Discrete Dynamics in Nature and Society

Volume 2008, Article ID 231710, 31 pages

http://dx.doi.org/10.1155/2008/231710

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

^{1}Department of Electricity and Electronics, Faculty of Science and Technology, University of Basque, Campus of Leioa (Bizkaia), Aptdo. 644, 48080 Bilbao, Spain^{2}Department 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., [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 th order linear time-varying dynamic system with internal (in general, incommensurate) known point delays: for any given bounded piecewise absolutely continuous function of initial conditions, where with , for some delays , of finite or infinite maximum allowable delays sizes , , where is the nonnegative real axis ; and , for all . 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 are piecewise continuous and uniformly bounded
for all .

*Assumption 2.2. *All
the eigenvalues of the matrix function satisfy for all , for some , that is, is a stability matrix for all .

*Assumption 2.3. *The matrix functions are almost everywhere time differentiable with
essentially bounded time derivative for all possessing eventual isolated bounded
discontinuities, then with being a -norm dependent nonnegative real constant and, furthermore, for
some for all , and
some fixed independent of .

At
time instants , where the time-derivative of some entry of does
not exist for any ,
the time derivative is defined in a distributional Dirac sense as what
equivalently means the presence of a discontinuity at in defined as

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 for any given piecewise absolutely continuous function of initial conditions. This follows from Picard-Lindelöff existence and uniqueness theorem. Assumption 2.2 establishes that 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 for all , 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 at times, where it is impulsive. An alternative assumption to Assumptions 2.3 which avoids the assumption of almost everywhere existence of a bounded , for all (see second part of Assumption 2.1) might be stated in terms of sufficiently smallness of for for all for all for some constant stability matrix whose eigenvalues satisfy . 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 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 for . At such times denote right values of the matrix function while is simply denoted by . A set of resetting systems of (2.1) is defined by the linear time-invariant systems: for some given for all for all for some given . 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 satisfy for all for all for all ;
that is, are constant stability matrices with
prescribed stability abscissa for
all .

The following definitions are then used.

*Definition 2.5. *The switching matrix function is a mapping from the nonnegative real axis to the set of
real 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). * is a switching instant if for some .

The set of switching instants generated by the switching law is denoted by . 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). * is a reset instant generated by the switching
law if and for some and some ,
provided that for some .

The set of reset instants generated by the switching law is denoted by . Note that 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, for some , .

*Definition 2.8 (reset-free switching instant). * is a switching reset-free instant generated by
the switching law if and for some ,
for all .

The set of reset-free instants is denoted by . 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 , , and 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 ,
the partial switching sequence ,
and the reset-free partial switching sequence ,
generated by the switching law up till any time ,
are defined, respectively, by , ,
and .

*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 , , ,
and for all : where is a Dirac impulse at . Note that Assumptions 2.3
imply for all ,
with the function satisfying with if exists with or if , 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) (i.e., a zero -matrix);

(ii) ;

(iii) ;

(iv) .

*Proof. *(i)

(since switching is not arbitrarily
fast), 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 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:

Theorem 2.12. *The
following properties hold. *

(i)*Assume the
following.(i.a) The matrix functions
for all are subject to Assumption 2.1.(i.b)The switching law is such that where ,
for
some time-differentiable real symmetric positive definite matrix function and some real symmetric positive definite matrices ,
where Thus,
the system (2.1) is globally asymptotically Lyapunov's stable for all delays , .
A necessary condition is , what implies that is a stability matrix of prescribed stability
abscissa on except eventually on a real subinterval of
finite measure of .*(ii)

*Assume the following(ii.a)*(iii)

*, , for some (eventually being dependent on t) satisfying Assumption 2.4.*(ii.b)*The switching law is such that (i.e., it generates reset switching instants only) with 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)*for some , , where 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 , . If (2.9) is replaced with , , and some then the state trajectory decays exponentially with .**There is a sufficiently small such that Property (i) holds for any , provided that all the delay-free resetting systems (2.4) 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 and

