⋯ℎ_𝑞𝑃𝑡𝐴_𝑞𝑡𝑀𝑡𝑀^𝑇𝑡⋮𝐴^𝑇1𝑡𝑃𝑡−0𝑅_1⋮0⋯0ℎ⋱⋮_𝑞𝑀^𝑇𝑡𝐴^𝑇𝑞𝑡𝑃𝑡0⋯−𝑅_𝑞⎤⎥⎥⎥⎥⎥⎥⎥⎦,(A.17), ℎ, where provided that 𝑡∈ST(𝜎) is sufficiently small and 𝑄𝑡⁺∶=⎡⎢⎢⎢⎢⎢⎢⎢⎣𝑃̇𝑡𝛿0ℎ¹ℎ₁𝑃𝑡𝐴₁𝑡⁺𝑀𝑡⁺⋯ℎ^𝑞𝑃𝑡𝐴_𝑞𝑡⁺𝑀𝑡⁺𝑀^𝑇𝑡⁺⋮𝐴^𝑇1𝑡⁺𝑃𝑡00_⋮0⋯0ℎ⋱⋮_𝑞𝑀^𝑇𝑡⁺𝐴^𝑇𝑞𝑡⁺𝑃𝑡0⋯0⎤⎥⎥⎥⎥⎥⎥⎥⎦,(A.18). On the other hand, one has for 𝛿(0) where 𝑃̇(𝑡)=𝑃̇(𝑡⁺)−𝑃̇(𝑡) is the unity scalar Dirac distribution centered at 𝑃𝑃̇𝑡𝑡⁺=^∞0𝑒^{∑𝑞𝑗=0𝐴𝑇𝑗}^𝑡+^𝜏𝑒^{∑𝑞𝑗=0𝐴𝑗}^𝑡+^𝜏𝑑𝜏,₊=^∞0𝑒^{∑𝑞𝑗=0𝐴𝑇𝑗}^𝑡+^𝜏((^𝑞+𝑃_𝑗=0𝑡𝐴̇^𝑇𝑗(_𝑞𝑡+Γ^{𝑇𝐴𝑑𝑗}𝑡)𝑃𝑡_𝑗=0𝐴̇^𝑗𝑡+Γ_𝐴𝑑𝑗𝑡))𝑒^{∑𝑞𝑗=0𝐴𝑗}^𝑡+^𝜏𝑑𝜏(A.19) and |||𝜆_max𝑃̇𝑡|||≤𝐾²⁰𝛾₂2𝜌²⁰;|||𝜆_max𝑃̇𝑡⁺|||≤𝐾²⁰𝛾²+∑^𝑞𝑗=0‖‖Γ_𝐴𝑑𝑗𝑡‖‖₂𝛿02𝜌²⁰0<𝜆_max,𝑃𝑡≤2𝐾𝜌⁰₀;0<𝜆_max𝑃𝑡⁺≤𝐾⁰1+∑^𝑞𝑗=0‖‖Γ_𝐴𝑑𝑗𝑡‖‖₂2𝜌₀,(A.20) subject to so that with 𝑛, or equivalently 𝐴_𝑗(𝑡), being a real 𝑡-matrix whose entries are zero if 𝑄𝑡⁺−≤𝑄⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣𝐾𝑡₂₀(∑^𝑞𝑗=0‖‖Γ_𝐴𝑑𝑗𝑡‖‖₂)2𝜌²⁰ℎ₁𝛿0𝐼_𝑛ℎ₁𝜗₁𝜆_max𝑃𝑡𝐸⋯ℎ^𝑞𝜗_𝑞𝜆_max𝑃𝑡𝐸𝜗₁𝜆_max⋮𝑃𝑡𝐸00_⋮0⋯0ℎ⋱⋮_𝑞𝜗_𝑞𝜆_max𝑃𝑡𝐸0⋯0⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦,(A.21) is continuous at time 𝜗_𝑖≤‖Γ_𝐴𝑑𝑖(𝑡)‖₂(∑^𝑞𝑗=0‖Γ_𝐴𝑑𝑗(𝑡)‖₂), and each nonzero entry is the amplitude of any impulse at the time derivative. Thus, one gets from A.18 where 𝐸𝑛×(𝑞+1)𝑛 and 𝑄(𝑡)<0 is a real ∑^𝑞𝑗=0Γ_𝐴𝑑𝑗(𝑡)≤0-matrix with all its entries being unity. Since forall𝑡∈ST(𝜎), it follows from A.21 that if the matrix ℎ, (∑^𝑞𝑗=0‖Γ_𝐴𝑑𝑗(𝑡)‖₂), then the (1,1) block-matrix in A.18 is semidefinite negative for any (sufficiently small) 𝜗∶=max(𝜗_𝑖∶𝑖∈𝑞∪{0}) compared to 2(𝜌₀/𝐾₀)²>𝛾₂≥esssup_{𝑡∈𝐑0+⧵ST(𝜎)}‖∑^𝑞𝑗=0𝐴̇_𝑗(𝑡)‖₂ and forall𝑡∈𝐑₀₊⧵ST(𝜎) provided that 𝑄𝑡≤⎡⎢⎢⎢⎢⎢⎢⎢⎣−𝐼^𝑛+𝐾²⁰2𝜌²⁰‖^𝑞_𝑗=0𝐴_𝑗̇𝑡‖₂𝐼_𝑛+^𝑞_𝑞𝑖=1_𝑗=0ℎ₁ℎ_𝑖𝑆_𝑖𝑗ℎ₁𝑃𝑡𝐴₁𝑡𝑀𝑡⋯ℎ_𝑞𝑃𝑡𝐴_𝑞𝑡𝑀𝑡𝑀^𝑇𝑡⋮𝐴^𝑇1𝑡𝑃𝑡−0𝑅_1⋮0⋯0ℎ⋱⋮_𝑞𝑀^𝑇𝑡𝐴^𝑇𝑞𝑡𝑃𝑡0⋯−𝑅_𝑞⎤⎥⎥⎥⎥⎥⎥⎥⎦(A.22), 𝑄^(𝑡)∶=𝑄(𝑡)−BlockDiag((𝐾²⁰/2𝜌²⁰‖∑^𝑞𝑗=0𝐴̇_𝑗(𝑡)‖−1)𝐼_𝑛,0,…,0). Thus, Property (i) has been proven.
