Volume 2008, Article ID 231710,31pages doi:10.1155/2008/231710

*Research Article*

**On the Global Asymptotic Stability of**

**Switched Linear Time-Varying Systems with** **Constant Point Delays**

**M. de la Sen**^{1}**and A. Ibeas**^{2}

*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 Vall´es, 08193 Bellaterra, Barcelona, Spain*

Correspondence should be addressed to M. de la Sen,manuel.delasen@ehu.es Received 22 July 2008; Accepted 25 September 2008

Recommended by Antonia Vecchio

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.

Copyrightq2008 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.

**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 stateinternal delaysor in the input or outputexternal 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 forthsee, e.g.,1–9. Time- delay models are useful for modeling both linear systems see, e.g.,1–4,10and certain nonlinear physical systems,see, e.g.,4,7–9,11. 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,2–4,6–13either 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 in19–21. Switched systems consist of a number of diﬀerent parameterizationsor distinct active systemssubject 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.

aThe 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.

bThe system parameters are changing so that the operation points change. Thus, a switched model which adjusts to several operation points may be useful19–21.

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 pair5,22.

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 rates23. 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 derivativewhich 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 forthsee, e.g.,24–28and references therein. However, it may be relevant in practical situations to distinguish a switch to prescribed time-invariant parameterizationssee the above situationsa–cfrom 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 3gives some extension for global asymptotic stability independent of the delays. Numerical examples are presented inSection 4, where switching through time in between distinct parameterizations is discussed. Finally, conclusions end the paper. Some mathematical derivations concerned with the results of Sections2 and 3 are derived inAppendix A.

**2. Asymptotic stability dependent on the delays**

Consider the *nth order linear time-varying dynamic system with* *q* internal in general,
incommensurateknown point delays:

*xt *˙ ^{q}

*j0*

*A**j*tx
*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

*h*

_{0}0, for some delays

*h*

*∈ 0,*

_{j}*h*

*, of finite or infinite maximum allowable delays sizes*

_{j}*h*

*j*∈

**R**

_{0 }, for all

*j*∈

*q*: {1,2, . . . , q}, where R0

**is the nonnegative real axis R**_{0 } : **R** ∪ {0} {0 ≤ *z* ∈ **R}; and***A*_{j}**: R**_{0 } → **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* **: R**_{0 } → **R*** ^{n×n}* are piecewise
continuous and uniformly bounded for all

*j*∈

*q*∪ {0}.

*Assumption 2.2. All the eigenvaluesλ**i**q*

*j0**A**j*t of the matrix function*q*

*j0**A**j*t satisfy
Re*λ**i**q*

*j0**A**j*t≤ −ρ0 *<*0 for all*t*∈**R**_{0 }, for all*i*∈*σ*σ≤*n*for some*ρ*0∈**R** :{0*< z*∈**R},**
that is,*q*

*j0**A** _{j}*tis a stability matrix for all

*t*∈

**R**

_{0 }.

*Assumptions 2.3. The matrix functions* *A**j* **: R**_{0 } → **R*** ^{n×n}* are almost everywhere time
diﬀerentiable with essentially bounded time derivative for all

*j*∈

*q*∪ {0}possessing eventual isolated bounded discontinuities, then ess sup

_{t∈R}

_{0 }*q*

*j0**A*˙* _{j}*t ≤

*γ*

_{}*<*∞with

*γ*

*being a*

_{}*-norm dependent nonnegative real constant and, furthermore,*

_{t T}

*t*

*A*˙* _{j}*τ

*dτ*≤

*μ*

_{j}*T*

*α*

*≤*

_{j}*μT*

*α*∀j∈

*q*∪ {0} 2.2 for some

*α*

_{j}*, μ*

_{j}*, α, μ*∈

**R**for all

*t*∈

**R**

_{0 }, and some fixed

*T*∈

**R**

_{0 }independent of

*t.*

At time instants*t, where the time-derivative of some entry ofA**j*tdoes not exist for
any*j* ∈*q∪{0}, the time derivative is defined in a distributional Dirac sense as ˙A** _{j}*t Γtδ0
what equivalently means the presence of a discontinuity at

*t*in

*A*

*tdefined as*

_{j}*A**j*

*t*
*A**j*

*t*^{−}

*ε→*lim0

_{ε}

−εΓ*j*τδt−*τdτ* *A**j*t Γ*j*t. 2.3

Assumption 2.1is relevant for existence and uniqueness of the solution of2.1. 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-Lindel ¨oﬀ existence and uniqueness theorem. Assumption 2.2 establishes that

*q*

*j0**A** _{j}*t is a stability matrix for all time what is known to be a necessary condition for
the global asymptotic stability of the system2.1for a set of prescribed maximum delays
in the time-invariant case. It is well known that even if

*A*

*≡ 0 for all*

_{j}*j*∈

*q, then the*resulting linear time-varying delay-free system cannot be proved to be stable without some additional assumptions, like for instance, Assumptions2.3. 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