Assume real constant symmetric matrices and , , so that where . 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 such that the time invariant system (2.1) defined with is globally asymptotically Lyapunov's stable and define a stability real -matrix . Decompose , where where If , then where what leads to where and (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 , , , ; , such that is a stability matrix. Then, Theorem 2.12(i) holds if for any switching law such that
*(1)*, ,*(2)* and are sufficiently small such that *

Corollary 2.14. *Consider
in (2.9) replacements with constant real matrices , , , for all , such that each is a stability matrix with being the parameterizations defining the
resetting systems (2.4). Assume that the system (2.1) is one of the resetting
systems (2.4) at . Then, Theorem 2.12(i) holds with a common Krasovsky-Lyapunov function for all those resetting systems
if for any switching law such that *

(1)*,*(2)* and are sufficiently small such that ,
provided that at time ,
the system (2.1)
coincides with at the 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 for all . If (2.19) is rewritten with the replacements then the reformulated weaker Corollary 2.14 is valid for all 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 by a Lyapunov matrix equality as follows. Consider a real -matrix such that and satisfying the Lyapunov equation as its unique solution. Note that for some , where is the stability abscissa of with for all . Define the decomposition , where Thus, the subsequent result follows from Corollary 2.13 and (2.20).

Corollary 2.15. *Consider
the matrices of Corollary 2.13 with being a stability matrix with stability abscissa which satisfies the Lyapunov equation .
Then, Theorem 2.12(i) holds if for any switching law such that *

##### 2.4. Sufficiency Type Asymptotic Stability Conditions Obtained for Time-Varying Symmetric Matrices ,

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 for all for
some and any switching law such that
(a) the switching instants are arbitrary;(b) is sufficiently small compared to the absolute
value of the prescribed stability abscissa of ;(c)the support testing matrix of
distributional derivatives of the same
matrices are semidefinite negative for all time instants, where the
conventional derivatives do not exist (i.e., ).*(ii)

*The switched system (2.1) is globally exponentially stable for any delays for all for some such that (a)*

*is sufficiently small compared to the absolute value of the prescribed stability abscissa of ;*(b)*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 for all , for some such that(a)*

*is sufficiently small compared to the absolute value of the prescribed stability abscissa of ;*(b)*the switching law is such that(a)**is sufficiently small compared to the lengths of time intervals between any two consecutive switching instants;*(b)*it exists a common Krasovsky-Lyapunov functional defined with constant matrices and , for all the time-invariant resetting systems (2.4) and some of the subsequent conditions hold under the resetting action ; :(b.1)**which is guaranteed, in particular, if ,*(b.2)*the tradeoff (a) is respected between sufficiently small norms of the matrices of distributional derivatives and the length , at any , if any, where the condition (b.1) is not satisfied, where .*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 , 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: whose time derivative along the state-trajectory solution of (2.1) is for all nonzero if 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 of the matrix function satisfy ; , for some ; that
is, is a stability matrix, .

*Assumption 3.2 b. * are almost everywhere time-differentiable with
essentially bounded time derivative, possessing eventual isolated bounded
discontinuities, then with being a -norm dependent nonnegative real constant and, furthermore, for some , ,
and some fixed independent of . 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 satisfy ; , , ;
that is, 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 , are subject to Assumption 2.1;(i.b)the switching law is such that ,
(3.3), for
some time-differentiable real symmetric positive definite matrix function and some real symmetric positive definite
matrix functions .
Thus, the system (2.1) is globally asymptotically Lyapunov's stable independent
of the delays (i.e., for all delays , ). A necessary condition is , what implies that is a stability matrix of prescribed stability
abscissa on except eventually on a real subinterval of
finite measure of .*(ii)

*Assume that(ii.a)*

*, , for some (eventually being dependent on ) satisfying Assumption 3.3;*(ii.b)*the switching law is such that (i.e., it generates reset switching instants only) with 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)*for some , . 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 . If (3.4) is replaced with , , and some then the state trajectory decays exponentially with .*Parallel results to Corollaries 2.13–2.15 are direct from Theorem 3.4 with the replacements (a constant stability matrix), , (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 , and any switching law such that (a) the switching instants are
arbitrary;(b) is sufficiently small compared to the absolute
value of the prescribed stability abscissa of ;(c)the support testing matrix of
distributional derivatives of the same
matrices are semidefinite negative for all time instants, where the
conventional derivatives do not exist (i.e., if) .*(ii)