(ii) From 2.9 and A.15, so that if forall𝑡∈𝐑⁺⁰⧵ST(𝜎), 𝑉̇(𝑡,𝑥_𝑡)≤0,forall𝑡(≥𝑡^),𝑡^∈𝐑₀₊, since 𝐑₊∋𝛽∶=inf_{𝑡∈𝐑0+}𝜆_min(𝑃(𝑡)) with 𝑃(𝑡)>0,forall𝑡∈𝐑₀₊ from the properties of the Krasovsky-Lyapunov functional Equation A.1 used in the proof of Theorem 2.12, where 𝑜 since 𝑂 and the “small-𝑓=𝑜𝑠⟺𝑓=𝑂𝑠⟺||𝑓||≤𝑘₁||𝑠||+𝑘₂,some𝑘_1,2∈𝐑₀₊∃lim_𝑠→0|||𝑓𝑠|||=0.(A.24)” and “big-𝛾₂≤2(𝜌₀/𝐾₀)(1−𝜀)” Landau's notations mean the following: Thus, if ℎ for some ‖‖𝑥𝑡⁺‖‖²²≤𝑉𝑡⁺,𝑥_𝑡+𝛽≤𝐾^𝑉≤𝐾_𝑉𝑒^{−𝜁𝑇𝑘}𝑉𝑡⁺−𝑇_𝑘,𝑥_{𝑡+−𝑇𝑘}≤𝐾_𝑉𝑒^{−𝜁𝑇𝑘}𝑉𝑡−𝑇^𝑘,𝑥_{𝑡−𝑇𝑘}+𝑉𝑡⁺−𝑇_𝑘,𝑥_{𝑡+−𝑇𝑘}−𝑉𝑡−𝑇^𝑘,𝑥_{𝑡−𝑇𝑘}𝐾𝑒^{−𝜁𝑡𝑘}(^𝑘_𝑖=1≤𝐾_𝑉𝜂𝑡_𝑖)max_{0≤𝑖≤𝑞}‖‖𝑥𝑡−𝑡_𝑘−ℎ_𝑖‖‖²²<∞𝐾𝑒^{−𝜁𝑡𝑘}1+𝜂^𝑘max_{0≤𝑖≤𝑞}‖‖𝑥𝑡−𝑡_𝑘−ℎ_𝑖‖‖²²<∞.(A.25) and 𝐾_𝑉,𝐾∈𝐑₊ is sufficiently small then one gets from A.23 that With the following.
(a) ST(𝜎)={𝑡_𝑖∶𝑖∈𝐍} being some constants independent of 𝑡∈(𝑡_𝑘,𝑡_𝑘+1]∩𝐑₊.
(b) 𝑡_𝑘<𝑡_𝑘+1∈ST(𝜎) is defined such that it is a strictly ordered set of switching instants, 𝑇_𝑘∶=𝑡_𝑘−𝑡_𝑘−1 provided that 𝑡_𝑘=∑^𝑘𝑖=1𝑇_𝑖 exists so that, 𝑡∈(𝑡_𝑘,∞) (i.e., 𝑇_𝑘∈𝐑₊), and 𝑘∈𝐍 for any 𝑡_𝑘 and some ST(𝜎), otherwise, (i.e., if 𝜂𝑡_𝑖≤1+𝜂with𝜂∶=max𝑉𝑡⁺−𝑇_𝑘,𝑥_{𝑡+−𝑇𝑘}−𝑉𝑡−𝑇^𝑘,𝑥_{𝑡−𝑇𝑘}∶𝑡∈𝐑⁰⁺.(A.26) is the existing maximal element in the set lim_𝑡→∞‖𝑥(𝑡)‖²²=0 provided that it is of finite cardinal) Thus, 𝜀₀(<1)∈𝐑₊ with exponential convergence rate for any admissible function of initial conditions of the state-trajectory solution −𝜁𝑡_𝑘(1+𝜂)^𝑘≤1−𝜀₀⇔1≤ln(1+𝜂)≤(𝜁𝑡_𝑘+ln(1−𝜀₀))/𝑘 provided that for some 𝑇_𝑘=𝑡_𝑘−𝑡_𝑘−1≥(1/𝜁)ln((1+𝜂)/(1−𝜀₀)): 𝑛 which is guaranteed, in particular, if 𝑆𝐼. Then, Property (ii) follows directly.
(iii) The proof is direct since either Condition (b.1) or Condition (b.2) allow to derive a similar proof as that proof Property (ii).

Acknowledgments

The authors are very grateful to the Spanish Ministry of Education for its partial support of this work through Project DPI2006-00714. They are also grateful to the Basque Government for its support through GIC07143-IT-269-07, SAIOTEK SPED06UN10, and SPE07UN04.