*q*

*j0**A** _{j}*tat times, where it is
impulsive. An alternative assumption to Assumptions2.3which avoids the assumption of
almost everywhere existence of a bounded ˙

*A*

*t, for all*

_{j}*j*∈

*q*∪ {0} see second part of Assumption 2.1might be stated in terms of suﬃciently smallness ofΔA

*j*tforΔA

*j*t:

*A*

*t−*

_{j}*A*

^{∗}

*for all*

_{j}*t*∈

**R**

_{0 }for all

*j*∈

*q*∪ {0}for some constant stability matrix

*q*

*j0**A*^{∗}* _{j}* whose
eigenvalues satisfy Re

*λ*

_{i}*q*

*j0**A*^{∗}* _{j}* ≤ −ρ0

*<*0. Such an alternative assumption guarantees also the global existence and uniqueness of the state-trajectory solution of2.1and 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 *q*

*j0**A**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
delaysseeTheorem 2.12i .

**2.2. Switching function, switching sequence, and basic assumptions on****the switching matrix function**

Assumption 2.1admits bounded discontinuities in the entries of*A**j*tfor*j* ∈*q*∪ {0}. At such
times*A** _{j}*t denote right values of the matrix function while

*A*

*t*

_{j}^{−}is simply denoted by

*A*

*j*t. A set of

*p*resetting systems of2.1is defined by the linear time-invariant systems:

˙
*z** _{j}*t

*q*
*i0*

*A*_{ij}

*z*_{j}*t*−*h*_{j}

2.4

for some given *A*_{ji}**R*** ^{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 2.2extends in a natural fashion to include the resetting systems as follows.

*Assumption 2.4. All the eigenvaluesλ**k**q*

*i0**A**ij*satisfy Reλ*k**q*

*i0**A**ij*≤ −ρ0 *<*0 for all*t*∈**R**_{0 }
for all*k* ∈ *σ** _{j0}* σ

*j0*≤

*n*for all

*j*∈

*p; that is,*

*q*

*i0**A** _{ij}* are constant stability matrices with
prescribed stability abscissa−ρ

_{0}

*<*0 for all

*j*∈

*p.*

The following definitions are then used.

*Definition 2.5. The switching matrix function is a mappingσ* **: R**_{0 } → {A*j*t −*A** _{j}*t,∀j ∈

*q*∪ {0}} ⊂

**R**

*from the nonnegative real axis to the set of real*

^{n×q 1n}*n*×q 1nmatrices.

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* ∈ **R**_{0 } 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 STσ.

Two kinds of switching instants, respectively, reset instants and reset-free switching instants defined in Definitions2.7and2.8are considered.

*Definition 2.7*reset instant. *t* ∈ **R**_{0 } is a reset instant generated by the switching law *σ*if
*t*∈STσand*A**i*t *A**ij*for some*i*∈*q*∪ {0}and some*j* ∈*p, provided thatA**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 ⊂ STσ from Definitions 2.6 and 2.7. Note also that the whole system
parameterization is driven to some of the prefixed resetting systems2.4when a reset instant
happens. Note that at reset instants, ˙*A** _{i}*t A

*ij*−

*A*

*tδ0for some*

_{i}*i*∈

*q*∪ {0},

*j*∈

*p.*

*Definition 2.8*reset-free switching instant. *t*∈**R**_{0 }is a switching reset-free instant generated
by the switching law*σ*if*t*∈STσand*A*_{ij}*/A** _{i}*t

*/A*

*tfor some*

_{i}*i*∈

*q*∪ {0}, for all

*j*∈

*p.*

The set of reset-free instants is denoted by *t* ∈ STrfσ. 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 parameterization2.4at the same time, then
the corresponding instant is considered a reset time instant. Note that STrfσ ⊂ STσ,
STσ ST*r*σ∪STrfσ, and ST*r*σ∩STrfσ ∅from Definitions2.6–2.8, 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 STσ, t, the partial switching sequence*
ST*r*σ, t, and the reset-free partial switching sequence STrfσ, t, generated by the switching
law *σ* **: R**_{0 } → {A*j*t−*A** _{j}*t

^{−}, ∀j ∈

*q*∪ {0}} ⊂

**R**

*up till any time*

^{n×q 1n}*t*∈

**R**

_{0 }, are defined, respectively, by STσ, t:{t

*i*∈STσ:

*t*

_{i}*< t}, ST*

*r*σ, t:{t

*i*∈ST

*σ:*

_{r}*t*

_{i}*< t}, and*STrfσ, t:{t

*i*∈STrfσ:

*t*

*i*

*< t}.*

*Remark 2.10. An interpretation of Assumptions*2.3is that the following conditions hold for
any given*-matrix norm for some nonnegative norm dependent real constantsμ** _{j}*,