*The switched system (2.1) is globally exponentially stable independent of the delays if (a)*

*is sufficiently small compared to the absolute value of the prescribed stability abscissa of ;*(b)*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 is sufficiently small compared to the absolute value of the prescribed stability abscissa of and, furthermore, the switching law is such that (a)*

*At reset-free switching instants, is sufficiently small compared to the time intervals in between any two consecutive reset switching instants;*(b)*there exists a common Krasovsky-Lyapunov functional defined with constant matrices and , 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 under the resetting action ; .*#### 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 , where second and each resetting
system (2.4) with is defined by with and ,
where denotes the largest integer not larger than
(*·*) and denotes the smallest integer not smaller than
(*·*). Note that and are
discontinuous functions at integer values of time. A graphical representation
of these functions is shown in Figures 1 and 2.

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.

Secondly, the time derivatives of the resetting systems defined above are given on each real interval with by with ; . Thus, the “sufficiently smallness condition” mentioned in Theorem 2.16(i) is fulfilled according to its proof in Appendix if , where for the existing unique symmetric positive definite solution of . Numerical computations lead to 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 and for all . Therefore, from Theorem 2.16(ii), the space-state trajectories of the solution asymptotically converge to zero as time evolves as Figure 4 shows.

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 .

##### 4.2. Delay-Independent Stability

This example is devoted to the delay independent stability ensured by Theorem 3.5(i). Thus, consider the system and the resetting systems with and . Again, this function possesses bounded discontinuities at integer values of time as Figure 6 shows.

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

##### 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 the strip of state-trajectory solution of the system 2.1 for . Consider the Krasovsky-Lyapunov functional candidate: which is nonnegative and radially unbounded since as , with , since all the eigenvalues of and positive and uniformly bounded from above and below . This also implies that cannot be neither positive definite nor negative definite . Direct calculations via 2.9 yield , if and only if . Then, and as for any given bounded function of initial conditions what imply that , and as . The global asymptotic stability has been proven. To prove the last part of Property (i), note that 2.9 implies Since , is impossible from the preceding part of the proof, it has to exist a nonnecessarily connected subinterval such that , . Now, proceed by contradiction to prove that 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) has finite measure so that the complement in is nonconnected with infinite measure with a component being necessarily of infinite measure (otherwise, has infinite measure). Thus, or indefinite . Since has finite measure and has a component of infinite measure, it exists a sufficiently large finite such that so that as is impossible what leads to a contradiction.(2)Both intervals and 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, is a stability matrix except possibly within an interval of finite measure.

(ii) Denote by the strip of state-trajectory solution , for and any resetting system. Consider the Krasovsky-Lyapunov functional candidate for all the resetting systems:

The real functional A.1 is a common Krasovsky-Lyapunov functional for all distinct resetting systems for all delays , provided that , and since the th resetting system 2.4 satisfies from 2.11 for all nonzero where and accordingly for all nonzero where for the switched system 2.1 since for any such that , where and 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: , for some finite , where . Then, from A.7 and A.3, and some , where and and denotes the (or spectral) vector norm or the corresponding induced ones for matrices. Then, the system 2.1 is globally exponentially stable for all delays , . The modification of 2.11-2.12 with , leads directly to an exponential decay of with rate from a similar slightly extended proof.

(iii) If Assumption 2.4 holds then for any . Thus, it exists a sufficiently small such that 2.9 holds , since ; .

If Property (i) holds then what implies since for any , . Thus, , and then , and sufficiently small since from 2.11 is a monotonically increasing function of the argument being zero if , . Then, Property (i) holds for a sufficiently small .

##### A.2. Derivation of the Inequality 2.17

It follows from the subsequent inequalities: