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

Authors: M. de la Sen; A. Ibeas
Journal: Discrete Dynamics in Nature and Society
 (2008)

---

## 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.


---

## Body

## 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
thenth order linear time-varying dynamic system with q internal (in
general, incommensurate) known point delays: (2.1)
x
˙
(
t
)
=
∑
j
=
0
q
A
j
(
t
)
x
(
t
−
h
j
) 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
j
∈
[
0
,
h
^
j
],
of finite or infinite maximum allowable delays sizes h
^
j
∈
R
0
+, for all
j
∈
q
¯
:=
{
1
,
2
,
…
,
q
},
where R
0
+ 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 functionsA
j
:
R
0
+
→
R
n
×
n are piecewise continuous and uniformly bounded
for all j
∈
q
¯
∪
{
0
}.Assumption 2.2.
All
the eigenvaluesλ
i
(
∑
j
=
0
q
A
j
(
t
)
) of the matrix function ∑
j
=
0
q
A
j
(
t
)  satisfy Re
λ
i
(
∑
j
=
0
q
A
j
(
t
)
)
≤
−
ρ
0
<
0 for all t
∈
R
0
+, for all
i
∈
σ
¯
(
σ
≤
n
) for some ρ
0
∈
R
+
:=
{
0
<
z
∈
R
}, that is, ∑
j
=
0
q
A
j
(
t
)  is a stability matrix  for all t
∈
R
0
+.Assumption 2.3.
The matrix functionsA
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
+
(
∥
∑
j
=
0
q
A
˙
j
(
t
)
∥
)
≤
γ
ℓ
<
∞ with γ
ℓ being a ℓ-norm dependent nonnegative real constant  and, furthermore, (2.2)
∫
t
t
+
T
∥
A
˙
j
(
τ
)
∥
d
τ
≤
μ
j
T
+
α
j
≤
μ
T
+
α
∀
j
∈
q
¯
∪
{
0
}  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 instantst, where the time-derivative of some entry of A
j
(
t
) does
not exist for any j
∈
q
¯
∪
{
0
},
the time derivative is defined in a distributional Dirac sense as A
˙
j
(
t
)
=
Γ
(
t
)
δ
(
0
) what
equivalently means the presence of a discontinuity at t in A
j
(
t
) defined as (2.3)
A
j
(
t
+
)
=
A
j
(
t
−
)
+
lim
ε
→
0
+
∫
−
ε
ε
Γ
j
(
τ
)
δ
(
t
−
τ
)
d
τ
=
A
j
(
t
)
+
Γ
j
(
t
)
.Assumption2.1
is relevant for existence and uniqueness of the solution of (2.1). The
differential system (2.1) has a unique state-trajectory solution for t
∈
R
+ for any given piecewise absolutely continuous
function φ
:
[
−
h
,
0
]
→
R
n of initial conditions. This follows from
Picard-Lindelöff existence and uniqueness theorem. Assumption 2.2
establishes that ∑
j
=
0
q
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
system (2.1) for a set of prescribed maximum delays in the time-invariant case.
It is well known that even if A
j
≡
0 for all j
∈
q
¯,
then the resulting linear time-varying delay-free system cannot be proved to be
stable without some additional assumptions, like for instance, Assumptions 2.3.
The latest assumption is related to the smallness of the time-derivative of the
delay-free system matrix everywhere it exists or generating sufficiently small
bounded discontinuities in ∑
j
=
0
q
A
j
(
t
) at times, where it is impulsive. An
alternative  assumption to Assumptions 2.3 which
avoids the assumption of almost everywhere existence of a bounded A
˙
j
(
t
),
for all j
∈
q
¯
∪
{
0
} (see second part of Assumption 2.1) might be
stated in terms of  sufficiently smallness
of ∥
Δ
A
j
(
t
)
∥ for Δ
A
j
(
t
)
:=
A
j
(
t
)
−
A
j
* for all t
∈
R
0
+ for
all j
∈
q
¯
∪
{
0
} for some constant stability matrix ∑
j
=
0
q
A
j
* whose eigenvalues satisfy Re
λ
i
(
∑
j
=
0
q
A
j
*
)
≤
−
ρ
0
<
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 (
∑
j
=
0
q
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 delays (see Theorem 2.12(i)).