*α*

*,*

_{j}*μ, and*

*α*for all

*j*∈

*q*∪ {0}:

ess sup

*t∈R*_{0}

*A*˙*j*t≤*μ**j*≤max ess sup

*t∈R*_{0}

*A*˙*j*t:*j*∈*q*∪ {0}

≤*μ,*

*τ∈*STσ∩t,t T

*A*˙* _{j}*τ

≤

*τ∈STσ∩t,t T*

Xτδ0 ≤*α*_{j}

≤max

*τ∈STσ∩t,t T*

*A*˙* _{j}*τ

:*j*∈*q*∪ {0}

≤*α,*

2.5

where*δ0*is a Dirac impulse at*t* 0. Note that Assumptions2.3imply|*q*

*j0**A** _{j}*t

*τ*−

*q*

*j0**A** _{j}*t| ≤

*μT*

*νt*

*τ*for all

*τ*∈ 0, T, with the function

*ν*

**: R**

_{0 }→

**R**

_{0 }satisfying

*νt*

*τ*≤

*α*

_{0}

*j*

*α*

_{1j}t

*τ*≤

*α*with

*α*

_{1j}t

*τ*0 if ˙

*A*

*t*

_{j}*τ*exists with

*A*˙

*t*

_{j}*τ ≤α*

*or*

_{j}*α*

*j*−

*α*0j ≥

*α*1jt

*τ*≥ Δ

*j*t

*τ*if ˙

*A*

*j*t

*τ*Δ

*j*t

*τδ0, that is, at least one of its entries*is impulsive.

*Note thatAssumption 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 2.11. The following properties are true irrespective of the switching function:*

i*t /*∈STσ⇔*σt* *σt *0* _{n×q 1n}*i.e., a zero

*n*×q 1n-matrix;

ii*t*∈STσ⇔*σt* −*σt/*0* _{n×q 1n}*;

iii*t /*∈STσ⇔STσ, t STσ, t;

iv*t*∈STσ⇔STσ, t */*STσ, t.

*Proof.* i*σt* *σt *0* _{n×q 1n}* ⇒

*t /*∈ST

*r*σ∧

*t /*∈STrfσ ⇔

*t /*∈STσ, t /∈ST ⇔

*σt*0 since switching is not arbitrarily fast,

∧
*σ*

*t*

−*σt *0_{n×q 1n}

⇐⇒*σ*
*t*

*σt *0_{n×q 1n}*.* 2.6

Propertyihas been proven. Propertyiiis the contrapositive logic proposition to Property i, and thus equivalent, since switching is not arbitrarily fast.

Propertiesiii-ivare also contrapositive logic propositions, then equivalent since

STσ, t:

*t** _{i}*∈STσ:

*t*

_{i}*< t*

⇐⇒ST
*σ, t*

:

*t** _{i}*∈STσ:

*t*

_{i}*< t*

*t** _{i}*∈STσ:

*t*

*≤*

_{i}*t*

⇐⇒STσ, t

⎧⎨

⎩

STσ, t if*t/*∈STσ,
*/*STσ, t if *t*∈ STσ

2.7

since switching cannot happen arbitrarily fast. Propertiesiii-ivhave been proven.

The subsequent global stability result is proven in Appendix A by guaranteeing that the Krasovsky-Lyapunov functional candidate below is indeed a Krasovsky-Lyapunov functional:

*V*t, x*t* *x** ^{T}*tPtxt

*q*

*i1*

*q*
*j0*

_{−h}_{j}

−h*i*−h*j*

_{t}

*t θ**x** ^{T}*τS

*ij*

*xτdτ dθ.*2.8

**Theorem 2.12. The following properties hold.**

i*Assume the following.*

i.a*The matrix functionsA** _{j}*t,

*for allj*∈

*q*∪ {0}

*are subject toAssumption 2.1.*

i.b*The switching lawσis such that*

*Qt*:

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎣

*H1* *h*1*P*tA1tMt · · · *h**q**P*tA*q*tMt
*h*1*M** ^{T}*tA

^{T}_{1}tPt −R1 0· · · 0

*...* 0

*...*

*. ..* *...*

*h**q**M** ^{T}*tA

^{T}*tPt 0 · · · −R*

_{q}*q*

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎦

*<*0,

∀t∈**R**_{0 }*,*
2.9

*whereH1 denotes**q*

*j0**A*^{T}* _{j}*tPt

*P*t

*q*

*j0**A** _{j}*t

*Pt*˙

*q*

*i1*

*q*

*j0**h*_{i}*S*_{ij}*, for*
*some time-diﬀerentiable real symmetric positive definite matrix functionP* **: R**_{0 } →
**R**^{n×}^{n}*and some real symmetric positive definite matricesS** _{ij}* ∈

**R**

*∀i ∈*

^{n×n}*q,*∀j ∈

*q*∪ {0}, where

*Mt*:

*A*_{0}t, . . . , A*q*t

∈**R**^{n×q 1n}*,*
*R** _{i}*:diag

*S*_{i0}*, S*_{i1}*, . . . , S*_{iq}

∈**R**^{q 1n×q 1n}; ∀i∈*q,* ∀t∈**R**_{0 }*.* 2.10
*Thus, the system*2.1*is globally asymptotically Lyapunov’s stable for all delaysh** _{i}*∈
0,

