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Discrete Dynamics in Nature and Society
Volume 2008 (2008), Article ID 231710, 31 pages
http://dx.doi.org/10.1155/2008/231710
Research Article

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

1Department of Electricity and Electronics, Faculty of Science and Technology, University of Basque, Campus of Leioa (Bizkaia), Aptdo. 644, 48080 Bilbao, Spain
2Department of Telecommunication and Systems Engineering, Engineering School, Autonomous University of Barcelona, Cerdanyola del Vallés, 08193 Bellaterra, Barcelona, Spain

Received 22 July 2008; Accepted 25 September 2008

Academic Editor: Antonia Vecchio

Copyright © 2008 M. de la Sen and A. Ibeas. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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., [19]). Time-delay models are useful for modeling both linear systems (see, e.g., [14, 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, 613] 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 [1417], 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 [1921]. 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 [1921].

(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., [2428] 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: 𝑡=̇𝑥𝑞𝑗=0𝐴𝑗𝑡𝑥𝑡𝑗(2.1) for any given bounded piecewise absolutely continuous function 𝜑[,0]𝐑𝑛 of initial conditions, where =max1𝑗𝑞(𝑗) with 0=0, for some delays 𝑗[0,𝑗], of finite or infinite maximum allowable delays sizes 𝑗𝐑0+, forall𝑗𝑞={1,2,,𝑞}, where 𝐑0+ is the nonnegative real axis 𝐑0+=𝐑+{0}={0𝑧𝐑}; and 𝐴𝑗𝐑0+𝐑𝑛×𝑛, for all 𝑗𝑞{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 𝐴𝑗𝐑0+𝐑𝑛×𝑛 are piecewise continuous and uniformly bounded for all 𝑗𝑞{0}.

Assumption 2.2. All the eigenvalues 𝜆𝑖(𝑞𝑗=0𝐴𝑗(𝑡)) of the matrix function 𝑞𝑗=0𝐴𝑗(𝑡) satisfy Re𝜆𝑖(𝑞𝑗=0𝐴𝑗(𝑡))𝜌0<0 for all 𝑡𝐑0+, forall𝑖𝜎(𝜎𝑛) for some 𝜌0𝐑+={0<𝑧𝐑}, that is, 𝑞𝑗=0𝐴𝑗(𝑡) is a stability matrix for all 𝑡𝐑0+.

Assumption 2.3. The matrix functions 𝐴𝑗𝐑0+𝐑𝑛×𝑛 are almost everywhere time differentiable with essentially bounded time derivative for all 𝑗𝑞{0} possessing eventual isolated bounded discontinuities, then esssup𝑡𝐑𝟎+(𝑞𝑗=0̇𝐴𝑗(𝑡))𝛾< with 𝛾 being a -norm dependent nonnegative real constant and, furthermore, 𝑡𝑡+𝑇̇𝐴𝑗𝜏𝑑𝜏𝜇𝑗𝑇+𝛼𝑗𝜇𝑇+𝛼𝑗𝑞{0}(2.2) for some 𝛼𝑗,𝜇𝑗,𝛼,𝜇𝐑+ for all 𝑡𝐑0+, and some fixed 𝑇𝐑0+ independent of 𝑡.
At time instants 𝑡, where the time-derivative of some entry of 𝐴𝑗(𝑡) does not exist for any 𝑗𝑞{0}, the time derivative is defined in a distributional Dirac sense as ̇𝐴𝑗(𝑡)=Γ(𝑡)𝛿(0) what equivalently means the presence of a discontinuity at 𝑡 in 𝐴𝑗(𝑡) defined as 𝐴𝑗𝑡+=𝐴𝑗𝑡+lim𝜀0+𝜀𝜀Γ𝑗𝜏𝛿𝑡𝜏𝑑𝜏=𝐴𝑗𝑡+Γ𝑗𝑡.(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 𝑡𝐑+ for any given piecewise absolutely continuous function 𝜑[,0]𝐑𝑛 of initial conditions. This follows from Picard-Lindelöff existence and uniqueness theorem. Assumption 2.2 establishes that 𝑞𝑗=0𝐴𝑗(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 𝐴𝑗0 for all 𝑗𝑞, 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=0𝐴𝑗(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 ̇𝐴𝑗(𝑡), for all 𝑗𝑞{0} (see second part of Assumption 2.1) might be stated in terms of sufficiently smallness of Δ𝐴𝑗(𝑡) for Δ𝐴𝑗(𝑡)=𝐴𝑗(𝑡)𝐴𝑗 for all 𝑡𝐑0+ for all 𝑗𝑞{0} for some constant stability matrix 𝑞𝑗=0𝐴𝑗 whose eigenvalues satisfy Re𝜆𝑖(𝑞𝑗=0𝐴𝑗)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=0𝐴𝑗(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 𝐴𝑗(𝑡) for 𝑗𝑞{0}. At such times 𝐴𝑗(𝑡+) denote right values of the matrix function while 𝐴𝑗(𝑡) is simply denoted by 𝐴𝑗(𝑡). A set of 𝑝 resetting systems of (2.1) is defined by the linear time-invariant systems: ̇𝑧𝑗𝑡=𝑞𝑖=0𝐴𝑖𝑗𝑧𝑗𝑡𝑗(2.4) for some given 𝐴𝑗𝑖𝐑𝑛×𝑛 for all 𝑗𝑝 for all 𝑖𝑞{0} for some given 𝑝𝐍. 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 𝜆𝑘(𝑞𝑖=0𝐴𝑖𝑗) satisfy Re𝜆𝑘(𝑞𝑖=0𝐴𝑖𝑗)𝜌0<0 for all 𝑡𝐑0+ for all 𝑘𝜎𝑗0(𝜎𝑗0𝑛) for all 𝑗𝑝; that is, 𝑞𝑖=0𝐴𝑖𝑗 are constant stability matrices with prescribed stability abscissa 𝜌0<0 for all 𝑗𝑝.

The following definitions are then used.

Definition 2.5. The switching matrix function is a mapping 𝜎𝐑0+{(𝐴𝑗(𝑡+)𝐴𝑗(𝑡)),𝑗𝑞{0}}𝐑𝑛×(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). 𝑡𝐑0+ is a switching instant if 𝐴𝑗(𝑡+)𝐴𝑗(𝑡) for some 𝑗𝑞{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). 𝑡𝐑0+ is a reset instant generated by the switching law 𝜎 if 𝑡ST(𝜎) and 𝐴𝑖(𝑡+)=𝐴𝑖𝑗 for some 𝑖𝑞{0} and some 𝑗𝑝, provided that 𝐴𝑖(𝑡)=𝐴𝑖𝑘𝐴𝑖𝑗 for some 𝑘(𝑗)𝑝.

The set of reset instants generated by the switching law 𝜎 is denoted by ST𝑟(𝜎). Note that ST𝑟(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, ̇𝐴𝑖(𝑡)=(𝐴𝑖𝑗𝐴𝑖(𝑡))𝛿(0) for some 𝑖𝑞{0}, 𝑗𝑝.

Definition 2.8 (reset-free switching instant). 𝑡𝐑0+ is a switching reset-free instant generated by the switching law 𝜎 if 𝑡ST(𝜎) and 𝐴𝑖𝑗𝐴𝑖(𝑡+)𝐴𝑖(𝑡) for some 𝑖𝑞{0}, for all 𝑗𝑝.

The set of reset-free instants is denoted by 𝑡STrf(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 STrf(𝜎)ST(𝜎), ST(𝜎)=ST𝑟(𝜎)STrf(𝜎), and ST𝑟(𝜎)STrf(𝜎)= from Definitions 2.62.8, 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 ST𝑟(𝜎,𝑡), and the reset-free partial switching sequence STrf(𝜎,𝑡), generated by the switching law 𝜎𝐑0+{(𝐴𝑗(𝑡)𝐴𝑗(𝑡)),𝑗𝑞{0}}𝐑𝑛×(𝑞+1)𝑛 up till any time 𝑡𝐑0+, are defined, respectively, by ST(𝜎,𝑡)={𝑡𝑖ST(𝜎)𝑡𝑖<𝑡}, ST𝑟(𝜎,𝑡)={𝑡𝑖ST𝑟(𝜎)𝑡𝑖<𝑡}, and STrf(𝜎,𝑡)={𝑡𝑖STrf(𝜎)𝑡𝑖<𝑡}.

Remark 2.10. An interpretation of Assumptions 2.3 is that the following conditions hold for any given -matrix norm for some nonnegative norm dependent real constants 𝜇𝑗, 𝛼𝑗, 𝜇, and 𝛼 for all 𝑗𝑞{0}: esssup𝑡𝐑+0̇𝐴𝑗𝑡𝜇𝑗maxesssup𝑡𝐑+0̇𝐴𝑗𝑡𝑗0𝑞𝜇,𝜏ST𝜎𝑡,𝑡+𝑇̇𝐴𝑗𝜏𝜏ST𝜎𝑡,𝑡+𝑇𝑋𝜏𝛿0𝛼𝑗max(𝜏ST𝜎𝑡,𝑡+𝑇̇𝐴𝑗𝜏𝑗0𝑞)𝛼,(2.5) where 𝛿(0) is a Dirac impulse at 𝑡=0. Note that Assumptions 2.3 imply |𝑞𝑗=0𝐴𝑗(𝑡+𝜏)𝑞𝑗=0𝐴𝑗(𝑡)|𝜇𝑇+𝜈(𝑡+𝜏) for all 𝜏[0,𝑇], with the function 𝜈𝐑0+𝐑0+ satisfying 𝜈(𝑡+𝜏)𝛼0𝑗+𝛼1𝑗(𝑡+𝜏)𝛼 with 𝛼1𝑗(𝑡+𝜏)=0 if ̇𝐴𝑗(𝑡+𝜏) exists with ̇𝐴𝑗(𝑡+𝜏)𝛼𝑗 or 𝛼𝑗𝛼0𝑗𝛼1𝑗(𝑡+𝜏)Δ𝑗(𝑡+𝜏) if ̇𝐴𝑗(𝑡+𝜏)=Δ𝑗(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) 𝑡ST(𝜎)𝜎(𝑡+)=𝜎(𝑡)=0𝑛×(𝑞+1)𝑛 (i.e., a zero 𝑛×(𝑞+1)𝑛-matrix);
(ii) 𝑡ST(𝜎)𝜎(𝑡+)𝜎(𝑡)0𝑛×(𝑞+1)𝑛;
(iii) 𝑡ST(𝜎)ST(𝜎,𝑡+)=ST(𝜎,𝑡);
(iv) 𝑡ST(𝜎)ST(𝜎,𝑡+)ST(𝜎,𝑡).

Proof. (i) 𝜎(𝑡+)=𝜎(𝑡)=0𝑛×(𝑞+1)𝑛𝑡ST𝑟(𝜎)𝑡STrf(𝜎)𝑡ST(𝜎),
𝑡ST𝜎(𝑡)=0 (since switching is not arbitrarily fast), 𝜎𝑡+𝑡𝜎=0𝑛×𝑞+1𝑛𝑡𝜎+𝑡=𝜎=0𝑛×𝑞+1𝑛.(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 𝑡ST𝜎,𝑡=𝑖𝜎ST𝑡𝑖<𝑡ST𝜎,𝑡+𝑡=𝑖𝜎ST𝑡𝑖<𝑡+=𝑡𝑖𝜎ST𝑡𝑖𝑡ST𝜎,𝑡+𝜎,𝜎=ST𝜎,𝑡if𝑡STST𝜎,𝑡if𝑡ST(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: 𝑉𝑡,𝑥𝑡=𝑥𝑇𝑡𝑃𝑡𝑥𝑡+𝑞𝑞𝑖=1𝑗=0𝑗𝑖𝑗𝑡𝑡+𝜃𝑥𝑇𝜏𝑆𝑖𝑗𝑥𝜏𝑑𝜏𝑑𝜃.(2.8)

Theorem 2.12. The following properties hold.
(i)Assume the following.(i.a)The matrix functions 𝐴𝑗(𝑡), for all 𝑗𝑞{0} are subject to Assumption 2.1.(i.b)The switching law 𝜎 is such that 𝑄𝑡=𝐻11𝑃𝑡𝐴1𝑡𝑀𝑡𝑞𝑃𝑡𝐴𝑞𝑡𝑀𝑡1𝑀𝑇𝑡𝐴𝑇1𝑡𝑃𝑡𝑅1000𝑞𝑀𝑇𝑡𝐴𝑇𝑞𝑡𝑃𝑡0𝑅𝑞<0,𝑡𝐑𝟎+,(2.9)where 𝐻1denotes(𝑞𝑗=0𝐴𝑇𝑗(𝑡))𝑃(𝑡)+𝑃(𝑡)(𝑞𝑗=0𝐴𝑗̇(𝑡))+𝑃(𝑡)+𝑞𝑖=1𝑞𝑗=0𝑖𝑆𝑖𝑗, for some time-differentiable real symmetric positive definite matrix function 𝑃𝐑0+𝐑𝑛×𝑛 and some real symmetric positive definite matrices 𝑆𝑖𝑗𝐑𝑛×𝑛(𝑖𝑞,𝑗𝑞{0}), where 𝑀𝑡𝐴=0𝑡,,𝐴𝑞𝑡𝐑𝑛×𝑞+1𝑛,𝑅𝑖𝑆=diag𝑖0,𝑆𝑖1,,𝑆𝑖𝑞𝐑𝑞+1𝑛×𝑞+1𝑛;𝑖𝑞,𝑡𝐑𝟎+.(2.10)Thus, the system (2.1) is globally asymptotically Lyapunov's stable for all delays 𝑖[0,𝑖], forall𝑖𝑞. A necessary condition is (𝑞𝑗=0𝐴𝑇𝑗(𝑡))𝑃(𝑡)+𝑃(𝑡)(𝑞𝑗=0𝐴𝑗̇(𝑡))+𝑃(𝑡)<0, forall𝑡𝐑0+ what implies that (𝑞𝑗=0𝐴𝑗(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)𝐴𝑖(𝑡)=𝐴𝑖𝑗, forall𝑖𝑞{0}, forall𝑡𝐑0+ for some 𝑗𝑝 (eventually being dependent on t) satisfying Assumption 2.4. (ii.b)The switching law 𝜎 is such that STrf(𝜎)= (i.e., it generates reset switching instants only) with ST𝑟(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)𝑄𝑖(=𝑞𝑗=0𝐴𝑇𝑗𝑖)𝑃+𝑃(𝑞𝑗=0𝐴𝑗𝑖)+𝑞𝑞𝑖=1𝑗=0𝑖𝑆𝑖𝑗1𝑃𝐴1𝑖𝑀1𝑞𝑃𝐴𝑞𝑖𝑀𝑞1𝑀1𝑇𝐴𝑇1𝑖𝑃𝑅1000𝑞𝑀𝑞𝑇𝐴𝑇𝑞𝑖𝑃0𝑅𝑞<0;𝑖𝑝(2.11)for some 𝐑𝑛×𝑛𝑃=𝑃𝑇>0, 𝐑𝑛×𝑛𝑆𝑖𝑗=𝑆𝑇𝑖𝑗>0(𝑖𝑞,𝑗𝑞{0}), where 𝑀𝑖𝐴=0𝑖,,𝐴𝑞𝑖𝐑𝑛×𝑞+1𝑛,𝑅𝑖𝑆=diag𝑖0,𝑆𝑖1,,𝐑(𝑞+1)𝑛×(𝑞+1)𝑛;𝑖𝑞(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 𝑖[0,𝑖], forall𝑖𝑞. If (2.9) is replaced with 𝑄𝑖2𝜀𝐼(𝑞+1)𝑛<0, forall𝑖𝑞, and some 𝜀𝐑+ then the state trajectory decays exponentially with rate(𝜀)<0.(iii) There is a sufficiently small =max𝑖𝑞𝑖 such that Property (i) holds for any 𝑖[0,𝑖], forall𝑖𝑞 provided that all the delay-free resetting systems (2.4)̇𝑧𝑗(𝑡)=(𝑞𝑖=0𝐴𝑖𝑗)𝑧𝑗(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 𝑆𝑖𝑗

Assume real constant symmetric matrices 𝑃(𝑡)=𝑃 and 𝑆𝑖𝑗, forall𝑖𝑞,forall𝑗𝑞{0}, forall𝑡𝐑0+ so that 𝑄𝑡=𝐻21𝑃𝐴1𝑡𝑀𝑡𝑞𝑃𝐴𝑞𝑡𝑀𝑡1𝑀𝑇𝑡𝐴𝑇1𝑡𝑃𝑅1000𝑞𝑀𝑇𝑡𝐴𝑇𝑞𝑡𝑃0𝑅𝑞<0,(2.13) where 𝐻2denotes(𝑞𝑗=0𝐴𝑇𝑗(𝑡))𝑃+𝑃(𝑞𝑗=0𝐴𝑗(𝑡))+𝑞𝑖=1𝑞𝑗=0𝑖𝑆𝑖𝑗. 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 𝐴𝑖(𝑖𝑞{0}) such that the time invariant system (2.1) defined with 𝐴𝑖(𝑡)𝐴𝑖 is globally asymptotically Lyapunov's stable and define a stability real 𝑛-matrix 𝐴=𝑞𝑖=0𝐴𝑖. Decompose 𝑄(𝑡)=𝑄+𝑄(𝑡), where 𝑄𝐴=𝑇𝑃+𝑃𝐴+𝑞𝑞𝑖=1𝑗=0𝑖𝑆𝑖𝑗1𝑃𝐴1𝑀𝑞𝑃𝐴𝑞𝑀1𝑀𝑇𝐴1𝑇𝑃𝑅1000𝑞𝑀𝑇𝐴𝑞𝑇𝑃0𝑅𝑞,𝑄𝑡𝐴=𝑇𝑡𝐴𝑡𝑃+𝑃1𝑃Δ1𝑡𝑞𝑃Δ𝑞𝑡1Δ𝑇1𝑡𝑃0000𝑞Δ𝑇𝑞𝑡,𝑃00(2.14)where 𝑀𝐴=diag0,𝐴1,,𝐴𝑞,𝑅𝑖𝑆=diag𝑖0,𝑆𝑖1,,𝑆𝑖𝑞,𝑀𝑡𝑡=𝑀𝑀,𝐴𝑖𝑡=𝐴𝑖𝑡𝐴,𝐴𝑡=(𝑞𝑗=0𝐴𝑗𝑖𝑡)𝐴,Δ𝑖𝑡𝐴=𝑖𝑡𝑀+𝐴𝑖𝑀𝑡+𝐴𝑖𝑡𝑀𝑡.(2.15) If 𝑡ST(𝜎), then 𝛿𝑄𝑡𝑄𝑡=+𝑄𝑡=𝐻31𝑃Δ1𝑡+Δ1𝑡𝑞𝑃Δ1𝑡+Δ1𝑡1Δ1𝑡+Δ1𝑡𝑇𝑃0000𝑞Δ1𝑡+Δ1𝑡𝑇,𝛿𝑄𝑡𝑃0022𝜆max𝑃[𝐴𝑇𝑡+𝐴𝑇𝑡2+𝑞𝑖=1𝑖Δ𝑖𝑡+Δ𝑖𝑡2]=2𝜆max𝑃[𝑞𝑖=0𝐴𝑇𝑖𝑡+𝐴𝑇𝑖𝑡2+𝑞𝑖=1𝑖Δ𝑖𝑡+Δ𝑖𝑡2],(2.16) where (𝐴𝐻3denotes𝑇(𝑡+𝐴)𝑇(𝑡))𝑃+𝑃((𝐴(𝑡+)𝐴(𝑡))), what leads to 𝛿𝑄𝑡22𝜆max𝑃[𝑎𝑞+1++𝑞(𝑎+(𝑞𝑗=0𝐴𝑗2+𝐴𝑞+1𝑖2+𝑞+1𝑎+𝑎+)+𝑞+12𝑎+𝑎+𝑎)],(2.17) where 𝑎=max𝑖𝑞{0}(𝐴𝑖2) and 𝑎+=sup𝑡ST(𝜎)max𝑖𝑞{0}(𝐴𝑖(𝑡+)𝐴𝑖2) (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=𝑄𝑇, 𝑃(𝑡)𝑃=𝑃𝑇>0, 𝐴𝑖(𝑡)𝐴𝑖, 𝐴(𝑡)𝐴=𝑞𝑖=0𝐴𝑖; forall𝑖𝑞,forall𝑗𝑞{0}, forall𝑡𝐑0+ such that 𝐴 is a stability matrix. Then, Theorem 2.12(i) holds if 𝑄<0 for any switching law 𝜎 such that (1)𝜆min(𝑄)=𝜆max(𝑄)>𝑄(𝑡)2=𝜆1/2max(𝑄𝑇(𝑡)𝑄(𝑡)), forall𝑡𝐑0+ST(𝜎),(2)𝑎=max𝑖𝑞{0}(𝐴𝑖2) and 𝑎+=sup𝑡ST(𝜎)max𝑖𝑞{0}(𝐴𝑖(𝑡+)𝐴𝑖2) are sufficiently small such that 𝜆min𝑄𝑄𝑡2>2𝜆max𝑃[𝑎𝑞+1++𝑞(𝑎+(𝑞𝑗=0𝐴𝑗2+𝑞+1max𝑖𝑞0𝐴𝑖2+𝑞+1𝑎+𝑎+)+𝑞+12𝑎+𝑎+𝜎.𝑎)],𝑡ST(2.18)

Corollary 2.14. Consider in (2.9) replacements with constant real matrices 𝑄(𝑡)𝑄𝑗=𝑄𝑗𝑇, 𝑃(𝑡)𝑃=𝑃𝑇>0, 𝐴𝑖𝑗(𝑡)𝐴𝑖𝑗, 𝐴(𝑡)𝐴𝑗=𝑞𝑖=0𝐴𝑖𝑗 for all 𝑗𝑝,forall𝑘𝑞{0}, forall𝑡𝐑0+ such that each 𝐴𝑗(𝑗𝑝) is a stability matrix with 𝐴𝑖𝑗;forall𝑖𝑞,forall𝑗𝑝 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 𝑄𝑗<0(𝑗𝑝) for any switching law 𝜎 such that
(1)𝜆min(𝑄)=𝜆max(𝑄)>𝑄(𝑡)2=𝜆1/2max(𝑄𝑇(𝑡)𝑄(𝑡)),forall𝑡𝐑0+ST(𝜎),(2)𝑎𝑗=max𝑖𝑞{0}(𝐴𝑗𝑖2) and 𝑎+𝑗=sup𝑡ST(𝜎)max𝑖𝑞{0}(𝐴𝑖(𝑡+)𝐴𝑗𝑖2) are sufficiently small such that 𝜆min𝑄𝑄𝑡2>2𝜆max𝑃[𝑎𝑞+1+𝑗+𝑞(𝑎+𝑗(𝑞𝑖=0𝐴𝑗𝑖2+𝑞+1max𝑖𝑞0𝐴𝑗𝑖2)+𝑎𝑞+1𝑗+𝑎+𝑗+𝑞+12𝑎𝑗+𝑎+𝑗𝑎+𝑗)](2.19)𝑡STfr(𝜎), provided that at time max(𝑡<𝑡𝑡ST𝑟(𝜎)), the system (2.1) coincides with at the 𝑗𝑝𝑟 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 𝑎𝑎𝑗,𝑎+𝑎+𝑗 for all 𝑗𝑝. If (2.19) is rewritten with the replacements 𝑎𝑗𝑎=max𝑗𝑝𝑎𝑗,𝑎+𝑗𝑎+=max𝑗𝑝𝑎+𝑗 then the reformulated weaker Corollary 2.14 is valid for all 𝑡STfr(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 𝑄0=𝑄0𝑇>0 such that 𝜆min(𝑄0)>𝜆max(𝑞𝑖=1𝑞𝑗=0𝑖𝑆𝑖𝑗) and 𝑃=0𝑒𝐴𝑇𝜏𝑄0𝑒𝐴𝜏𝑑𝜏 satisfying the Lyapunov equation 𝐴𝑇𝑃+𝑃𝐴=𝑄0<0 as its unique solution. Note that 𝜆max(𝑃)𝐾𝜆max(𝑄0)/2𝜌 for some 𝐾𝐑+, where 𝜌<0 is the stability abscissa of 𝐴 with 𝑒𝐴𝑇𝑡𝐾𝑒𝜌𝑡 for all 𝑡𝐑. Define the decomposition 𝑄(𝑡)=𝑄+𝑄(𝑡), where 𝑄=BlockDiag(𝑄0+𝑞𝑞𝑖=1𝑗=0𝑖𝑆𝑖𝑗,𝑅1,,𝑅𝑞),𝑄𝑡𝐴=𝑇𝑡𝐴𝑡𝑃+𝑃1𝑃𝐴1𝑀+Δ1𝑡𝑞𝑃𝐴𝑞𝑀+Δ𝑞𝑡1𝐴1𝑀+Δ1𝑡𝑇𝑃0000𝑞𝐴𝑞𝑀+Δ𝑞𝑡𝑇𝑃00𝑄𝑡2𝐾(𝜆max𝑄0𝜌𝐴)(𝑇𝑡2+𝑞𝑖=1𝑖𝐴𝑖𝑀+Δ𝑖𝑡2).(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 𝐴𝑇𝑃+𝑃𝐴=𝑄0<0. Then, Theorem 2.12(i) holds if 𝑄<0 for any switching law 𝜎 such that 𝜆min𝑄=𝜆max𝑄>𝐾𝜆max𝑄0𝜌(𝐴𝑇𝑡2+𝑞𝑖=1𝑖𝐴𝑖𝑀+Δ𝑖𝑡2),𝑡𝐑0+,𝑡ST(𝜎).(2.21)

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

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

Theorem 2.16. Under Assumptions 2.12.3, the following properties hold.
(i)The switched system (2.1) is globally asymptotically Lyapunov's for any delays 𝑖[0,𝑖] for all 𝑖𝑞 for some =max𝑖𝑞(𝑖) and any switching law 𝜎 such that (a)the switching instants are arbitrary;(b)̇𝐴max(esssup𝑗(𝑡)𝑡𝐑0+,𝑗𝑝) is sufficiently small compared to the absolute value of the prescribed stability abscissa of 𝑞𝑗=0𝐴𝑗(𝑡);(c)the support testing matrix of distributional derivatives Γ𝐴𝑑𝑗(𝑡) of the same matrices are semidefinite negative for all time instants, where the conventional derivatives do not exist (i.e., ̇𝐴𝑗(𝑡)=Γ𝐴𝑑𝑗(𝑡)(𝑡)𝛿(0)).(ii)The switched system (2.1) is globally exponentially stable for any delays 𝑖[0,𝑖] for all 𝑖𝑞 for some =max𝑖𝑞(𝑖) such that (a)̇𝐴max(esssup𝑗(𝑡)𝑡𝐑0+,𝑗𝑝) is sufficiently small compared to the absolute value of the prescribed stability abscissa of 𝑞𝑗=0𝐴𝑗(𝑡);(b)max(Γ𝐴𝑑𝑗̇𝐴(𝑡)(𝑡)𝑗(𝑡)=Γ𝐴𝑑𝑗(𝑡)(𝑡)𝛿(0),𝑡ST(𝜎),𝑗𝑝) is sufficiently small compared to the timeintervals in between any two consecutive switching instants. Furthermore, if Assumptions 2.12.4 hold, then (iii)the switched system (2.1) is globally exponentially stable for any delays 𝑖[0,𝑖] for all 𝑖𝑞, for some =max𝑖𝑞(𝑖) such that(a)̇𝐴max(esssup𝑗(𝑡)𝑡𝐑0+,𝑗𝑝) is sufficiently small compared to the absolute value of the prescribed stability abscissa of 𝑞𝑗=0𝐴𝑗(𝑡);(b)the switching law 𝜎 is such that(a)max(Γ𝐴𝑑𝑗̇𝐴(𝑡)𝑗(𝑡)=Γ𝐴𝑑𝑗(𝑡)𝛿(0),𝑡STrf(𝜎),𝑗𝑝) 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,𝑥𝑡) defined with constant matrices 𝑃=𝑃𝑇>0 and 𝑆𝑖𝑗=𝑆𝑇𝑖𝑗>0, forall(𝑖,𝑗)(𝑞{0})×𝑞 for all the time-invariant resetting systems (2.4) and some of the subsequent conditions hold forall𝑡ST𝑟(𝜎) under the resetting action 𝑃(𝑡+)=𝑃; forall(𝑖,𝑗)(𝑞{0})×𝑞:(b.1)𝑉(𝑡+,𝑥𝑡)𝑉(𝑡,𝑥𝑡) which is guaranteed, in particular, if 𝑃(𝑡+)=𝑃𝑃(𝑡),(b.2)the tradeoff (a) is respected between sufficiently small norms of the matrices of distributional derivatives and the length |𝑡𝑡|, at any 𝑡ST𝑟(𝜎), if any, where the condition (b.1) is not satisfied, where 𝑡=max(𝜏𝐑0+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 𝑖𝐑0+, forall𝑖𝑞 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: 𝑉𝑡,𝑥𝑡=𝑥𝑇𝑡𝑃𝑡𝑥𝑡+𝑞𝑖=1𝑡𝑡𝑖𝑥𝑇𝜏𝑆𝑖𝜏𝑥𝜏𝑑𝜏(3.1) whose time derivative along the state-trajectory solution of (2.1) is ̇𝑉𝑡,𝑥𝑡=𝑥𝑇𝑡(𝐴𝑇𝑡𝑃𝑡𝑡𝐴𝑡++𝑃𝑞𝑖=1𝑆𝑖𝑡+̇𝑃𝑡𝑡)𝑥+2𝑥𝑇𝑡𝑞𝑖=1𝑃𝐴𝑖𝑡𝑥𝑡𝑖𝑞𝑖=1𝑥𝑇𝑡𝑖𝑆𝑖𝑡𝑖𝑥𝑡𝑖=𝑥𝑇𝑡𝑄𝑡𝑥𝑡<0(3.2) for all nonzero 𝑥𝑇(𝑡)=(𝑥𝑇(𝑡),𝑥𝑇(𝑡1),,𝑥𝑇(𝑡𝑞)) if 𝑄𝑡𝐴=𝑇𝑡𝑃𝑡𝑡𝐴𝑡++𝑃𝑞𝑖=1𝑆𝑖𝑡+̇𝑃𝑡𝑃𝐴1𝑡𝑡𝐴𝑃𝐴𝑞𝑇1𝑡𝑃𝑆1𝑡1𝐴00𝑇𝑞𝑡𝑃0𝑆𝑞𝑡𝑞<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 𝜆𝑖(𝐴0(𝑡)) of the matrix function 𝐴0(𝑡) satisfy Re𝜆𝑖(𝐴0(𝑡))𝜌00<0; forall𝑡𝐑0+, forall𝑖𝜎(𝜎𝑛) for some 𝜌00𝐑+={0<𝑧𝐑}; that is, 𝐴0(𝑡) is a stability matrix, forall𝑡𝐑0+.