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

Assumption2.1
admits bounded discontinuities in the entries of A
j
(
t
) for j
∈
q
¯
∪
{
0
}.
At such times A
j
(
t
+
) denote right values of the matrix function
while A
j
(
t
−
) is simply denoted by A
j
(
t
).
A set of p resetting systems of (2.1)
is defined by the linear time-invariant
systems: (2.4)
z
˙
j
(
t
)
=
(
∑
i
=
0
q
A
i
j
)
z
j
(
t
−
h
j
) for
some given A
j
i
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 2.2 extends in a natural fashion to include
the resetting systems as follows.Assumption 2.4.
All the eigenvaluesλ
k
(
∑
i
=
0
q
A
i
j
) satisfy Re
λ
k
(
∑
i
=
0
q
A
i
j
)
≤
−
ρ
0
<
0 for all t
∈
R
0
+ for all k
∈
σ
¯
j
0
(
σ
j
0
≤
n
) for all j
∈
p
¯;
that is, ∑
i
=
0
q
A
i
j 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
n
×
(
q
+
1
)
n from the nonnegative real axis to the set of
real n
×
(
q
+
1
)
n matrices.The
trivial switching matrix function is that being identically zero so that no
switch occurs. If some switch occurs then the switching matrix function is
nonzero. The switching matrix function is colloquially referred to in the
following asthe 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 lawσ is denoted by ST
(
σ
).
Two kinds of switching instants, respectively, reset instants and reset-free
switching instants defined in Definitions 2.7 and 2.8 are considered.Definition 2.7  (reset instant).
t
∈
R
0
+ is a reset instant generated by the switching
law σ if t
∈
ST
(
σ
) and A
i
(
t
+
)
=
A
i
j for some i
∈
q
¯
∪
{
0
} and some j
∈
p
¯,
provided that A
i
(
t
)
=
A
i
k
≠
A
i
j for some k
(
≠
j
)
∈
p
¯.The
set of reset instants generated by the switching lawσ is denoted by ST
r
(
σ
).
Note that ST
r
(
σ
)
⊂
ST
(
σ
) from Definitions 2.6 and 2.7. Note also that the whole
system parameterization is driven to some of the prefixed resetting systems (2.4)
when a reset instant happens. Note that at reset instants, A
˙
i
(
t
)
=
(
A
i
j
−
A
i
(
t
)
)
δ
(
0
) for some 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
∈
ST
(
σ
) and A
i
j
≠
A
i
(
t
+
)
≠
A
i
(
t
) for some i
∈
q
¯
∪
{
0
},
for all j
∈
p
¯.The
set of reset-free instants is denoted byt
∈
ST
rf
(
σ
).
Note that at reset-free switching instants some of the switched system
parameters suffer an undriven bounded discontinuity. If all the parameters jump
to a parameterization (2.4)
at the same time, then the corresponding instant is
considered a reset time instant. Note that ST
rf
(
σ
)
⊂
ST
(
σ
), ST
(
σ
)
=
ST
r
(
σ
)
∪
ST
rf
(
σ
),
and ST
r
(
σ
)
∩
ST
rf
(
σ
)
=
∅ from  Definitions 2.6–2.8, that is, the whole set of
switching instants is the disjoint union of the sets of reset and reset-free
switching instants.Definition 2.9.
The
partial switching sequenceST
(
σ
,
t
),
the partial switching sequence ST
r
(
σ
,
t
),
and the reset-free partial switching sequence ST
rf
(
σ
,
t
),
generated by the switching law σ
:
R
0
+
→
{
(
A
j
(
t
)
−
A
j
(
t
−
)
)
,
∀
j
∈
q
¯
∪
{
0
}
}
⊂
R
n
×
(
q
+
1
)
n up till any time t
∈
R
0
+,
are defined, respectively, by ST
(
σ
,
t
)
:=
{
t
i
∈
ST
(
σ
)
:
t
i
<
t
}, ST
r
(
σ
,
t
)
:=
{
t
i
∈
ST
r
(
σ
)
:
t
i
<
t
},
and ST
rf
(
σ
,
t
)
:=
{
t
i
∈
ST
rf
(
σ
)
:
t
i
<
t
}.Remark 2.10.
An
interpretation of Assumptions2.3 is that the following conditions hold for any
given ℓ-matrix norm for some nonnegative norm
dependent real constants μ
j, α
j, μ,
and α for all j
∈
q
¯
∪
{
0
}: (2.5)
ess
sup
t
∈
R
0
+
∥
A
˙
j
(
t
)
∥
≤
μ
j
≤
max
(
ess
sup
t
∈
R
0
+
∥
A
˙
j
(
t
)
∥
:
j
∈
q
¯
∪
{
0
}
)
≤
μ
,
∥
∑
τ
∈
ST
(
σ
)
∩
[
t
,
t
+
T
)
A
˙
j
(
τ
)
∥
≤
∑
τ
∈
ST
(
σ
)
∩
[
t
,
t
+
T
)
∥
X
(
τ
)
δ
(
0
)
∥
≤
α
j
≤
max
(
∥
∑
τ
∈
ST
(
σ
)
∩
[
t
,
t
+
T
)
A
˙
j
(
τ
)
∥
:
j
∈
q
¯
∪
{
0
}
)
≤
α
,   where  δ
(
0
) is a Dirac impulse at  t
=
0. Note that Assumptions  2.3
imply |
∥
∑
j
=
0
q
A
j
(
t
+
τ
)
∥
−
∥
∑
j
=
0
q
A
j
(
t
)
∥
|
≤
μ
T
+
ν
(
t
+
τ
) for all τ
∈
[
0
,
T
],
with the function ν
:
R
0
+
→
R
0
+ satisfying ν
(
t
+
τ
)
≤
α
0
j
+
α
1
j
(
t
+
τ
)
≤
α with α
1
j
(
t
+
τ
)
=
0 if A
˙
j
(
t
+
τ
) exists with ∥
A
˙
j
(
t
+
τ
)
∥
≤
α
j or α
j
−
α
0
j
≥
α
1
j
(
t
+
τ
)
≥
∥
Δ
j
(
t
+
τ
)
∥ if A
˙
j
(
t
+
τ
)
=
Δ
j
(
t
+
τ
)
δ
(
0
), that
is, at least one of its entries is impulsive.Note that Assumption2.1 implies that
switching does not happen arbitrarily fast neither to reset parameters nor to
reset-free ones . The subsequent result is direct.Assertions 1.
The
following properties are true irrespective of the switching function:
(i)t
∉
ST
(
σ
)
⇔
σ
(
t
+
)
=
σ
(
t
)
=
0
n
×
(
q
+
1
)
n (i.e., a zero n
×
(
q
+
1
)
n-matrix);
(ii)t
∈
ST
(
σ
)
⇔
σ
(
t
+
)
−
σ
(
t
)
≠
0
n
×
(
q
+
1
)
n;
(iii)t
∉
ST
(
σ
)
⇔
ST
(
σ
,
t
+
)
=
ST
(
σ
,
t
);
(iv)t
∈
ST
(
σ
)
⇔
ST
(
σ
,
t
+
)
≠
ST
(
σ
,
t
).Proof.
(i)σ
(
t
+
)
=
σ
(
t
)
=
0
n
×
(
q
+
1
)
n
⇒
t
∉
ST
r
(
σ
)
⋀
t
∉
ST
rf
(
σ
)
⇔
t
∉
ST
(
σ
)
,
t
∉
ST
⇔
σ
(
t
)
=
0 (since switching is not arbitrarily
fast), (2.6)
⋀
(
σ
(
t
+
)
−
σ
(
t
)
=
0
n
×
(
q
+
1
)
n
)
⟺
σ
(
t
+
)
=
σ
(
t
)
=
0
n
×
(
q
+
1
)
n
. 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(2.7)
ST
(
σ
,
t
)
:=
{
t
i
∈
ST
(
σ
)
:
t
i
<
t
}
⟺
ST
(
σ
,
t
+
)
:=
{
t
i
∈
ST
(
σ
)
:
t
i
<
t
+
}
=
{
t
i
∈
ST
(
σ
)
:
t
i
≤
t
}
⟺
ST
(
σ
,
t
+
)
{
=
ST
(
σ
,
t
)
if
t
∉
ST
(
σ
)
,
≠
ST
(
σ
,
t
)
if
t
∈
ST
(
σ
)   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: (2.8)
V
(
t
,
x
t
)
=
x
T
(
t
)
P
(
t
)
x
(
t
)
+
∑
i
=
1
q
∑
j
=
0
q
∫
−
h
i
−
h
j
−
h
j
∫
t
+
θ
t
x
T
(
τ
)
S
i
j
x
(
τ
)
d
τ
d
θ
.Theorem 2.12.
The
following properties hold.
(i)
Assume the
following.(i.a)
The matrix functionsA
j
(
t
)
,
for all j
∈
q
¯
∪
{
0
} are subject to Assumption 2.1.
(i.b)
The switching lawσ is such  that (2.9)
Q
(
t
)
:=
[
H
1
h
^
1
P
(
t
)
A
1
(
t
)
M
(
t
)
⋯
h
^
q
P
(
t
)
A
q
(
t
)
M
(
t
)
h
^
1
M
T
(
t
)
A
1
T
(
t
)
P
(
t
)
−
R
1
0
⋯
0
⋮
0
⋮
⋱
⋮
h
^
q
M
T
(
t
)
A
q
T
(
t
)
P
(
t
)
0
⋯
−
R
q
]
<
0
,
∀
t
∈
R
0
+
,where H
1
denotes
(
∑
j
=
0
q
A
j
T
(
t
)
)
P
(
t
)
+
P
(
t
)
(
∑
j
=
0
q
A
j
(
t
)
)
+
P
˙
(
t
)
+
∑
i
=
1
q
∑
j
=
0
q
h
i
^
S
i
j,
for
some time-differentiable real symmetric positive definite matrix function P
:
R
0
+
→
R
n
×
n  and some real symmetric positive definite matrices S
i
j
∈
R
n
×
n
(
∀
i
∈
q
¯
,
∀
j
∈
q
¯
∪
{
0
}
),
where (2.10)
M
(
t
)
:=
[
A
0
(
t
)
,
…
,
A
q
(
t
)
]
∈
R
n
×
(
q
+
1
)
n
,
R
i
:=
diag
(
S
i
0
,
S
i
1
,
…
,
S
i
q
)
∈
R
(
q
+
1
)
n
×
(
q
+
1
)
n
;
∀
i
∈
q
,
∀
t
∈
R
0
+
.Thus,
the system (2.1) is globally asymptotically Lyapunov's stable for all delays h
i
∈
[
0
,
h
^
i
], for
all
i
∈
q
¯.
A necessary condition is (
∑
j
=
0
q
A
j
T
(
t
)
)
P
(
t
)
+
P
(
t
)
(
∑
j
=
0
q
A
j
(
t
)
)
+
P
˙
(
t
)
<
0, for
all
t
∈
R
0
+  what implies that (
∑
j
=
0
q
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
i
j,  for
all
i
∈
q
¯
∪
{
0
}, for
all
t
∈
R
0
+  for some  j
∈
p
¯ (eventually being dependent on t) satisfying
Assumption   2.4.
(ii.b)
The switching lawσ 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)
(2.11)
Q
i
*
:=
[
(
∑
j
=
0
q
A
j
i
T
)
P
*
+
P
*
(
∑
j
=
0
q
A
j
i
)
+
∑
i
=
1
q
∑
j
=
0
q
h
^
i
S
i
j
*
h
^
1
P
*
A
1
i
M
1
*
⋯
h
^
q
P
*
A
q
i
M
q
*
h
^
1
M
1
*
T
A
1
i
T
P
*
−
R
1
*
0
⋯
0
⋮
0
⋮
⋱
⋮
h
^
q
M
q
*
T
A
q
i
T
P
*
0
⋯
−
R
q
*
]
<
0
;
∀
i
∈
p
¯for
some R
n
×
n
∋
P
*
=
P
*
T
>
0, R
n
×
n
∋
S
i
j
*
=
S
i
j
*
T
>
0
(
∀
i
∈
q
¯
,
∀
j
∈
q
¯
∪
{
0
}
),
where (2.12)
M
i
:=
[
A
0
i
,
…
,
A
q
i
]
∈
R
n
×
(
q
+
1
)
n
,
R
i
*
:=
diag
(
S
i
0
*
,
S
i
1
*
,
…
,
)
∈
R
(
q
+
1
)
n
×
(
q
+
1
)
n
;
∀
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 for all delays h
i
∈
[
0
,
h
^
i
], for all
i
∈
q
¯.
If (2.9) is replaced with Q
i
*
≤
−
2
ε
I
(
q
+
1
)
n
<
0, for all
i
∈
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 any h
i
∈
[
0
,
h
^
i
], for
all
i
∈
q
¯ provided that all the delay-free resetting systems
(2.4)z
˙
j
(
t
)
=
(
∑
i
=
0
q
A
i
j
)
z
j
(
t
) fulfil Assumption 2.4, that is, they are
globally exponentially stable.It
is of interest to discuss particular cases easy to test, guaranteeing Theorem2.12 (i).