*h*

*i*, for all

*i*∈

*q. A necessary condition is*

*q*

*j0**A*^{T}* _{j}*tPt

*P*t

*q*

*j0**A**j*t
*P*˙t *<* *0, for allt* ∈ **R**_{0 } *what implies that* *q*

*j*0*A**j*t *is a stability matrix of*
**prescribed stability abscissa on R**_{0 } *except eventually on a real subinterval of finite*
**measure of R**_{0 }*.*

ii*Assume the following*

ii.a*A**i*t *A**ij**, for alli* ∈ *q*∪ {0}, for all*t* ∈ **R**_{0 } *for some* *j* ∈ *p* *(eventually being*
*dependent on t) satisfyingAssumption 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}

*j0*

*A*^{T}_{ji}

*P*^{∗} P^{∗}
_{q}

*j0*

*A*_{ji}_{q}

*i1*

*q*
*j0*

*h*_{i}*S*^{∗}_{ij}*h*_{1}*P*^{∗}*A*_{1i}*M*_{1}^{∗} · · · *h*_{q}*P*^{∗}*A*_{qi}*M*^{∗}_{q}*h*_{1}*M*^{∗}_{1}^{T}*A*^{T}_{1i}*P*^{∗} −R^{∗}_{1} 0· · · 0

*...* 0

*...*

*. ..* *...*
*h*_{q}*M*^{∗}_{q}^{T}*A*^{T}_{qi}*P*^{∗} 0 · · · −R^{∗}_{q}

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎦

*<*0;

∀i∈*p*
2.11

**for some R**^{n×n}*P*^{∗}*P*^{∗T} *> 0, R*

^{n×n}*S*

^{∗}

_{ij}*S*

^{∗}

_{ij}

^{T}*>*0∀i∈

*q,*∀j∈

*q*∪ {0}, where

*M*

*i*:

*A*0i*, . . . , A**qi*

∈**R**^{n×q 1n}*,*

*R*^{∗}* _{i}* :diag

*S*^{∗}_{i0}*, S*^{∗}_{i1}*, . . . ,*

∈**R**^{q 1n×q 1n}; ∀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*

*, for all*

_{i}*i*∈

*q. If*2.9

*is replaced with*

*Q*

^{∗}

*≤*

_{i}−2εI_{q 1n} *<* *0, for alli* ∈ *q, and some* *ε* ∈ **R** *then the state trajectory decays*
*exponentially with rate* −ε *<* *0.*

iii*There is a suﬃciently smallh*:max_{i∈q}*h*_{i}*such that Property (i) holds for anyh** _{i}*∈0,

*h*

*,*

_{i}*for alli*∈

*qprovided that all the delay-free resetting systems*2.4

*z*˙

*t*

_{j}*q*

*i0**A** _{ij}*z

*j*t

*fulfilAssumption 2.4, that is, they are globally exponentially stable.*

It is of interest to discuss particular cases easy to test, guaranteeingTheorem 2.12i.
**2.3. Sufficiency type asymptotic stability conditions obtained for**

**constant symmetric matrices***P* **and**S_{ij}

Assume real constant symmetric matrices*Pt * *P* and*S**ij*, for all*i* ∈ *q ,*for all*j* ∈ *q*∪ {0},
for all*t* ∈ **R**_{0 }so that

*Qt *

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎣

*H2* *h*1*P A*1tMt · · · *h**q**P A**q*tMt
*h*1*M** ^{T}*tA

^{T}_{1}tP −R1 0· · · 0

... 0

...

. .. ...
*h*_{q}*M** ^{T}*tA

^{T}*tP 0 · · · −R*

_{q}*q*

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎦

*<*0, 2.13

where*H2 denotes**q*

*j0**A*^{T}* _{j}*tP

*P*

*q*

*j0**A** _{j}*t

*q*

*i1*

*q*

*j0**h*_{i}*S** _{ij}*. In this case, the Krasovsky-
Lyapunov functional used in the proof of Theorem 2.12i 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
2.13to hold follows. Consider*A*^{∗}* _{i}* i ∈

*q*∪ {0} such that the time invariant system2.1 defined with

*A*

*i*t →

*A*

^{∗}

*is globally asymptotically Lyapunov’s stable and define a stability real*

_{i}*n-matrixA*

^{∗}:

*q*

*i0**A*^{∗}* _{i}*. Decompose

*Qt Q*

^{∗}

*Qt, where*

*Q*^{∗}:

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎣

*A*^{∗}^{T}*P* *P A*^{∗}
*q*

*i1*

*q*
*j0*

*h*_{i}*S*_{ij}*h*_{1}*P A*^{∗}_{1}*M*^{∗} · · · *h*_{q}*P A*^{∗}_{q}*M*^{∗}
*h*1*M*^{∗T}*A*^{∗}_{1}^{T}*P* −R^{∗}_{1} 0· · · 0

... 0

...