Assumption 3.2 b. 𝐴𝑗𝐑0+𝐑𝑛×𝑛 are almost everywhere time-differentiable with essentially bounded time derivative, forall𝑗𝑞{0} possessing eventual isolated bounded discontinuities, then esssup𝑡𝐑0+̇𝐴(0(𝑡))𝛾0< with 𝛾0 being a -norm dependent nonnegative real constant and, furthermore, 𝑡𝑡+𝑇̇𝐴0(𝜏)𝑑𝜏𝜇0𝑇+𝛼0 for some 𝛼0,𝜇0𝐑+, forall𝑡𝐑0+, and some fixed 𝑇𝐑0+ 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 𝜆𝑘(𝐴0𝑗) satisfy Re𝜆𝑘(𝐴𝑗0)𝜌00<0; forall𝑡𝐑0+, forall𝑘𝜎0𝑗(𝜎0𝑗𝑛), forall𝑗𝑝; that is, 𝐴0𝑗 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 𝐴𝑗(𝑡), forall𝑗𝑞{0} are subject to Assumption 2.1;(i.b)the switching law 𝜎 is such that 𝑄(𝑡)<0, (3.3), forall𝑡𝐑0+for some time-differentiable real symmetric positive definite matrix function 𝑃𝐑0+𝐑𝑛×𝑛 and some real symmetric positive definite matrix functions 𝑆𝑖𝐑0+𝐑𝑛×𝑛(𝑖𝑞). Thus, the system (2.1) is globally asymptotically Lyapunov's stable independent of the delays (i.e., for all delays 𝑖[0,), forall𝑖𝑞). A necessary condition is 𝐴𝑇0(𝑡)𝑃(𝑡)+𝑃(𝑡)𝐴0̇(𝑡)+𝑃(𝑡)<0, forall𝑡𝐑0+ 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)𝐴𝑗(𝑡)=𝐴𝑗𝑖, forall𝑗𝑞{0}, forall𝑡𝐑0+ for some 𝑖𝑝 (eventually being dependent on 𝑡) satisfying Assumption 3.3; (ii.b)the switching law 𝜎 is such that STrf(𝜎)= (i.e., it generates reset switching instants only) with ST𝑟(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)𝑄𝑖𝐴=𝑇0𝑖𝑃+𝑃𝐴0𝑖+𝑞𝑖=1𝑆𝑖𝑃𝐴1𝑖𝑃𝐴𝑞𝑖𝐴𝑇1𝑖𝑃𝑆1000𝐴𝑇𝑞𝑖𝑃0𝑆𝑞<0;𝑖𝑝(3.4)for some 𝐑𝑛×𝑛𝑃=𝑃𝑇>0, 𝐑𝑛×𝑛𝑆𝑖=𝑆𝑖𝑇>0(𝑖𝑞). 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 forall𝑖𝑞. If (3.4) is replaced with 𝑄𝑖2𝜀𝐼(𝑞+1)𝑛<0, forall𝑖𝑞, and some 𝜀𝐑+ then the state trajectory decays exponentially with rate(𝜀)<0.

Parallel results to Corollaries 2.132.15 are direct from Theorem 3.4 with the replacements 𝐴0(𝑡)𝐴0 (a constant stability matrix), 𝐴0𝑗(𝑡)𝐴0𝑗, forall𝑗𝑝 (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 2.22.4 by Assumptions 3.13.3.

Theorem 3.5. Under Assumptions 2.1 and 3.13.2, 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 𝑖[0,), forall𝑖𝑝 and any switching law 𝜎 such that (a)the switching instants are arbitrary;(b)̇𝐴max(esssup0(𝑡)𝑡𝐑0+) is sufficiently small compared to the absolute value of the prescribed stability abscissa of 𝐴0(𝑡);(c)the support testing matrix of distributional derivatives Γ𝐴𝑑0(𝑡) of the same matrices are semidefinite negative for all time instants, where the conventional derivatives do not exist (i.e., iḟ𝐴0(𝑡)=Γ𝐴𝑑0(𝑡)(𝑡)𝛿(0)) .(ii)The switched system (2.1) is globally exponentially stable independent of the delays if (a)̇𝐴max(esssup0(𝑡)𝑡𝐑0+) is sufficiently small compared to the absolute value of the prescribed stability abscissa of 𝐴0(𝑡);(b)max(Γ𝐴𝑑0̇𝐴(𝑡)(𝑡)0(𝑡)=Γ𝐴𝑑0(𝑡)(𝑡)𝛿(0),𝑡ST(𝜎)) is sufficiently small compared to the time intervals in between any two consecutive switching instants. If Assumptions 2.1 and 3.13.3 hold, then (iii)the switched system (2.1) is globally exponentially stable independent of the delays if ̇𝐴max(esssup0(𝑡)𝑡𝐑0+) 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(Γ𝐴𝑑0̇𝐴(𝑡)(𝑡)0(𝑡)=Γ𝐴𝑑0(𝑡)(𝑡)𝛿(0),𝑡STrf(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,𝑥𝑡) defined with constant matrices 𝑃(𝑡)𝑃=𝑃𝑇>0 and 𝑆𝑖(𝑡)𝑆𝑖=𝑆𝑇𝑖>0, forall𝑖𝑞{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 forall𝑡ST𝑟(𝜎) under the resetting action 𝑃(𝑡+)=𝑃; forall𝑖𝑞{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 ̇𝑥(𝑡)=𝐴0(𝑡)𝑥(𝑡)+𝐴1(𝑡)𝑥(𝑡), where =0.75 second and each resetting system (2.4) with 𝑝=2 is defined by 𝐴01(𝑡)=1.3cos𝑡402.5,𝐴11,𝐴(𝑡)=2.2𝑎(𝑡)02.102(𝑡)=1sin𝑡402,𝐴12(𝑡)=1.5𝑏(𝑡)02(4.1) with 𝑎(𝑡)=1/(𝑡+𝑡+1) and 𝑏(𝑡)=𝑡/(𝑡2+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.

231710.fig.001
Figure 1: Graphical representation of 𝑎(𝑡).
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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.

231710.fig.003
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 𝑘𝑁 by ̇𝐴01=0sin𝑡4,̇𝐴0011=,̇𝐴0̇𝑎(𝑡)0002=0cos𝑡4,̇𝐴0012=0̇𝑏(𝑡)00(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>esssup𝑡𝑅0+/ST(𝜎)1𝑗=0𝐴𝑗, where 𝜆max(𝑃(𝑡))𝐾0/2𝜌0 for the existing unique symmetric positive definite solution 𝑃(𝑡) of (1𝑗=0𝐴𝑇𝑗)𝑃(𝑡)+𝑃(𝑡)(1𝑗=0𝐴𝑗)=𝐼𝑛. Numerical computations lead to 10.33=2(𝜌0/𝐾0)2>esssup𝑡𝑅0+/ST(𝜎)1𝑗=0𝐴𝑗=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 𝑡𝑁. Therefore, from Theorem 2.16(ii), the space-state trajectories of the solution asymptotically converge to zero as time evolves as Figure 4 shows.

231710.fig.004
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].

231710.fig.005
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 ̇𝑥(𝑡)=𝐴0(𝑡)𝑥(𝑡)+𝐴1(𝑡)𝑥(𝑡) and the resetting systems 𝐴01=11203,𝐴02=2.5201.6,𝐴1=2.5𝑎(𝑡)01.6(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.

231710.fig.006
Figure 6: Graphical representation of 𝑎(𝑡).

Furthermore, the conditions of Theorem 3.5(i) are especially easy to verify since the resetting matrices 𝐴0𝑖 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.

231710.fig.007
Figure 7: State trajectories evolution for a delay of 1.75 seconds.
231710.fig.008
Figure 8: State trajectories evolution for a delay of 7.25 seconds.
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Figure 9: State trajectories evolution for a delay of 15.8 seconds.
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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., [2535]), 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 𝑥𝑡 the strip of state-trajectory solution 𝑥(𝑡+𝜏) of the system 2.1 for 𝜏[,0]. Consider the Krasovsky-Lyapunov functional candidate: 𝑉𝑡,𝑥𝑡=𝑥𝑇𝑡𝑃𝑡𝑥𝑡+𝑞𝑞𝑖=1𝑗=0𝑗𝑖𝑗𝑡𝑡+𝜃𝑥𝑇𝜏𝑆𝑖𝑗𝑥𝜏𝑑𝜏𝑑𝜃(A.1) which is nonnegative and radially unbounded since 𝑉(𝑡,𝑥𝑡) as 𝑥𝑡=sup𝜏0𝑥(𝜏), with =max1𝑖𝑞(𝑖), since all the eigenvalues of 𝑃(𝑡) and 𝑆𝑖𝑗(𝑡) positive and uniformly bounded from above and below forall𝑡𝐑0+. This also implies that ̇𝑃(𝑡) cannot be neither positive definite nor negative definite forall𝑡𝐑0+. Direct calculations via 2.9 yield ̇𝑉(𝑡,𝑥𝑡)𝑥𝑇(𝑡)𝑄(𝑡)𝑥(𝑡)<0, forall𝑡𝐑0+ if and only if 𝑥(𝑡)=(𝑥𝑇(𝑡),𝑥𝑇(𝑡1),,𝑥𝑇(𝑡𝑞))𝑇0. Then, 𝑉(𝑡,𝑥𝑡)𝑉(0,𝜑)< and 𝑉(𝑡,𝑥𝑡)0 as 𝑡 for any given bounded function of initial conditions what imply that 𝑥𝑡<, forall𝑡𝐑0+ and 𝑥𝑡0 as 𝑡. The global asymptotic stability has been proven. To prove the last part of Property (i), note that 2.9 implies (𝑞𝑗=0𝐴𝑇𝑗𝑡𝑡𝑡()𝑃+𝑃𝑞𝑗=0𝐴𝑗𝑡̇𝑃𝑡+)+𝑞𝑞𝑖=1𝑗=0𝑖𝑆𝑖𝑗<0,𝑡𝐑0+(𝑞𝑗=0𝐴𝑇𝑗𝑡𝑡𝑡()𝑃+𝑃𝑞𝑗=0𝐴𝑗𝑡̇𝑃𝑡)+<0.(A.2) Since ̇𝑃(𝑡)<0, forall𝑡𝐑0+ is impossible from the preceding part of the proof, it has to exist a nonnecessarily connected subinterval 𝑆𝑅𝐑0+ such that (𝑞𝑗=0𝐴𝑇𝑗(𝑡))𝑃(𝑡)+𝑃(𝑡)(𝑞𝑗=0𝐴𝑖(𝑡))<0, forall𝑡𝑆𝑅. Now, proceed by contradiction to prove that 𝑆𝑅 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)𝑆𝑅 has finite measure so that the complement 𝑆𝑅 in 𝐑0+ is nonconnected with infinite measure with a component being necessarily of infinite measure (otherwise, 𝑆𝑅 has infinite measure). Thus, (𝑞𝑗=0𝐴𝑇𝑗(𝑡))𝑃(𝑡)+𝑃(𝑡)(𝑞𝑗=0𝐴𝑗(𝑡))0 or indefinite forall𝑡𝑆𝑅. Since 𝑆𝑅 has finite measure and 𝑆𝑅 has a component of infinite measure, it exists a sufficiently large finite 𝑡0𝐑0+ such that 𝑆𝑅𝑡𝑡0 so that 𝑥𝑡0 as 𝑡 is impossible what leads to a contradiction.(2)Both intervals 𝑆𝑅 and 𝑆𝑅 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=0𝐴𝑗(𝑡)) is a stability matrix forall𝑡𝐑0+ except possibly within an interval of finite measure.