### 2.3. Sufficiency Type Asymptotic Stability Conditions Obtained for Constant Symmetric MatricesP and S
i
j

Assume real constant symmetric matricesP
(
t
)
=
P and S
i
j, for
all
i
∈
q
¯
,
for
all
j
∈
q
¯
∪
{
0
}, for
all
t
∈
R
0
+ so that (2.13)
Q
(
t
)
=
[
H
2
h
^
1
P
A
1
(
t
)
M
(
t
)
⋯
h
^
q
P
A
q
(
t
)
M
(
t
)
h
^
1
M
T
(
t
)
A
1
T
(
t
)
P
−
R
1
0
⋯
0
⋮
0
⋮
⋱
⋮
h
^
q
M
T
(
t
)
A
q
T
(
t
)
P
0
⋯
−
R
q
]
<
0, where H
2
denotes
(
∑
j
=
0
q
A
j
T
(
t
)
)
P
+
P
(
∑
j
=
0
q
A
j
(
t
)
)
+
∑
i
=
1
q
∑
j
=
0
q
h
^
i
S
i
j. In
this case, the  Krasovsky-Lyapunov
functional used in the proof of Theorem 2.12(i) holds defined with constant
matrices for all time irrespective of being a switching-free instant or any
switching instant (independently of its nature: reset time or reset-free
switching instant). A practical test for (2.13) to hold follows. Consider A
i
*
(
i
∈
q
¯
∪
{
0
}
)  such that the time invariant system (2.1)
defined with A
i
(
t
)
→
A
i
* is globally asymptotically Lyapunov's stable
and define  a stability real  n-matrix A
*
:=
∑
i
=
0
q
A
i
*.
Decompose Q
(
t
)
=
Q
*
+
Q
˜
(
t
),
where (2.14)
Q
*
:=
[
A
*
T
P
+
P
A
*
+
∑
i
=
1
q
∑
j
=
0
q
h
^
i
S
i
j
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
*
]
,
Q
˜
(
t
)
:=
[
A
˜
T
(
t
)
P
+
P
A
˜
(
t
)
h
^
1
P
Δ
˜
1
(
t
)
⋯
h
^
q
P
Δ
˜
q
(
t
)
h
^
1
Δ
˜
1
T
(
t
)
P
0
0
⋯
0
⋮
0
⋮
⋱
⋮
h
^
q
Δ
˜
q
T
(
t
)
P
0
⋯
0
]
,where (2.15)
M
*
:=
diag
(
A
0
*
,
A
1
*
,
…
,
A
q
*
)
,
R
i
*
:=
diag
(
S
i
0
*
,
S
i
1
*
,
…
,
S
i
q
*
)
,
M
˜
(
t
)
:=
M
(
t
)
−
M
*
,
A
˜
i
(
t
)
:=
A
i
(
t
)
−
A
*
,
A
˜
(
t
)
:=
(
∑
j
=
0
q
A
j
i
(
t
)
)
−
A
*
,
Δ
˜
i
(
t
)
:=
(
A
˜
i
(
t
)
M
*
+
A
i
*
M
˜
(
t
)
+
A
˜
i
(
t
)
M
↔
(
t
)
)
. If t
∈
ST
(
σ
),
then (2.16)
δ
Q
˜
(
t
)
:=
Q
˜
(
t
+
)
−
Q
˜
(
t
)
=
[
H
3
h
^
1
P
(
Δ
˜
1
(
t
+
)
−
Δ
˜
1
(
t
)
)
⋯
h
^
q
P
(
Δ
˜
1
(
t
+
)
−
Δ
˜
1
(
t
)
)
h
^
1
(
Δ
˜
1
(
t
+
)
−
Δ
˜
1
(
t
)
)
T
P
0
0
⋯
0
⋮
0
⋮
⋱
⋮
h
^
q
(
Δ
˜
1
(
t
+
)
−
Δ
˜
1
(
t
)
)
T
P
0
⋯
0
]
,
∥
δ
Q
˜
(
t
)
∥
2
≤
2
λ
max
(
P
)
[
∥
A
˜
T
(
t
+
)
−
A
˜
T
(
t
)
∥
2
+
∑
i
=
1
q
h
^
i
∥
Δ
˜
i
(
t
+
)
−
Δ
˜
i
(
t
)
∥
2
]
=
2
λ
max
(
P
)
[
∥
∑
i
=
0
q
(
A
i
T
(
t
+
)
−
A
i
T
(
t
)
)
∥
2
+
∑
i
=
1
q
h
^
i
∥
Δ
˜
i
(
t
+
)
−
Δ
˜
i
(
t
)
∥
2
]
,    where H
3
denotes
(
A
˜
T
(
t
+
)
−
A
˜
T
(
t
)
)
P
+
P
(
(
A
˜
(
t
+
)
−
A
˜
(
t
)
)
)
, what
leads to (2.17)
∥
δ
Q
˜
(
t
)
∥
2
≤
2
λ
max
(
P
)
[
(
q
+
1
)
a
+
+
h
^
q
(
a
+
(
∑
j
=
0
q
∥
A
j
*
∥
2
+
(
q
+
1
)
∥
A
i
*
∥
2
+
(
q
+
1
)
(
a
+
a
+
)
)
+
(
q
+
1
)
(
2
a
+
a
+
)
a
)
]
, where a
:=
max
i
∈
q
¯
∪
{
0
}
(
∥
A
i
∥
2
) and a
+
:=
sup
t
∈
ST
(
σ
)
max
i
∈
q
¯
∪
{
0
}
(
∥
A
i
(
t
+
)
−
A
i
∥
2
) (see A.9 in  Appendix ). Direct  results from Theorem 2.12 which follow from (2.13)
to (2.17) are given below.Corollary 2.13.
Consider
in (2.9) replacements with constant real matrices Q
(
t
)
→
Q
*
=
Q
*
T, P
(
t
)
→
P
=
P
T
>
0, A
i
(
t
)
→
A
i
*, A
(
t
)
→
A
*
:=
∑
i
=
0
q
A
i
*;   for
all
i
∈
q
¯
,
for
all
j
∈
q
¯
∪
{
0
}, for
all
t
∈
R
0
+ such that A
* is a stability matrix. Then, Theorem  2.12(i) holds if Q
*
<
0 for any switching law σ such that
(1)
λ
min
(
−
Q
*
)
=
−
λ
max
(
Q
*
)
>
∥
Q
˜
(
t
)
∥
2
:=
λ
max
1
/
2
(
Q
˜
T
(
t
)
Q
˜
(
t
)
), for
all
t
∈
R
0
+
∖
ST
(
σ
),
(2)
a
:=
max
i
∈
q
¯
∪
{
0
}
(
∥
A
i
∥
2
) and a
+
:=
sup
t
∈
ST
(
σ
)
max
i
∈
q
¯
∪
{
0
}
(
∥
A
i
(
t
+
)
−
A
i
∥
2
) are sufficiently small such that (2.18)
λ
min
(
−
Q
*
)
−
∥
Q
˜
(
t
)
∥
2
>
2
λ
max
(
P
)
[
(
q
+
1
)
a
+
+
h
^
q
(
a
+
(
∑
j
=
0
q
∥
A
j
*
∥
2
+
(
q
+
1
)
max
i
∈
q
¯
∪
{
0
}
∥
A
i
*
∥
2
+
(
q
+
1
)
(
a
+
a
+
)
)
+
(
q
+
1
)
(
2
a
+
a
+
)
a
)
]
,
∀
t
∈
ST
(
σ
)
.Corollary 2.14.
Consider
in (2.9) replacements with constant real matrices Q
(
t
)
→
Q
j
*
=
Q
j
*
T, P
(
t
)
→
P
=
P
T
>
0, A
i
j
(
t
)
→
A
i
j
*, A
(
t
)
→
A
j
*
:=
∑
i
=
0
q
A
i
j
* for all j
∈
p
¯
,
for
all
k
∈
q
¯
∪
{
0
},  for
all
t
∈
R
0
+ such that each A
j
*
(
∀
j
∈
p
¯
)      is a stability matrix with A
i
j
*
;
for
all
i
∈
q
¯
,
for
all
j
∈
p
¯  being the parameterizations defining the
resetting systems  (2.4). Assume that the system  (2.1) is one of the resetting
systems  (2.4) at t
=
0. Then, Theorem  2.12(i) holds with a common Krasovsky-Lyapunov function for all those resetting systems
if Q
j
*
<
0
(
∀
j
∈
p
¯
)        for any switching law σ such that
(1)
λ
min
(
−
Q
*
)
=
−
λ
max
(
Q
*
)
>
∥
Q
˜
(
t
)
∥
2
:=
λ
max
1
/
2
(
Q
˜
T
(
t
)
Q
˜
(
t
)
)
,
for
all
t
∈
R
0
+
∖
ST
(
σ
),
(2)
a
j
:=
max
i
∈
q
¯
∪
{
0
}
(
∥
A
j
i
∥
2
) and a
j
+
:=
sup
t
∈
ST
(
σ
)
max
i
∈
q
¯
∪
{
0
}
(
∥
A
i
(
t
+
)
−
A
j
i
∥
2
) are sufficiently small such that (2.19)
λ
min
(
−
Q
*
)
−
∥
Q
˜
(
t
)
∥
2
>
2
λ
max
(
P
)
[
(
q
+
1
)
a
j
+
+
h
^
q
(
a
j
+
(
∑
i
=
0
q
∥
A
j
i
∥
2
+
(
q
+
1
)
max
i
∈
q
¯
∪
{
0
}
∥
A
j
i
∥
2
)
+
(
q
+
1
)
(
a
j
+
a
j
+
)
+
(
q
+
1
)
(
2
a
j
+
a
j
+
)
a
j
+
)
]
t
∈
ST
fr
(
σ
),
provided that at time max
(
t
′
<
t
:
t
′
∈
ST
r
(
σ
)
),
the system (2.1)
coincides with at the j
∈
p
¯
r  setting system (2.4).The
proof of Corollary2.14 is close to that of Corollary 2.13 from A.9 in Appendix
with the replacements a
→
a
j
,
a
+
→
a
j
+ for all j
∈
p
¯.
If (2.19) is rewritten with the replacements a
j
→
a
¯
:=
max
j
∈
p
¯
a
j
,
a
j
+
→
a
→
+
:=
max
j
∈
p
¯
a
j
+  then the reformulated weaker Corollary 2.14 is valid for all t
∈
ST
fr
(
σ
) irrespective of the preceding reset switching.
A  result which guarantees Corollary 2.13,
and then Theorem 2.12(i),  is now  obtained by replacing the  (1,1) block matrix of Q
* by a Lyapunov matrix equality as follows.
Consider a real n-matrix Q
0
*
=
Q
0
*
T
>
0 such that λ
min
(
Q
0
*
)
>
λ
max
(
∑
i
=
1
q
∑
j
=
0
q
h
^
i
S
i
j
) and P
:=
∫
0
∞
e
A
*
T
τ
Q
0
*
e
A
*
τ
d
τ satisfying the Lyapunov equation A
*
T
P
+
P
A
*
=
−
Q
0
*
<
0 as its unique solution.  Note that λ
max
(
P
)
≤
K
*
λ
max
(
Q
0
*
)
/
2
ρ
* for some K
*
∈
R
+,
where −
ρ
*
<
0 is the stability abscissa of A
* with ∥
e
A
*
T
t
∥
≤
K
*
e
−
ρ
*
t for all t
∈
R.
Define the decomposition Q
(
t
)
=
Q
¯
*
+
Q
¯
˜
(
t
),
where (2.20)
Q
¯
*
:=
Block
Diag
(
−
Q
0
*
+
∑
i
=
1
q
∑
j
=
0
q
h
^
i
S
i
j
,
−
R
1
*
,
…
,
−
R
q
*
)
,
Q
¯
˜
(
t
)
:=
[
A
˜
T
(
t
)
P
+
P
A
˜
(
t
)
h
^
1
P
(
A
1
*
M
*
+
Δ
˜
1
(
t
)
)
⋯
h
^
q
P
(
A
q
*
M
*
+
Δ
˜
q
(
t
)
)
h
^
1
(
A
1
*
M
*
+
Δ
˜
1
(
t
)
)
T
P
0
0
⋯
0
⋮
0
⋮
⋱
⋮
h
^
q
(
A
q
*
M
*
+
Δ
˜
q
(
t
)
)
T
P
0
⋯
0
]
⇒
∥
Q
¯
˜
(
t
)
∥
2
≤
(
K
*
λ
max
(
Q
0
*
)
ρ
*
)
(
∥
A
˜
T
(
t
)
∥
2
+
∑
i
=
1
q
h
^
i
∥
A
i
*
M
*
+
Δ
˜
i
(
t
)
∥
2
)
.                                       Thus,
the subsequent result follows from Corollary 2.13 and (2.20).Corollary 2.15.
Consider
the matrices of Corollary2.13 with A
* being a stability matrix with stability abscissa (
−
ρ
*
)
<
0 which satisfies the Lyapunov equation A
*
T
P
+
P
A
*
=
−
Q
0
*
<
0.
Then, Theorem  2.12(i) holds if Q
*
<
0 for any switching law σ such that (2.21)
λ
min
(
−
Q
¯
*
)
=
−
λ
max
(
Q
¯
*
)
>
(
K
*
λ
max
(
Q
0
*
)
ρ
*
)
(
∥
A
˜
T
(
t
)
∥
2
+
∑
i
=
1
q
h
^
i
∥
A
i
*
M
*
+
Δ
˜
i
(
t
)
∥
2
)
,
∀
t
∈
R
0
+
,
∀
t
∈
ST
(
σ
)
.