. .. ...
*h*_{q}*M*^{∗}^{T}*A*^{∗}_{q}^{T}*P* 0 · · · −R^{∗}_{q}

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎦
*,*

*Qt* :

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎣

*A** ^{T}*tP

*PAt*

*h*

_{1}

*P*Δ1t · · ·

*h*

_{q}*P*Δ

*q*t

*h*

_{1}Δ

^{T}_{1}tP 0 0· · · 0

... 0

...

. .. ...
*h** _{q}*Δ

^{T}*tP 0 · · · 0*

_{q}⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎦
*,*

2.14

where

*M*^{∗}:diag

*A*^{∗}_{0}*, A*^{∗}_{1}*, . . . , A*^{∗}_{q}

*,* *R*^{∗}* _{i}* :diag

*S*^{∗}_{i0}*, S*^{∗}_{i1}*, . . . , S*^{∗}_{iq}

*,* *Mt* :*Mt*−*M*^{∗}*,*
*A**i*t:*A**i*t−*A*^{∗}*,* *At* :

_{q}

*j0*

*A**ji*t

−*A*^{∗}*,*
Δ*i*t: *A**i*tM^{∗} *A*^{∗}_{i}*Mt * *A**i*t*M*^{↔} t

*.*

2.15

If*t* ∈ STσ, then

*δQt* :*Q*
*t*

−*Qt*

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎣

*H3* *h*1*P* Δ1

*t*

−Δ1t

· · · *h**q**P* Δ1

*t*

−Δ1t
*h*1 Δ1

*t*

−Δ1t_{T}

*P* 0 0· · · 0

... 0

...

. .. ...

*h**q* Δ1

*t*

−Δ1t_{T}

*P* 0 · · · 0

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎦
*,*

*δQt*

2≤2λmaxP

*A*^{T}*t*

−*A** ^{T}*t

2

*q*
*i1*

*h**i*Δ*i*

*t*

−Δ*i*t

2

2λmaxP

*q*
*i0*

*A*^{T}_{i}*t*

−*A*^{T}* _{i}*t

2

*q*
*i1*

*h**i*Δ*i*

*t*

−Δ*i*t

2

*,* 2.16

where*H3 denotes* *A** ^{T}*t −

*A*

*tP*

^{T}*PAt*−

*At,*what leads to

*δQt*

2≤2λ_{max}P

q 1a *hq*

*a*
_{q}

*j0*

A^{∗}* _{j}*2 q 1A

^{∗}

*2 q 1a*

_{i}*a*

q 12a *a* a

*,*

2.17

where *a* : max* _{i∈q∪{0}}*A

*i*2 and

*a*: sup

*max*

_{t∈STσ}*A*

_{i∈q∪{0}}*i*t −

*A*

_{i}_{2}see A.9 in Appendix A. Direct results fromTheorem 2.12which follow from2.13to2.17are given below.

* Corollary 2.13. Consider in* 2.9

*replacements with constant real matricesQt*→

*Q*

^{∗}

*Q*

^{∗}

^{T}*,*

*Pt*→

*P*

*P*

^{T}*>*

*0,A*

*i*t →

*A*

^{∗}

_{i}*,At*→

*A*

^{∗}:

*q*

*i0**A*^{∗}_{i}*; for alli* ∈ *q ,for allj* ∈*q*∪ {0},
*for allt* ∈ **R**_{0 } *such that* *A*^{∗} *is a stability matrix. Then,* *Theorem 2.12(i) holds if* *Q*^{∗} *<* *0 for any*
*switching lawσsuch that*

1*λ*min−Q^{∗} −λmaxQ^{∗}*>Qt* _{2}:*λ*^{1/2}_{max}*Q** ^{T}*t

*Qt, for all*

*t*∈

**R**

_{0 }\STσ, 2

*a*: max

*A*

_{i∈q∪{0}}*i*2

*anda*: sup

*max*

_{t∈STσ}*A*

_{i∈q∪{0}}*i*t −

*A*

*2*

_{i}*are suﬃciently*

*small such that*

*λ*min

−*Q*^{∗}

− *Qt*

2

*>*2λ_{max}P

q 1a *hq*

*a*
_{q}

*j0*

*A*^{∗}_{j}

2 q 1max

*i∈q∪{0}**A*^{∗}_{i}

2 q 1a *a*

q 12a *a* a

*,* ∀t∈STσ.

2.18

* Corollary 2.14. Consider in* 2.9

*replacements with constant real matricesQt*→

*Q*

^{∗}

_{j}*Q*

_{j}^{∗}

^{T}*,*

*Pt*→

*P*

*P*

^{T}*>*

*0,A*

*t →*

_{ij}*A*

^{∗}

_{ij}*,At*→

*A*

^{∗}

*:*

_{j}*q*

*i0**A*^{∗}_{ij}*for allj* ∈ *p,for allk* ∈ *q*∪ {0},
*for allt* ∈ **R**0 *such that eachA*^{∗}* _{j}* ∀j ∈

*pis a stability matrix withA*

^{∗}

_{ij}*; for alli*∈

*q,*

*for allj*∈

*p*

*being the parameterizations defining the resetting systems*2.4. Assume that the system2.1

*is one of*

*the resetting systems*2.4

*att0. Then,Theorem 2.12(i) holds with a common Krasovsky-Lyapunov*

*function for all those resetting systems ifQ*

^{∗}

_{j}*<*0 ∀j∈

*pfor any switching lawσsuch that*

1*λ*min−Q^{∗} −λmaxQ^{∗}*>Qt* _{2}:*λ*^{1/2}_{max}*Q** ^{T}*t

*Qt,*

*for all*

*t*∈

**R**

_{0 }\STσ, 2

*a*

*j*:max

*A*

_{i∈q∪{0}}*ji*

_{2}

*anda*

*:sup*

_{j}*max*

_{t∈STσ}*A*

_{i∈q∪{0}}*i*t −

*A*

*ji*

_{2}

*are suﬃciently*

*small such that*

*λ*min

−*Q*^{∗}

− *Qt*

2

*>*2λ_{max}P

q 1a_{j}*hq*

*a*_{j}_{q}

*i0*

A*ji*_{2} q 1max

*i∈q*∪{0}A*ji*_{2}

*q* 1

*a*_{j}*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 system2.1*coincides*
*with at thej* ∈ *prsetting system*2.4.

The proof ofCorollary 2.14is close to that ofCorollary 2.13fromA.9inAppendix A
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∈p}*a*

*j*

*, a*

*→*

_{j}*a*: max

_{j∈}*p*

*a*

*then the reformulated weaker Corollary 2.14is valid for all*

_{j}*t*∈STfrσirrespective of the preceding reset switching. A result which guarantees Corollary 2.13, and then Theorem 2.12i, is now obtained by replacing the1,1block matrix of

*Q*

^{∗}by a Lyapunov matrix equality as follows. Consider a real

*n-*matrix

*Q*

^{∗}

_{0}

*Q*

^{∗T}

_{0}

*>*0 such that

*λ*minQ

^{∗}

_{0}

*> λ*max

*q*

*i1*

*q*

*j0**h**i**S**ij*and*P* : _{∞}

0 *e*^{A}^{∗T}^{τ}*Q*_{0}^{∗}*e*^{A}^{∗}^{τ}*dτ*
satisfying the Lyapunov equation*A*^{∗T}*P* *P A*^{∗} −Q^{∗}_{0} *<* 0 as its unique solution. Note that
*λ*maxP ≤ *K*^{∗}*λ*maxQ^{∗}_{0}/2ρ^{∗}for some*K*^{∗} ∈**R** , where−ρ^{∗} *<* 0 is the stability abscissa of*A*^{∗}
withe^{A∗}^{T}* ^{t}* ≤√

*K*^{∗}*e*^{−ρ}^{∗}* ^{t}*for all

*t*∈

**R. Define the decomposition**

*Qt*

*Q*

^{∗}

*Qt, where*

*Q*^{∗}: Block Diag

−*Q*^{∗}_{0}
*q*

*i1*

*q*
*j0*

*h*_{i}*S*_{ij}*,*−R^{∗}_{1}*, . . . ,*−R^{∗}_{q}

*,*

*Qt* :

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎢⎣

*A** ^{T}*tP

*PA*

*t*

*h*

_{1}

*P*

*A*^{∗}_{1}*M*^{∗} Δ1t

· · · *h*_{q}*P*

*A*^{∗}_{q}*M*^{∗} Δ*q*t
*h*_{1}

*A*^{∗}_{1}*M*^{∗} Δ1t*T*

*P* 0 0· · · 0

... 0

...

. .. ...

*h*_{q}

*A*^{∗}_{q}*M*^{∗} Δ*q*t*T*

*P* 0 · · · 0

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎦

⇒*Qt*

2≤

*K*^{∗}*λ*_{max}
*Q*_{0}^{∗}
*ρ*^{∗}

*A** ^{T}*t

2

*q*
*i1*

*h*_{i}*A*^{∗}_{i}*M*^{∗} Δ*i*t

2

*.*

2.20

Thus, the subsequent result follows fromCorollary 2.13and2.20.

**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 equationA*^{∗}^{T}*P* *P A*^{∗} −Q_{0}^{∗} *<* *0. Then,*
*Theorem 2.12(i) holds ifQ*^{∗}*<0 for any switching lawσsuch that*

*λ*_{min}

−*Q*^{∗}

−λmax

*Q*^{∗}

*>* *K*^{∗}*λ*max

*Q*^{∗}_{0}
*ρ*^{∗}

*A** ^{T}*t

2

*q*
*i1*

*h**i**A*^{∗}_{i}*M*^{∗} Δ*i*t

2

*,*

∀t∈**R**_{0 }*,* ∀t∈STσ.

2.21

**2.4. Sufficiency type asymptotic stability conditions obtained for time-varying*** symmetric matricesP*t,

*S*

*t*

_{ij}*S*

_{ij}The following result, which is proven inAppendix A, 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 delaysh** _{i}* ∈0,

*h*

_{i}*for all*

*i*∈

*qfor someh*:max

_{i∈q}*h*

*i*

*and any switching lawσsuch that*

a*the switching instants are arbitrary;*

bmaxess sup*A*˙* _{j}*t:

*t*∈

**R**

_{0 }

*, j*∈

*pis suﬃciently small compared to the absolute*

*value of the prescribed stability abscissa of*

*q*

*j*0*A*_{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., ˙A** _{j}*t Γ

*Adj*ttδ0).

ii*The switched system*2.1*is globally exponentially stable for any delaysh**i*∈0,*h**i**for all*
*i*∈*qfor someh*:max_{i∈q}*h*_{i}*such that*

amaxess sup*A*˙*j*t:*t*∈**R**_{0 }*, j* ∈*pis suﬃciently small compared to the absolute*
*value of the prescribed stability abscissa of**q*

*j*0*A*_{j}*t;*

bmaxΓ*Adj*tt: ˙*A**j*t Γ*Adj*ttδ0, ∀t∈STσ, j ∈*pis suﬃciently small*
*compared to the timeintervals in between any two consecutive switching instants.*

*Furthermore, if Assumptions2.1–2.4hold, then*

iii*the switched system*2.1*is globally exponentially stable for any delaysh** _{i}*∈0,

*h*

_{i}*for all*

*i*∈

*q, for someh*:max

_{i∈q}*h*

*i*

*such that*

amaxess sup*A*˙*j*t:*t*∈**R**_{0 }*, j* ∈*pis suﬃciently small compared to the absolute*
*value of the prescribed stability abscissa of**q*

*j0**A**j*t;

b*the switching lawσis such that*

amaxΓ*Adj*t: ˙*A** _{j}*t Γ

*Adj*tδ0,∀t∈ST

*σ, j ∈*

_{rf}*pis 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 matricesP* *P*^{T}*>0 andS**ij**S*^{T}_{ij}*>0, for all*i, j∈q∪{0}×q*for all*
*the time-invariant resetting systems*2.4*and some of the subsequent conditions*
*hold for allt* ∈ ST*r*σ *under the resetting action* *Pt* *P; for all*i, j ∈
q∪ {0}×*q:*

b.1*V*t *, x** _{t}*≤

*V*t, x

*t*

*which is guaranteed, in particular, ifP*t

*P*≤

*Pt,*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, wheret*

^{}maxτ ∈

**R**

_{0 }: STσ

*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 amplitudein 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 indepen-
dent of the delays, that is, for any *h**i* ∈ **R**_{0 }, for all*i* ∈ *q* of the switched system2.1are
obtained under very close guidelines as those involved in the results on stability dependent
of the delays given inSection 2. The Krasovsky-Lyapunov functional candidate ofSection 2
andAppendix Ais modified as follows:

*V*t, x*t* *x** ^{T}*tPtxt

^{q}*i1*

_{t}

*t−h**i*

*x** ^{T}*τS

*i*τxτdτ 3.1

whose time derivative along the state-trajectory solution of2.1is

*V*˙t, x*t* *x** ^{T}*t

*A** ^{T}*tPt

*PtAt*

^{q}*i1*

*S**i*t *P*˙t

*xt*

2x* ^{T}*t

*q*

*i1*

*P A**i*tx
*t*−*h**i*

−
*q*

*i1*

*x*^{T}*t*−*h**i*

*S**i*

*t*−*h**i*

*x*
*t*−*h**i*

*x** ^{T}*tQ

^{}txt

*<*0

3.2

for all nonzero*x** ^{T}*t x

*t, x*

^{T}*t−*

^{T}*h*1, . . . , x

*t−*

^{T}*h*

*q*if

*Q*^{}t:

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎣

*A** ^{T}*tPt

*PtAt*

*q*

*i1*

*S**i*t *Pt*˙ *P A*1t · · · *P Aqt*

*A*^{T}_{1}tP −S1

*t*−*h*1

0· · · 0

... ... ...

*A*^{T}* _{q}*tP 0 −S

*q*

*t*−*h**q*

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎦

*<*0. 3.3

Assumption 2.1ofSection 2remains unchanged while Assumptions2.2and2.4ofSection 2 are modified under similar justifications as follows.

*Assumption 3.1. All the eigenvalues* *λ** _{i}*A0t of the matrix function

*A*

_{0}t satisfy Re

*λ*

*i*A0t≤ −ρ00

*<*0; for all

*t*∈

**R**

_{0 }, for all

*i*∈

*σ*σ≤

*n*for some

*ρ*00 ∈

**R**:{0

*< z*∈

**R};**

that is,*A*_{0}tis a stability matrix, for all*t*∈**R**_{0 }.

*Assumptions 3.2.* *A*_{j}**: R**_{0 } → **R**^{n×}* ^{n}* are almost everywhere time-diﬀerentiable with
essentially bounded time derivative, for all

*j*∈

*q*∪ {0}possessing eventual isolated bounded discontinuities, then ess sup

_{t∈R}0 *A*˙_{0}t ≤ *γ*_{0} *<* ∞ with *γ*_{0} being a *-norm dependent*
nonnegative real constant and, furthermore,_{t}_{T}

*t* *A*˙0*τd τ* ≤ *μ*0*T* *α*0 for some*α*0*, μ*0 ∈
**R** , for all*t* ∈ **R**_{0 }, and some fixed*T* ∈ **R**_{0 }independent of*t. If the time derivative does not*
exist then it is defined in the distributional sense as in Assumptions2.3.

*Assumption 3.3*for the resetting systems. All the eigenvalues*λ**k*A0jsatisfy Re*λ**k*A*j0* ≤

−ρ00*<*0; for all*t*∈**R**_{0 }, for all*k*∈*σ*0jσ0j≤*n, for allj*∈*p; that is,A*0*j*are constant stability
matrices with prescribed stability abscissa.

A parallel result toTheorem 2.12i-iiis the following.

**Theorem 3.4. The subsequent properties hold.**

i*Assume that*

i.a*the matrix functionsA**j*t, for all*j* ∈ *q* ∪ {0}*are subject toAssumption 2.1;*

i.b*the switching lawσis such thatQ*^{}*t* *<* *0,*3.3, for all*t* ∈ **R**_{0 }

*for some time-diﬀerentiable real symmetric positive definite matrix function* *P* **: R**_{0 } →
**R**^{n×}^{n}*and some real symmetric positive definite matrix functionsS*_{i}**: R**_{0 } → **R*** ^{n×n}*∀i∈

*q. Thus, the system*2.1

*is globally asymptotically Lyapunov’s stable independent of the*

*delays (i.e., for all delaysh*

*i*∈0,∞, for all

*i*∈

*q). A necessary condition isA*

^{T}_{0}tPt

*P*tA0t

*P*˙t

*<*

*0, for all*

*t*∈

**R**

_{0 }

*what implies that*

*A*

_{0}t

*is a stability matrix of*

**prescribed stability abscissa on R**_{0 }

*except eventually on a real subinterval of finite measure*

**of R**_{0 }

*.*

ii*Assume that*

ii.a*A**j*t *A**ji**, for allj* ∈ *q*∪ {0}, for all*t* ∈ **R**_{0 } *for some* *i* ∈ *p(eventually being*
*dependent ont) satisfyingAssumption 3.3;*

ii.b*the switching lawσis 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*^{T}_{0i}*P*^{∗} *P*^{∗}*A*_{0i}
*q*

*i1*

*S*^{∗}_{i}*P*^{∗}*A*_{1i} · · · *P*^{∗}*A*_{qi}*A*^{T}_{1i}*P*^{∗} −S^{∗}_{1} 0· · · 0

*...* 0

*...*

*. ..* *...*
*A*^{T}_{qi}*P*^{∗} 0 · · · −S^{∗}_{q}

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎦

*<*0; ∀i∈*p* 3.4

**for some R**^{n×n}*P*^{∗} *P*^{∗T} *>* **0, R**^{n×n}*S*^{∗}_{i}*S*^{∗}_{i}^{T}*>* 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 for alli* ∈ *q. If* 3.4*is replaced withQ*^{∗}* _{i}* ≤ −2εI

_{q 1n}

*<0,*

*for alli*∈

*q, and someε*∈

**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*A*_{0}t → *A*^{∗}_{0}a constant stability matrix,*A*_{0j}t → *A*^{∗}_{0j}, for all *j* ∈ *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 toTheorem 2.16, follows by replacing Assumptions2.2–2.4by Assumptions3.1–3.3.

**Theorem 3.5. Under Assumptions**2.1and3.1–3.2, the following properties hold.

i*The switched system*2.1*is globally asymptotically Lyapunov’s stable independent of the*
*delays, that is, for any delaysh**i*∈0,∞, for all*i* ∈ *pand any switching lawσsuch that*

a*the switching instants are arbitrary;*

bmaxess sup*A*˙0t:*t*∈**R**_{0 }*is suﬃciently small compared to the absolute value*
*of the prescribed stability abscissa ofA*0t;

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., if ˙A*_{0}t Γ*Ad0*ttδ0) .

ii*The switched system*2.1*is globally exponentially stable independent of the delays if*
amaxess sup*A*˙_{0}t:*t*∈**R**_{0 }*is suﬃciently small compared to the absolute value*

*of the prescribed stability abscissa ofA*0t;

bmaxΓ*Ad0*tt : ˙*A*0t Γ*Ad0*ttδ0, ∀t ∈ STσ *is suﬃciently small*
*compared to the time intervals in between any two consecutive switching instants.*