(ii) Denote by 𝑧𝑡 the strip of state-trajectory solution 𝑧(𝑡+𝜏), for 𝜏[,0] and any resetting system. Consider the Krasovsky-Lyapunov functional candidate for all the resetting systems: 𝑉𝑡,𝑧𝑡=𝑧𝑇𝑡𝑃𝑧𝑡+𝑞𝑞𝑖=1𝑗=0𝑗𝑖𝑗𝑡𝑡+𝜃𝑧𝑇𝜏𝑆𝑖𝑗𝑧𝜏𝑑𝜏𝑑𝜃.(A.3)

The real functional A.1 is a common Krasovsky-Lyapunov functional for all 𝑝 distinct resetting systems for all delays 𝑖[0,𝑖](𝑖𝑞), provided that 𝑧𝑡𝑧(𝑡+𝜏), forall𝜏[,0] and =max1𝑖𝑞(𝑖) since the 𝑖th resetting system 2.4 satisfies from 2.11̇𝑉𝑖𝑡,𝑧𝑡𝑧𝑇𝑡𝑄𝑖𝑧𝑡<0;𝑖𝑞,𝑡𝐑0+(A.4) for all nonzero 𝑧(𝑡) where 𝑧(𝑡)=(𝑧𝑇(𝑡),𝑧𝑇(𝑡1),,𝑧𝑇(𝑡𝑞))𝑇 and accordingly ̇𝑉𝑡,𝑥𝑡|||min𝑖𝑞𝑥𝑇𝑡𝑄𝑖𝑥𝑡|||0,𝑡𝐑0+(A.5) for all nonzero 𝑥(𝑡) where 𝑥(𝑡)=(𝑥𝑇(𝑡),𝑥𝑇(𝑡1),,𝑥𝑇(𝑡𝑞))𝑇 for the switched system 2.1 since STrf𝜎=ST𝑟𝜎=ST𝑟𝜎(max𝑖𝑝𝜆max𝑄𝑖)𝑧𝑇𝑡𝑧𝑡̇𝑉𝑡,𝑧𝑡(min𝑖𝑝𝜆min𝑄𝑖𝑖)𝑧𝑇𝑡𝑧𝑡<0(A.6) for any 𝑡𝐑0+ such that 𝑧(𝑡)0, where 𝜆max() and 𝜆min() 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.3A.6: ̇𝑉𝑡,𝑥𝑡𝑉𝑡,𝑥𝑡𝛿<0𝑉𝑡,𝑥𝑡𝑒𝛿𝑇𝑉𝑡𝑇,𝑥𝑡𝑇𝐾𝑒𝛿𝑡𝑉0,𝜑(A.7)forall𝑡𝐑0+, for some finite 𝐾𝐑+, where 𝛿=|max𝑖𝑝𝜆max(𝑄𝑖)|/𝜆min(𝑃)>0. Then, from A.7 and A.3𝜆min𝑃𝑥𝑇𝑡𝑥𝑡𝑉𝑡,𝑥𝑡𝑒𝛿𝑇𝑉𝑡𝑇,𝑥𝑡𝑇𝐾𝑒𝛿𝑡𝑉0,𝜑𝐾𝐾1𝑒𝛿𝑡sup𝜏0𝜑𝜏22𝑥𝑡2𝐾𝐾1𝑒(𝛿/2)𝑡sup𝜏0𝜑𝜏2(A.8)forall𝑡𝐑0+, and some 𝐾𝐑+, where 𝐾1=𝜆max(𝑃)+(𝑞+1)max𝑖𝑝,𝑗𝑞{0}𝜆max(𝑆𝑖𝑗) and =max𝑖𝑞𝑖 and 2 denotes the 2 (or spectral) vector norm or the corresponding induced ones for matrices. Then, the system 2.1 is globally exponentially stable for all delays 𝑖[0,𝑖], forall𝑖𝑞. The modification of 2.11-2.12 with 𝑄𝑖2𝜀𝐼(𝑞+1)𝑛<0, forall𝑖𝑞 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=0𝐴𝑇𝑗𝑖)𝑃+𝑃(𝑞𝑗=0𝐴𝑗𝑖)<0 for any 𝑃=𝑃𝑇>0. Thus, it exists a sufficiently small =max𝑖𝑞𝑖 such that 2.9 holds forall𝑖[0,𝑖], since 𝑅𝑖>0; forall𝑖𝑞.

If Property (i) holds then (𝑞𝑗=0𝐴𝑇𝑗𝑖)𝑃+𝑃(𝑞𝑗=0𝐴𝑗𝑖)+𝑞𝑖=1𝑞𝑗=0𝑖𝑆𝑖𝑗<0 what implies (𝑞𝑗=0𝐴𝑇𝑗𝑖)𝑃+𝑃(𝑞𝑗=0𝐴𝑗𝑖)<0 since 𝑞𝑖=1𝑞𝑗=0𝑖𝑆𝑖𝑗0 for any 𝑖0, forall𝑖𝑞. Thus, 𝑄id=BlockDiag((𝑞𝑗=0𝐴𝑇𝑗𝑖)𝑃+𝑃(𝑞𝑗=0𝐴𝑗𝑖)+𝑞𝑖=1𝑞𝑗=0𝑖𝑆𝑖𝑗<0,𝑅1,,𝑅𝑞)<0, forall𝑖𝑞 and then 𝑄𝑖<0, forall𝑖[0,𝑖] and sufficiently small =max𝑖𝑞𝑖 since from 2.11𝑄𝑖𝑄id is a monotonically increasing function of the argument (1,,𝑞) being zero if 𝑖=0, forall𝑖𝑞. Then, Property (i) holds for a sufficiently small .

A.2. Derivation of the Inequality 2.17

It follows from the subsequent inequalities: 𝑀2𝑞𝑖=0𝐴𝑖2,𝑀(𝑡)2𝑞𝑖=0𝐴𝑖(𝑡)2,𝑀𝑡+𝑀(𝑡)2𝑞𝑖=0𝐴𝑖𝑡+𝐴𝑖(𝑡)2,𝑀𝑡+𝑀2𝑞𝑖=0𝐴𝑖𝑡+𝐴𝑖2,Δ𝑖𝑡+Δ𝑖(𝑡)2=𝐴𝑖𝑡+𝐴𝑖𝑀(𝑡)+𝐴𝑖𝑀𝑡++𝐴𝑀(𝑡)𝑖𝑡+𝑀𝑡+𝑀+𝐴𝑖(𝑡)𝑀(𝑡)𝑀2𝐴𝑖𝑡+𝐴𝑖(𝑡)2(𝑞𝑗=0𝐴𝑗2𝐴)+𝑖2(𝑞𝑗=0𝐴𝑗𝑡+𝐴𝑗(𝑡)2)2+𝐴𝑖𝑡+