### 2.4. Sufficiency Type Asymptotic Stability Conditions Obtained for Time-Varying Symmetric MatricesP
(
t
), S
i
j
(
t
)
=
S
i
j

The
following result, which is proven in Appendix, holds.Theorem 2.16.
Under Assumptions2.1–2.3, the following properties
hold.
(i)
The switched system (2.1) is globally asymptotically Lyapunov's for any delays h
i
∈
[
0
,
h
^
i
] for all  i
∈
q
¯   for
some h
^
:=
max
i
∈
q
¯
(
h
^
i
) and any switching law  σ such that
(a)
the switching instants are arbitrary;
(b)
max
(
ess
sup
∥
A
˙
j
(
t
)
∥
:
t
∈
R
0
+
,
j
∈
p
¯
)  is sufficiently small compared to the absolute
value of the prescribed stability abscissa of  ∑
j
=
0
q
A
j
(
t
);
(c)
the support testing matrix of
distributional derivativesΓ
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
(
t
)
(
t
)
δ
(
0
)).
(ii)
The switched system (2.1) is globally exponentially stable for any delays h
i
∈
[
0
,
h
^
i
] for all i
∈
q
¯ for some h
^
:=
max
i
∈
q
¯
(
h
^
i
) such that
(a)
max
(
ess
sup
∥
A
˙
j
(
t
)
∥
:
t
∈
R
0
+
,
j
∈
p
¯
) is sufficiently small compared to the absolute value of the prescribed stability abscissa of ∑
j
=
0
q
A
j
(
t
);
(b)
max
(
∥
Γ
A
d
j
(
t
)
(
t
)
∥
:
A
˙
j
(
t
)
=
Γ
A
d
j
(
t
)
(
t
)
δ
(
0
)
,
∀
t
∈
ST
(
σ
)
,
j
∈
p
¯
)  is sufficiently small compared to the timeintervals in between any two consecutive switching instants.
Furthermore, if Assumptions 2.1–2.4 hold, then
(iii)
the switched system (2.1) is globally
exponentially stable for any delays h
i
∈
[
0
,
h
^
i
] for all i
∈
q
¯, for some h
^
:=
max
i
∈
q
¯
(
h
^
i
) such that(a)
max
(
ess
sup
∥
A
˙
j
(
t
)
∥
:
t
∈
R
0
+
,
j
∈
p
¯
) is sufficiently small compared to the absolute
value of the prescribed stability abscissa of ∑
j
=
0
q
A
j
(
t
);
(b)
the switching lawσ  is such that(a)
max
(
∥
Γ
A
d
j
(
t
)
∥
:
A
˙
j
(
t
)
=
Γ
A
d
j
(
t
)
δ
(
0
)
,
∀
t
∈
ST
rf
(
σ
)
,
j
∈
p
¯
)  is sufficiently small compared to the lengths
of  time intervals between any two
consecutive switching instants;
(b)
it exists a common Krasovsky-Lyapunov
functionalV
(
t
,
x
t
) defined with constant matrices P
¯
=
P
¯
T
>
0 and S
¯
i
j
=
S
¯
i
j
T
>
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
all
t
∈
ST
r
(
σ
) under the resetting action P
(
t
+
)
=
P
¯; for
all
(
i
,
j
)
∈
(
q
¯
∪
{
0
}
)
×
q
¯:(b.1)
V
(
t
+
,
x
t
)
≤
V
(
t
,
x
t
)  which is guaranteed, in particular, if P
(
t
+
)
=
P
¯
≤
P
(
t
),
(b.2)
the tradeoff (a) is respected between sufficiently small norms of the matrices
of distributional derivatives and the length|
t
−
t
′
|,
at any t
∈
ST
r
(
σ
),  if any,  where the condition (b.1)  is not satisfied, where t
′
=
max
(
τ
∈
R
0
+
:
ST
(
σ
)
∋
τ
<
t
).The
characterization of the “sufficient smallness” of the involved magnitudes in
Theorem2.16 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.


## 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 functionsA
j
:
R
0
+
→
R
n
×
n are piecewise continuous and uniformly bounded
for all j
∈
q
¯
∪
{
0
}.Assumption 2.2.
All
the eigenvaluesλ
i
(
∑
j
=
0
q
A
j
(
t
)
) of the matrix function ∑
j
=
0
q
A
j
(
t
)  satisfy Re
λ
i
(
∑
j
=
0
q
A
j
(
t
)
)
≤
−
ρ
0
<
0 for all t
∈
R
0
+, for all
i
∈
σ
¯
(
σ
≤
n
) for some ρ
0
∈
R
+
:=
{
0
<
z
∈
R
}, that is, ∑
j
=
0
q
A
j
(
t
)  is a stability matrix  for all t
∈
R
0
+.Assumption 2.3.
The matrix functionsA
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
+
(
∥
∑
j
=
0
q
A
˙
j
(
t
)
∥
)
≤
γ
ℓ
<
∞ with γ
ℓ being a ℓ-norm dependent nonnegative real constant  and, furthermore, (2.2)
∫
t
t
+
T
∥
A
˙
j
(
τ
)
∥
d
τ
≤
μ
j
T
+
α
j
≤
μ
T
+
α
∀
j
∈
q
¯
∪
{
0
}  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 instantst, where the time-derivative of some entry of A
j
(
t
) does
not exist for any j
∈
q
¯
∪
{
0
},
the time derivative is defined in a distributional Dirac sense as A
˙
j
(
t
)
=
Γ
(
t
)
δ
(
0
) what
equivalently means the presence of a discontinuity at t in A
j
(
t
) defined as (2.3)
A
j
(
t
+
)
=
A
j
(
t
−
)
+
lim
ε
→
0
+
∫
−
ε
ε
Γ
j
(
τ
)
δ
(
t
−
τ
)
d
τ
=
A
j
(
t
)
+
Γ
j
(
t
)
.Assumption2.1
is relevant for existence and uniqueness of the solution of (2.1). The
differential system (2.1) has a unique state-trajectory solution for t
∈
R
+ for any given piecewise absolutely continuous
function φ
:
[
−
h
,
0
]
→
R
n of initial conditions. This follows from
Picard-Lindelöff existence and uniqueness theorem. Assumption 2.2
establishes that ∑
j
=
0
q
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
system (2.1) for a set of prescribed maximum delays in the time-invariant case.
It is well known that even if A
j
≡
0 for all j
∈
q
¯,
then the resulting linear time-varying delay-free system cannot be proved to be
stable without some additional assumptions, like for instance, Assumptions 2.3.
The latest assumption is related to the smallness of the time-derivative of the
delay-free system matrix everywhere it exists or generating sufficiently small
bounded discontinuities in ∑
j
=
0
q
A
j
(
t
) at times, where it is impulsive. An
alternative  assumption to Assumptions 2.3 which
avoids the assumption of almost everywhere existence of a bounded A
˙
j
(
t
),
for all j
∈
q
¯
∪
{
0
} (see second part of Assumption 2.1) might be
stated in terms of  sufficiently smallness
of ∥
Δ
A
j
(
t
)
∥ for Δ
A
j
(
t
)
:=
A
j
(
t
)
−
A
j
* for all t
∈
R
0
+ for
all j
∈
q
¯
∪
{
0
} for some constant stability matrix ∑
j
=
0
q
A
j
* whose eigenvalues satisfy Re
λ
i
(
∑
j
=
0
q
A
j
*
)
≤
−
ρ
0
<
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 (
∑
j
=
0
q
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 delays (see Theorem 2.12(i)).


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

Assumption2.1
admits bounded discontinuities in the entries of A
j
(
t
) for j
∈
q
¯
∪
{
0
}.
At such times A
j
(
t
+
) denote right values of the matrix function
while A
j
(
t
−
) is simply denoted by A
j
(
t
).
A set of p resetting systems of (2.1)
is defined by the linear time-invariant
systems: (2.4)
z
˙
j
(
t
)
=
(
∑
i
=
0
q
A
i
j
)
z
j
(
t
−
h
j
) for
some given A
j
i
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 2.2 extends in a natural fashion to include
the resetting systems as follows.Assumption 2.4.
All the eigenvaluesλ
k
(
∑
i
=
0
q
A
i
j
) satisfy Re
λ
k
(
∑
i
=
0
q
A
i
j
)
≤
−
ρ
0
<
0 for all t
∈
R
0
+ for all k
∈
σ
¯
j
0
(
σ
j
0
≤
n
) for all j
∈
p
¯;
that is, ∑
i
=
0
q
A
i
j 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
n
×
(
q
+
1
)
n from the nonnegative real axis to the set of
real n
×
(
q
+
1
)
n matrices.The
trivial switching matrix function is that being identically zero so that no
switch occurs. If some switch occurs then the switching matrix function is
nonzero. The switching matrix function is colloquially referred to in the
following asthe 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 lawσ is denoted by ST
(
σ
).
Two kinds of switching instants, respectively, reset instants and reset-free
switching instants defined in Definitions 2.7 and 2.8 are considered.Definition 2.7  (reset instant).
t
∈
R
0
+ is a reset instant generated by the switching
law σ if t
∈
ST
(
σ
) and A
i
(
t
+
)
=
A
i
j for some i
∈
q
¯
∪
{
0
} and some j
∈
p
¯,
provided that A
i
(
t
)
=
A
i
k
≠
A
i
j for some k
(
≠
j
)
∈
p
¯.The
set of reset instants generated by the switching lawσ is denoted by ST
r
(
σ
).
Note that ST
r
(
σ
)
⊂
ST
(
σ
) from Definitions 2.6 and 2.7. Note also that the whole
system parameterization is driven to some of the prefixed resetting systems (2.4)
when a reset instant happens. Note that at reset instants, A
˙
i
(
t
)
=
(
A
i
j
−
A
i
(
t
)
)
δ
(
0
) for some 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
∈
ST
(
σ
) and A
i
j
≠
A
i
(
t
+
)
≠
A
i
(
t
) for some i
∈
q
¯
∪
{
0
},
for all j
∈
p
¯.The
set of reset-free instants is denoted byt
∈
ST
rf
(
σ
).
Note that at reset-free switching instants some of the switched system
parameters suffer an undriven bounded discontinuity. If all the parameters jump
to a parameterization (2.4)
at the same time, then the corresponding instant is
considered a reset time instant. Note that ST
rf
(
σ
)
⊂
ST
(
σ
), ST
(
σ
)
=
ST
r
(
σ
)
∪
ST
rf
(
σ
),
and ST
r
(
σ
)
∩
ST
rf
(
σ
)
=
∅ from  Definitions 2.6–2.8, that is, the whole set of
switching instants is the disjoint union of the sets of reset and reset-free
switching instants.Definition 2.9.
The
partial switching sequenceST
(
σ
,
t
),
the partial switching sequence ST
r
(
σ
,
t
),
and the reset-free partial switching sequence ST
rf
(
σ
,
t
),
generated by the switching law σ
:
R
0
+
→
{
(
A
j
(
t
)
−
A
j
(
t
−
)
)
,
∀
j
∈
q
¯
∪
{
0
}
}
⊂
R
n
×
(
q
+
1
)
n up till any time t
∈
R
0
+,
are defined, respectively, by ST
(
σ
,
t
)
:=
{
t
i
∈
ST
(
σ
)
:
t
i
<
t
}, ST
r
(
σ
,
t
)
:=
{
t
i
∈
ST
r
(
σ
)
:
t
i
<
t
},
and ST
rf
(
σ
,
t
)
:=
{
t
i
∈
ST
rf
(
σ
)
:
t
i
<
t
}.Remark 2.10.
An
interpretation of Assumptions2.3 is that the following conditions hold for any
given ℓ-matrix norm for some nonnegative norm
dependent real constants μ
j, α
j, μ,
and α for all j
∈
q
¯
∪
{
0
}: (2.5)
ess
sup
t
∈
R
0
+
∥
A
˙
j
(
t
)
∥
≤
μ
j
≤
max
(
ess
sup
t
∈
R
0
+
∥
A
˙
j
(
t
)
∥
:
j
∈
q
¯
∪
{
0
}
)
≤
μ
,
∥
∑
τ
∈
ST
(
σ
)
∩
[
t
,
t
+
T
)
A
˙
j
(
τ
)
∥
≤
∑
τ
∈
ST
(
σ
)
∩
[
t
,
t
+
T
)
∥
X
(
τ
)
δ
(
0
)
∥
≤
α
j
≤
max
(
∥
∑
τ
∈
ST
(
σ
)
∩
[
t
,
t
+
T
)
A
˙
j
(
τ
)
∥
:
j
∈
q
¯
∪
{
0
}
)
≤
α
,   where  δ
(
0
) is a Dirac impulse at  t
=
0. Note that Assumptions  2.3
imply |
∥
∑
j
=
0
q
A
j
(
t
+
τ
)
∥
−
∥
∑
j
=
0
q
A
j
(
t
)
∥
|
≤
μ
T
+
ν
(
t
+
τ
) for all τ
∈
[
0
,
T
],
with the function ν
:
R
0
+
→
R
0
+ satisfying ν
(
t
+
τ
)
≤
α
0
j
+
α
1
j
(
t
+
τ
)
≤
α with α
1
j
(
t
+
τ
)
=
0 if A
˙
j
(
t
+
τ
) exists with ∥
A
˙
j
(
t
+
τ
)
∥
≤
α
j or α
j
−
α
0
j
≥
α
1
j
(
t
+
τ
)
≥
∥
Δ
j
(
t
+
τ
)
∥ if A
˙
j
(
t
+
τ
)
=
Δ
j
(
t
+
τ
)
δ
(
0
), that
is, at least one of its entries is impulsive.Note that Assumption2.1 implies that
switching does not happen arbitrarily fast neither to reset parameters nor to
reset-free ones . The subsequent result is direct.Assertions 1.
The
following properties are true irrespective of the switching function:
(i)t
∉
ST
(
σ
)
⇔
σ
(
t
+
)
=
σ
(
t
)
=
0
n
×
(
q
+
1
)
n (i.e., a zero n
×
(
q
+
1
)
n-matrix);
(ii)t
∈
ST
(
σ
)
⇔
σ
(
t
+
)
−
σ
(
t
)
≠
0
n
×
(
q
+
1
)
n;
(iii)t
∉
ST
(
σ
)
⇔
ST
(
σ
,
t
+
)
=
ST
(
σ
,
t
);
(iv)t
∈
ST
(
σ
)
⇔
ST
(
σ
,
t
+
)
≠
ST
(
σ
,
t
).Proof.
(i)σ
(
t
+
)
=
σ
(
t
)
=
0
n
×
(
q
+
1
)
n
⇒
t
∉
ST
r
(
σ
)
⋀
t
∉
ST
rf
(
σ
)
⇔
t
∉
ST
(
σ
)
,
t
∉
ST
⇔
σ
(
t
)
=
0 (since switching is not arbitrarily
fast), (2.6)
⋀
(
σ
(
t
+
)
−
σ
(
t
)
=
0
n
×
(
q
+
1
)
n
)
⟺
σ
(
t
+
)
=
σ
(
t
)
=
0
n
×
(
q
+
1
)
n
. 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(2.7)
ST
(
σ
,
t
)
:=
{
t
i
∈
ST
(
σ
)
:
t
i
<
t
}
⟺
ST
(
σ
,
t
+
)
:=
{
t
i
∈
ST
(
σ
)
:
t
i
<
t
+
}
=
{
t
i
∈
ST
(
σ
)
:
t
i
≤
t
}
⟺
ST
(
σ
,
t
+
)
{
=
ST
(
σ
,
t
)
if
t
∉
ST
(
σ
)
,
≠
ST
(
σ
,
t
)
if
t
∈
ST
(
σ
)   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: (2.8)
V
(
t
,
x
t
)
=
x
T
(
t
)
P
(
t
)
x
(
t
)
+
∑
i
=
1
q
∑
j
=
0
q
∫
−
h
i
−
h
j
−
h
j
∫
t
+
θ
t
x
T
(
τ
)
S
i
j
x
(
τ
)
d
τ
d
θ
.Theorem 2.12.
The
following properties hold.
(i)
Assume the
following.(i.a)
The matrix functionsA
j
(
t
)
,
for all j
∈
q
¯
∪
{
0
} are subject to Assumption 2.1.
(i.b)
The switching lawσ is such  that (2.9)
Q
(
t
)
:=
[
H
1
h
^
1
P
(
t
)
A
1
(
t
)
M
(
t
)
⋯
h
^
q
P
(
t
)
A
q
(
t
)
M
(
t
)
h
^
1
M
T
(
t
)
A
1
T
(
t
)
P
(
t
)
−
R
1
0
⋯
0
⋮
0
⋮
⋱
⋮
h
^
q
M
T
(
t
)
A
q
T
(
t
)
P
(
t
)
0
⋯
−
R
q
]
<
0
,
∀
t
∈
R
0
+
,where H
1
denotes
(
∑
j
=
0
q
A
j
T
(
t
)
)
P
(
t
)
+
P
(
t
)
(
∑
j
=
0
q
A
j
(
t
)
)
+
P
˙
(
t
)
+
∑
i
=
1
q
∑
j
=
0
q
h
i
^
S
i
j,
for
some time-differentiable real symmetric positive definite matrix function P
:
R
0
+
→
R
n
×
n  and some real symmetric positive definite matrices S
i
j
∈
R
n
×
n
(
∀
i
∈
q
¯
,
∀
j
∈
q
¯
∪
{
0
}
),
where (2.10)
M
(
t
)
:=
[
A
0
(
t
)
,
…
,
A
q
(
t
)
]
∈
R
n
×
(
q
+
1
)
n
,
R
i
:=
diag
(
S
i
0
,
S
i
1
,
…
,
S
i
q
)
∈
R
(
q
+
1
)
n
×
(
q
+
1
)
n
;
∀
i
∈
q
,
∀
t
∈
R
0
+
.Thus,
the system (2.1) is globally asymptotically Lyapunov's stable for all delays h
i
∈
[
0
,
h
^
i
], for
all
i
∈
q
¯.
A necessary condition is (
∑
j
=
0
q
A
j
T
(
t
)
)
P
(
t
)
+
P
(
t
)
(
∑
j
=
0
q
A
j
(
t
)
)
+
P
˙
(
t
)
<
0, for
all
t
∈
R
0
+  what implies that (
∑
j
=
0
q
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
i
j,  for
all
i
∈
q
¯
∪
{
0
}, for
all
t
∈
R
0
+  for some  j
∈
p
¯ (eventually being dependent on t) satisfying
Assumption   2.4.
(ii.b)
The switching lawσ 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)
(2.11)
Q
i
*
:=
[
(
∑
j
=
0
q
A
j
i
T
)
P
*
+
P
*
(
∑
j
=
0
q
A
j
i
)
+
∑
i
=
1
q
∑
j
=
0
q
h
^
i
S
i
j
*
h
^
1
P
*
A
1
i
M
1
*
⋯
h
^
q
P
*
A
q
i
M
q
*
h
^
1
M
1
*
T
A
1
i
T
P
*
−
R
1
*
0
⋯
0
⋮
0
⋮
⋱
⋮
h
^
q
M
q
*
T
A
q
i
T
P
*
0
⋯
−
R
q
*
]
<
0
;
∀
i
∈
p
¯for
some R
n
×
n
∋
P
*
=
P
*
T
>
0, R
n
×
n
∋
S
i
j
*
=
S
i
j
*
T
>
0
(
∀
i
∈
q
¯
,
∀
j
∈
q
¯
∪
{
0
}
),
where (2.12)
M
i
:=
[
A
0
i
,
…
,
A
q
i
]
∈
R
n
×
(
q
+
1
)
n
,
R
i
*
:=
diag
(
S
i
0
*
,
S
i
1
*
,
…
,
)
∈
R
(
q
+
1
)
n
×
(
q
+
1
)
n
;
∀
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 for all delays h
i
∈
[
0
,
h
^
i
], for all
i
∈
q
¯.
If (2.9) is replaced with Q
i
*
≤
−
2
ε
I
(
q
+
1
)
n
<
0, for all
i
∈
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 any h
i
∈
[
0
,
h
^
i
], for
all
i
∈
q
¯ provided that all the delay-free resetting systems
(2.4)z
˙
j
(
t
)
=
(
∑
i
=
0
q
A
i
j
)
z
j
(
t
) fulfil Assumption 2.4, that is, they are
globally exponentially stable.It
is of interest to discuss particular cases easy to test, guaranteeing Theorem2.12 (i).


## 2.3. Sufficiency Type Asymptotic Stability Conditions Obtained for Constant Symmetric MatricesP and S
i
j

Assume real constant symmetric matricesP
(
t
)
=
P and S
i
j, for
all
i
∈
q
¯
,
for
all
j
∈
q
¯
∪
{
0
}, for
all
t
∈
R
0
+ so that (2.13)
Q
(
t
)
=
[
H
2
h
^
1
P
A
1
(
t
)
M
(
t
)
⋯
h
^
q
P
A
q
(
t
)
M
(
t
)
h
^
1
M
T
(
t
)
A
1
T
(
t
)
P
−
R
1
0
⋯
0
⋮
0
⋮
⋱
⋮
h
^
q
M
T
(
t
)
A
q
T
(
t
)
P
0
⋯
−
R
q
]
<
0, where H
2
denotes
(
∑
j
=
0
q
A
j
T
(
t
)
)
P
+
P
(
∑
j
=
0
q
A
j
(
t
)
)
+
∑
i
=
1
q
∑
j
=
0
q
h
^
i
S
i
j. In
this case, the  Krasovsky-Lyapunov
functional used in the proof of Theorem 2.12(i) holds defined with constant
matrices for all time irrespective of being a switching-free instant or any
switching instant (independently of its nature: reset time or reset-free
switching instant). A practical test for (2.13) to hold follows. Consider A
i
*
(
i
∈
q
¯
∪
{
0
}
)  such that the time invariant system (2.1)
defined with A
i
(
t
)
→
A
i
* is globally asymptotically Lyapunov's stable
and define  a stability real  n-matrix A
*
:=
∑
i
=
0
q
A
i
*.
Decompose Q
(
t
)
=
Q
*
+
Q
˜
(
t
),
where (2.14)
Q
*
:=
[
A
*
T
P
+
P
A
*
+
∑
i
=
1
q
∑
j
=
0
q
h
^
i
S
i
j
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
*
]
,
Q
˜
(
t
)
:=
[
A
˜
T
(
t
)
P
+
P
A
˜
(
t
)
h
^
1
P
Δ
˜
1
(
t
)
⋯
h
^
q
P
Δ
˜
q
(
t
)
h
^
1
Δ
˜
1
T
(
t
)
P
0
0
⋯
0
⋮
0
⋮
⋱
⋮
h
^
q
Δ
˜
q
T
(
t
)
P
0
⋯
0
]
,where (2.15)
M
*
:=
diag
(
A
0
*
,
A
1
*
,
…
,
A
q
*
)
,
R
i
*
:=
diag
(
S
i
0
*
,
S
i
1
*
,
…
,
S
i
q
*
)
,
M
˜
(
t
)
:=
M
(
t
)
−
M
*
,
A
˜
i
(
t
)
:=
A
i
(
t
)
−
A
*
,
A
˜
(
t
)
:=
(
∑
j
=
0
q
A
j
i
(
t
)
)
−
A
*
,
Δ
˜
i
(
t
)
:=
(
A
˜
i
(
t
)
M
*
+
A
i
*
M
˜
(
t
)
+
A
˜
i
(
t
)
M
↔
(
t
)
)
. If t
∈
ST
(
σ
),
then (2.16)
δ
Q
˜
(
t
)
:=
Q
˜
(
t
+
)
−
Q
˜
(
t
)
=
[
H
3
h
^
1
P
(
Δ
˜
1
(
t
+
)
−
Δ
˜
1
(
t
)
)
⋯
h
^
q
P
(
Δ
˜
1
(
t
+
)
−
Δ
˜
1
(
t
)
)
h
^
1
(
Δ
˜
1
(
t
+
)
−
Δ
˜
1
(
t
)
)
T
P
0
0
⋯
0
⋮
0
⋮
⋱
⋮
h
^
q
(
Δ
˜
1
(
t
+
)
−
Δ
˜
1
(
t
)
)
T
P
0
⋯
0
]
,
∥
δ
Q
˜
(
t
)
∥
2
≤
2
λ
max
(
P
)
[
∥
A
˜
T
(
t
+
)
−
A
˜
T
(
t
)
∥
2
+
∑
i
=
1
q
h
^
i
∥
Δ
˜
i
(
t
+
)
−
Δ
˜
i
(
t
)
∥
2
]
=
2
λ
max
(
P
)
[
∥
∑
i
=
0
q
(
A
i
T
(
t
+
)
−
A
i
T
(
t
)
)
∥
2
+
∑
i
=
1
q
h
^
i
∥
Δ
˜
i
(
t
+
)
−
Δ
˜
i
(
t
)
∥
2
]
,    where H
3
denotes
(
A
˜
T
(
t
+
)
−
A
˜
T
(
t
)
)
P
+
P
(
(
A
˜
(
t
+
)
−
A
˜
(
t
)
)
)
, what
leads to (2.17)
∥
δ
Q
˜
(
t
)
∥
2
≤
2
λ
max
(
P
)
[
(
q
+
1
)
a
+
+
h
^
q
(
a
+
(
∑
j
=
0
q
∥
A
j
*
∥
2
+
(
q
+
1
)
∥
A
i
*
∥
2
+
(
q
+
1
)
(
a
+
a
+
)
)
+
(
q
+
1
)
(
2
a
+
a
+
)
a
)
]
, where a
:=
max
i
∈
q
¯
∪
{
0
}
(
∥
A
i
∥
2
) and a
+
:=
sup
t
∈
ST
(
σ
)
max
i
∈
q
¯
∪
{
0
}
(
∥
A
i
(
t
+
)
−
A
i
∥
2
) (see A.9 in  Appendix ). Direct  results from Theorem 2.12 which follow from (2.13)
to (2.17) are given below.Corollary 2.13.
Consider
in (2.9) replacements with constant real matrices Q
(
t
)
→
Q
*
=
Q
*
T, P
(
t
)
→
P
=
P
T
>
0, A
i
(
t
)
→
A
i
*, A
(
t
)
→
A
*
:=
∑
i
=
0
q
A
i
*;   for
all
i
∈
q
¯
,
for
all
j
∈
q
¯
∪
{
0
}, for
all
t
∈
R
0
+ such that A
* is a stability matrix. Then, Theorem  2.12(i) holds if Q
*
<
0 for any switching law σ such that
(1)
λ
min
(
−
Q
*
)
=
−
λ
max
(
Q
*
)
>
∥
Q
˜
(
t
)
∥
2
:=
λ
max
1
/
2
(
Q
˜
T
(
t
)
Q
˜
(
t
)
), for
all
t
∈
R
0
+
∖
ST
(
σ
),
(2)
a
:=
max
i
∈
q
¯
∪
{
0
}
(
∥
A
i
∥
2
) and a
+
:=
sup
t
∈
ST
(
σ
)
max
i
∈
q
¯
∪
{
0
}
(
∥
A
i
(
t
+
)
−
A
i
∥
2
) are sufficiently small such that (2.18)
λ
min
(
−
Q
*
)
−
∥
Q
˜
(
t
)
∥
2
>
2
λ
max
(
P
)
[
(
q
+
1
)
a
+
+
h
^
q
(
a
+
(
∑
j
=
0
q
∥
A
j
*
∥
2
+
(
q
+
1
)
max
i
∈
q
¯
∪
{
0
}
∥
A
i
*
∥
2
+
(
q
+
1
)
(
a
+
a
+
)
)
+
(
q
+
1
)
(
2
a
+
a
+
)
a
)
]
,
∀
t
∈
ST
(
σ
)
.Corollary 2.14.
Consider
in (2.9) replacements with constant real matrices Q
(
t
)
→
Q
j
*
=
Q
j
*
T, P
(
t
)
→
P
=
P
T
>
0, A
i
j
(
t
)
→
A
i
j
*, A
(
t
)
→
A
j
*
:=
∑
i
=
0
q
A
i
j
* for all j
∈
p
¯
,
for
all
k
∈
q
¯
∪
{
0
},  for
all
t
∈
R
0
+ such that each A
j
*
(
∀
j
∈
p
¯
)      is a stability matrix with A
i
j
*
;
for
all
i
∈
q
¯
,
for
all
j
∈
p
¯  being the parameterizations defining the
resetting systems  (2.4). Assume that the system  (2.1) is one of the resetting
systems  (2.4) at t
=
0. Then, Theorem  2.12(i) holds with a common Krasovsky-Lyapunov function for all those resetting systems
if Q
j
*
<
0
(
∀
j
∈
p
¯
)        for any switching law σ such that
(1)
λ
min
(
−
Q
*
)
=
−
λ
max
(
Q
*
)
>
∥
Q
˜
(
t
)
∥
2
:=
λ
max
1
/
2
(
Q
˜
T
(
t
)
Q
˜
(
t
)
)
,
for
all
t
∈
R
0
+
∖
ST
(
σ
),
(2)
a
j
:=
max
i
∈
q
¯
∪
{
0
}
(
∥
A
j
i
∥
2
) and a
j
+
:=
sup
t
∈
ST
(
σ
)
max
i
∈
q
¯
∪
{
0
}
(
∥
A
i
(
t
+
)
−
A
j
i
∥
2
) are sufficiently small such that (2.19)
λ
min
(
−
Q
*
)
−
∥
Q
˜
(
t
)
∥
2
>
2
λ
max
(
P
)
[
(
q
+
1
)
a
j
+
+
h
^
q
(
a
j
+
(
∑
i
=
0
q
∥
A
j
i
∥
2
+
(
q
+
1
)
max
i
∈
q
¯
∪
{
0
}
∥
A
j
i
∥
2
)
+
(
q
+
1
)
(
a
j
+
a
j
+
)
+
(
q
+
1
)
(
2
a
j
+
a
j
+
)
a
j
+
)
]
t
∈
ST
fr
(
σ
),
provided that at time max
(
t
′
<
t
:
t
′
∈
ST
r
(
σ
)
),
the system (2.1)
coincides with at the j
∈
p
¯
r  setting system (2.4).The
proof of Corollary2.14 is close to that of Corollary 2.13 from A.9 in Appendix
with the replacements a
→
a
j
,
a
+
→
a
j
+ for all j
∈
p
¯.
If (2.19) is rewritten with the replacements a
j
→
a
¯
:=
max
j
∈
p
¯
a
j
,
a
j
+
→
a
→
+
:=
max
j
∈
p
¯
a
j
+  then the reformulated weaker Corollary 2.14 is valid for all t
∈
ST
fr
(
σ
) irrespective of the preceding reset switching.
A  result which guarantees Corollary 2.13,
and then Theorem 2.12(i),  is now  obtained by replacing the  (1,1) block matrix of Q
* by a Lyapunov matrix equality as follows.
Consider a real n-matrix Q
0
*
=
Q
0
*
T
>
0 such that λ
min
(
Q
0
*
)
>
λ
max
(
∑
i
=
1
q
∑
j
=
0
q
h
^
i
S
i
j
) and P
:=
∫
0
∞
e
A
*
T
τ
Q
0
*
e
A
*
τ
d
τ satisfying the Lyapunov equation A
*
T
P
+
P
A
*
=
−
Q
0
*
<
0 as its unique solution.  Note that λ
max
(
P
)
≤
K
*
λ
max
(
Q
0
*
)
/
2
ρ
* for some K
*
∈
R
+,
where −
ρ
*
<
0 is the stability abscissa of A
* with ∥
e
A
*
T
t
∥
≤
K
*
e
−
ρ
*
t for all t
∈
R.
Define the decomposition Q
(
t
)
=
Q
¯
*
+
Q
¯
˜
(
t
),
where (2.20)
Q
¯
*
:=
Block
Diag
(
−
Q
0
*
+
∑
i
=
1
q
∑
j
=
0
q
h
^
i
S
i
j
,
−
R
1
*
,
…
,
−
R
q
*
)
,
Q
¯
˜
(
t
)
:=
[
A
˜
T
(
t
)
P
+
P
A
˜
(
t
)
h
^
1
P
(
A
1
*
M
*
+
Δ
˜
1
(
t
)
)
⋯
h
^
q
P
(
A
q
*
M
*
+
Δ
˜
q
(
t
)
)
h
^
1
(
A
1
*
M
*
+
Δ
˜
1
(
t
)
)
T
P
0
0
⋯
0
⋮
0
⋮
⋱
⋮
h
^
q
(
A
q
*
M
*
+
Δ
˜
q
(
t
)
)
T
P
0
⋯
0
]
⇒
∥
Q
¯
˜
(
t
)
∥
2
≤
(
K
*
λ
max
(
Q
0
*
)
ρ
*
)
(
∥
A
˜
T
(
t
)
∥
2
+
∑
i
=
1
q
h
^
i
∥
A
i
*
M
*
+
Δ
˜
i
(
t
)
∥
2
)
.                                       Thus,
the subsequent result follows from Corollary 2.13 and (2.20).Corollary 2.15.
Consider
the matrices of Corollary2.13 with A
* being a stability matrix with stability abscissa (
−
ρ
*
)
<
0 which satisfies the Lyapunov equation A
*
T
P
+
P
A
*
=
−
Q
0
*
<
0.
Then, Theorem  2.12(i) holds if Q
*
<
0 for any switching law σ such that (2.21)
λ
min
(
−
Q
¯
*
)
=
−
λ
max
(
Q
¯
*
)
>
(
K
*
λ
max
(
Q
0
*
)
ρ
*
)
(
∥
A
˜
T
(
t
)
∥
2
+
∑
i
=
1
q
h
^
i
∥
A
i
*
M
*
+
Δ
˜
i
(
t
)
∥
2
)
,
∀
t
∈
R
0
+
,
∀
t
∈
ST
(
σ
)
.


## 2.4. Sufficiency Type Asymptotic Stability Conditions Obtained for Time-Varying Symmetric MatricesP
(
t
), S
i
j
(
t
)
=
S
i
j

The
following result, which is proven in Appendix, holds.Theorem 2.16.
Under Assumptions2.1–2.3, the following properties
hold.
(i)
The switched system (2.1) is globally asymptotically Lyapunov's for any delays h
i
∈
[
0
,
h
^
i
] for all  i
∈
q
¯   for
some h
^
:=
max
i
∈
q
¯
(
h
^
i
) and any switching law  σ such that
(a)
the switching instants are arbitrary;
(b)
max
(
ess
sup
∥
A
˙
j
(
t
)
∥
:
t
∈
R
0
+
,
j
∈
p
¯
)  is sufficiently small compared to the absolute
value of the prescribed stability abscissa of  ∑
j
=
0
q
A
j
(
t
);
(c)
the support testing matrix of
distributional derivativesΓ
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
(
t
)
(
t
)
δ
(
0
)).
(ii)
The switched system (2.1) is globally exponentially stable for any delays h
i
∈
[
0
,
h
^
i
] for all i
∈
q
¯ for some h
^
:=
max
i
∈
q
¯
(
h
^
i
) such that
(a)
max
(
ess
sup
∥
A
˙
j
(
t
)
∥
:
t
∈
R
0
+
,
j
∈
p
¯
) is sufficiently small compared to the absolute value of the prescribed stability abscissa of ∑
j
=
0
q
A
j
(
t
);
(b)
max
(
∥
Γ
A
d
j
(
t
)
(
t
)
∥
:
A
˙
j
(
t
)
=
Γ
A
d
j
(
t
)
(
t
)
δ
(
0
)
,
∀
t
∈
ST
(
σ
)
,
j
∈
p
¯
)  is sufficiently small compared to the timeintervals in between any two consecutive switching instants.
Furthermore, if Assumptions 2.1–2.4 hold, then
(iii)
the switched system (2.1) is globally
exponentially stable for any delays h
i
∈
[
0
,
h
^
i
] for all i
∈
q
¯, for some h
^
:=
max
i
∈
q
¯
(
h
^
i
) such that(a)
max
(
ess
sup
∥
A
˙
j
(
t
)
∥
:
t
∈
R
0
+
,
j
∈
p
¯
) is sufficiently small compared to the absolute
value of the prescribed stability abscissa of ∑
j
=
0
q
A
j
(
t
);
(b)
the switching lawσ  is such that(a)
max
(
∥
Γ
A
d
j
(
t
)
∥
:
A
˙
j
(
t
)
=
Γ
A
d
j
(
t
)
δ
(
0
)
,
∀
t
∈
ST
rf
(
σ
)
,
j
∈
p
¯
)  is sufficiently small compared to the lengths
of  time intervals between any two
consecutive switching instants;
(b)
it exists a common Krasovsky-Lyapunov
functionalV
(
t
,
x
t
) defined with constant matrices P
¯
=
P
¯
T
>
0 and S
¯
i
j
=
S
¯
i
j
T
>
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
all
t
∈
ST
r
(
σ
) under the resetting action P
(
t
+
)
=
P
¯; for
all
(
i
,
j
)
∈
(
q
¯
∪
{
0
}
)
×
q
¯:(b.1)
V
(
t
+
,
x
t
)
≤
V
(
t
,
x
t
)  which is guaranteed, in particular, if P
(
t
+
)
=
P
¯
≤
P
(
t
),
(b.2)
the tradeoff (a) is respected between sufficiently small norms of the matrices
of distributional derivatives and the length|
t
−
t
′
|,
at any t
∈
ST
r
(
σ
),  if any,  where the condition (b.1)  is not satisfied, where t
′
=
max
(
τ
∈
R
0
+
:
ST
(
σ
)
∋
τ
<
t
).The
characterization of the “sufficient smallness” of the involved magnitudes in
Theorem2.16 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 anyh
i
∈
R
0
+, for
all
i
∈
q
¯ of the switched system (2.1) are obtained  under very close guidelines as those involved
in the results on stability dependent of
the delays given in Section 2. The Krasovsky-Lyapunov functional candidate of
Section 2 and Appendix   is modified as follows: (3.1)
V
(
t
,
x
t
)
=
x
T
(
t
)
P
(
t
)
x
(
t
)
+
∑
i
=
1
q
∫
t
−
h
i
t
x
T
(
τ
)
S
i
(
τ
)
x
(
τ
)
d
τ whose
time derivative along  the
state-trajectory solution of (2.1) is (3.2)
V
˙
(
t
,
x
t
)
=
x
T
(
t
)
(
A
T
(
t
)
P
(
t
)
+
P
(
t
)
A
(
t
)
+
∑
i
=
1
q
S
i
(
t
)
+
P
˙
(
t
)
)
x
(
t
)
+
2
x
T
(
t
)
∑
i
=
1
q
P
A
i
(
t
)
x
(
t
−
h
i
)
−
∑
i
=
1
q
x
T
(
t
−
h
i
)
S
i
(
t
−
h
i
)
x
(
t
−
h
i
)
=
x
¯
T
(
t
)
Q
′
(
t
)
x
¯
(
t
)
<
0 for all nonzero x
¯
T
(
t
)
=
(
x
T
(
t
)
,
x
T
(
t
−
h
1
)
,
…
,
x
T
(
t
−
h
q
)
) if (3.3)
Q
′
(
t
)
:=
[
A
T
(
t
)
P
(
t
)
+
P
(
t
)
A
(
t
)
+
∑
i
=
1
q
S
i
(
t
)
+
P
˙
(
t
)
P
A
1
(
t
)
⋯
P
A
q
(
t
)
A
1
T
(
t
)
P
−
S
1
(
t
−
h
1
)
0
⋯
0
⋮
⋮
⋮
A
q
T
(
t
)
P
0
−
S
q
(
t
−
h
q
)
]
<
0. 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 eigenvaluesλ
i
(
A
0
(
t
)
) of the matrix function A
0
(
t
) satisfy Re
λ
i
(
A
0
(
t
)
)
≤
−
ρ
0
0
<
0; for
all
t
∈
R
0
+, for
all
i
∈
σ
¯
(
σ
≤
n
) for some ρ
0
0
∈
R
+
:=
{
0
<
z
∈
R
}; that
is, A
0
(
t
) is a stability matrix, for
all
t
∈
R
0
+.Assumption 3.2 b.
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
ℓ
<
∞ with γ
0
ℓ being a ℓ-norm dependent nonnegative real constant  and, furthermore, ∫
t
t
+
T
∥
A
˙
0
(
τ
)
∥
d
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 Assumptions 2.3.Assumption 3.3 b (for the resetting systems).
All the eigenvaluesλ
k
(
A
0
j
) satisfy Re
λ
k
(
A
j
0
)
≤
−
ρ
0
0
<
0; for
all
t
∈
R
0
+, for
all
k
∈
σ
¯
0
j
(
σ
0
j
≤
n
), for
all
j
∈
p
¯;
that is, A
0
j are constant stability matrices with
prescribed stability abscissa.A
parallel result to
Theorem2.12(i)-(ii)  is
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 to Assumption  2.1;
(i.b)
the switching lawσ is such that Q
′
(
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 functions  S
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 delays  h
i
∈
[
0
,
∞
), for
all
i
∈
q
¯). A necessary condition is A
0
T
(
t
)
P
(
t
)
+
P
(
t
)
A
0
(
t
)
+
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
j
i,  for
all
j
∈
q
¯
∪
{
0
}, for
all
t
∈
R
0
+  for some i
∈
p
¯ (eventually being dependent on t)  satisfying
Assumption 3.3;
(ii.b)
the switching lawσ is such that ST
rf
(
σ
)
=
∅  (i.e., it generates reset switching
instants only) with ST
r
(
σ
) being arbitrary, namely, the set of reset
times is either any arbitrary strictly increasing sequence of nonnegative real
values (i.e., the resetting switching never ends) or any finite set of strictly
ordered nonnegative real numbers with a finite maximal (i.e., the resetting
switching ends in finite time);
(ii.c)
(3.4)
Q
i
*
′
:=
[
A
0
i
T
P
*
+
P
*
A
0
i
+
∑
i
=
1
q
S
i
*
P
*
A
1
i
⋯
P
*
A
q
i
A
1
i
T
P
*
−
S
1
*
0
⋯
0
⋮
0
⋮
⋱
⋮
A
q
i
T
P
*
0
⋯
−
S
q
*
]
<
0
;
∀
i
∈
p
¯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
all
i
∈
q
¯.
If  (3.4) is replaced with Q
i
*
′
≤
−
2
ε
I
(
q
+
1
)
n
<
0, for
all
i
∈
q
¯,
and some ε
∈
R
+  then the state trajectory decays   exponentially  with rate
(
−
ε
)
<
0.Parallel
results to Corollaries2.13–2.15 are direct from Theorem 3.4 with the replacements A
0
(
t
)
→
A
0
* (a constant stability matrix), A
0
j
(
t
)
→
A
0
j
*, 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 to Theorem 2.16, follows by replacing Assumptions 2.2–2.4 by
Assumptions 3.1–3.3.Theorem 3.5.
Under Assumptions2.1 and 3.1–3.2, the following
properties hold.
(i)
The switched
system (2.1) is globally asymptotically Lyapunov's stable independent of the
delays, that is, for any delays h
i
∈
[
0
,
∞
), for
all
i
∈
p
¯ and any switching law σ such that (a)
the switching instants are
arbitrary;
(b)
max
(
ess
sup
∥
A
˙
0
(
t
)
∥
:
t
∈
R
0
+
)  is sufficiently small compared to the absolute
value of the prescribed stability abscissa of A
0
(
t
);
(c)
the support testing matrix of
distributional derivativesΓ
A
d
0
(
t
) of the same
matrices are semidefinite negative for all time instants, where the
conventional derivatives do not exist (i.e., ifA
˙
0
(
t
)
=
Γ
A
d
0
(
t
)
(
t
)
δ
(
0
)) .
(ii)
The
switched system (2.1) is globally exponentially stable independent of the delays
if (a)
max
(
ess
sup
∥
A
˙
0
(
t
)
∥
:
t
∈
R
0
+
) is sufficiently small compared to the absolute
value of the prescribed stability abscissa of A
0
(
t
);
(b)
max
(
∥
Γ
A
d
0
(
t
)
(
t
)
∥
:
A
˙
0
(
t
)
=
Γ
A
d
0
(
t
)
(
t
)
δ
(
0
)
,
∀
t
∈
ST
(
σ
)
) is sufficiently small compared to the time
intervals in between any two consecutive switching instants.
If Assumptions 2.1
and 3.1–3.3 hold, then
(iii)
the switched system (2.1) is globally
exponentially stable independent of the delays if max
(
ess
sup
∥
A
˙
0
(
t
)
∥
:
t
∈
R
0
+
) is sufficiently small compared to the absolute
value of the prescribed stability abscissa of A
0
(
t
) and, furthermore, the switching law σ is such that (a)
At reset-free switching
instants,max
(
∥
Γ
A
d
0
(
t
)
(
t
)
∥
:
A
˙
0
(
t
)
=
Γ
A
d
0
(
t
)
(
t
)
δ
(
0
)
,
∀
t
∈
ST
rf
(
σ
)
)  is sufficiently small compared to the time
intervals in between any two consecutive reset switching instants;
(b)
there
exists a common Krasovsky-Lyapunov functionalV
(
t
,
x
t
) defined with constant matrices P
(
t
)
→
P
¯
=
P
¯
T
>
0 and S
i
(
t
)
→
S
¯
i
=
S
¯
i
T
>
0, for
all
i
∈
q
¯
∪
{
0
} in (3.1)
for all the time-invariant resetting systems (2.4) and some of the
conditions (b.1)-(b.2) of Theorem 2.16 hold for
all
t
∈
ST
r
(
σ
) under the resetting action P
(
t
+
)
=
P
¯; for
all
i
∈
q
¯
∪
{
0
}.


## 4. Simulation Examples and Potential Future Research

In
this section, some simulation examples showing numerically the application of
the results introduced below are carried out. The section contains two examples:
one related to the delay-dependent stability property introduced in Section2
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 systemx
˙
(
t
)
=
A
0
(
t
)
x
(
t
)
+
A
1
(
t
)
x
(
t
−
h
), where h
=
0.75 second and each resetting
system  (2.4) with p
=
2 is defined by (4.1)
A
01
(
t
)
=
(
−
1.3
cos
t
4
0
−
2.5
)
,
A
11
(
t
)
=
(
−
2.2
a
(
t
)
0
−
2.1
)
,
A
02
(
t
)
=
(
−
1
sin
t
4
0
−
2
)
,
A
12
(
t
)
=
(
−
1.5
b
(
t
)
0
−
2
) with a
(
t
)
=
1
/
(
⌊
t
⌋
+
t
+
1
) and b
(
t
)
=
t
/
(
⌈
t
⌉
2
+
1
),
where ⌊
·
⌋ denotes the  largest integer not larger than
(·) and ⌈
·
⌉ denotes the smallest integer not smaller than
(·). Note that a
(
t
) and b
(
t
) are
discontinuous functions at integer values of time. A graphical representation
of these functions is shown in Figures 1 and 2.Figure 1
Graphical representation ofa
(
t
).Figure 2
Graphical representation ofb
(
t
).Initially,
it will be checked that Theorem2.16(ii) conditions
hold. Firstly, the switching instants between resetting systems have been
selected arbitrarily and defined by Figure 3.Figure 3
Sequence of switching instants between resetting systems.Secondly,
the time derivatives of the resetting systems defined above are given on each
real interval(
k
,
k
+
1
)
⊂
R with k
∈
N by (4.2)
A
˙
01
=
(
0
−
sin
t
4
0
0
)
,
A
˙
11
=
(
0
a
˙
(
t
)
0
0
)
,
A
˙
02
=
(
0
cos
t
4
0
0
)
,
A
˙
12
=
(
0
b
˙
(
t
)
0
0
) 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
(
ρ
0
/
K
0
)
2
>
ess
sup
t
∈
R
0
+
/
ST
(
σ
)
∥
∑
j
=
0
1
A
j
∥,
where λ
max
(
P
(
t
)
)
≤
K
0
/
2
ρ
0 for the existing
unique symmetric positive definite solution P
(
t
) of (
∑
j
=
0
1
A
j
T
)
P
(
t
)
+
P
(
t
)
(
∑
j
=
0
1
A
j
)
=
−
I
n. Numerical
computations lead to 10.33
=
2
(
ρ
0
/
K
0
)
2
>
ess
sup
t
∈
R
0
+
/
ST
(
σ
)
∥
∑
j
=
0
1
A
j
∥
=
1.25 which guarantees the fulfilment the second
item of the theorem. Finally, from Figures 1 and 2 above it becomes apparent
that the distributional derivative at integer time instants (where the ordinary
derivative does not exist) is negative since a
(
t
)
>
a
(
t
+
) and b
(
t
)
>
b
(
t
+
) for all t
∈
N.
Therefore, from Theorem 2.16(ii),  the
space-state trajectories of the solution asymptotically converge to zero as
time evolves as Figure 4 shows.Figure 4
Convergence to zero of the state-trajectories.The
phase plane is shown in Figure5, where it can be appreciated the convergence
of the state evolution to the origin. Note that delay-dependent stability is
achieved. As simulations show, the system becomes unstable as delay exceeds a
certain threshold. Global asymptotic stability is guaranteed within the delay
variation interval [
0
,
1
].Figure 5
Phase portrait of the evolution of the system.


### 4.2. Delay-Independent Stability

This
example is devoted to the delay independent stability ensured by
Theorem3.5(i).
Thus, consider the system x
˙
(
t
)
=
A
0
(
t
)
x
(
t
)
+
A
1
(
t
)
x
(
t
−
h
) and the resetting systems (4.3)
A
01
=
(
−
1
1
2
0
−
3
)
,
A
02
=
(
−
2.5
2
0
−
1.6
)
,
A
1
=
(
−
2.5
a
(
t
)
0
−
1.6
) with a
(
t
)
=
2
+
1
/
(
1
+
t
)
n
/
10 and n
=
⌊
t
⌋.
Again, this function possesses bounded discontinuities at integer values of
time as Figure 6 shows.Figure 6
Graphical representation ofa
(
t
).Furthermore,
the conditions of Theorem3.5(i) are especially easy to verify since the resetting
matrices A
0
i are time-invariant and hence its time-derivatives
are identically zero. The switching sequence is the same as depicted
in Figure 3. Figures
7,
8,
9,
and  10 show the convergence of the state trajectories of the
system to zero for different values of the delay showing the delay independence
property.Figure 7
State trajectories evolution for a delay of 1.75 seconds.Figure 8
State trajectories evolution for a delay of 7.25 seconds.Figure 9
State trajectories evolution for a delay of 15.8 seconds.Figure 10
State trajectories evolution for a delay of 35.8 seconds.


#### 4.3. Potential Future Research

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


### 5. Conclusions

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


## 4.1. Delay-Dependent Stability

Consider
the delay systemx
˙
(
t
)
=
A
0
(
t
)
x
(
t
)
+
A
1
(
t
)
x
(
t
−
h
), where h
=
0.75 second and each resetting
system  (2.4) with p
=
2 is defined by (4.1)
A
01
(
t
)
=
(
−
1.3
cos
t
4
0
−
2.5
)
,
A
11
(
t
)
=
(
−
2.2
a
(
t
)
0
−
2.1
)
,
A
02
(
t
)
=
(
−
1
sin
t
4
0
−
2
)
,
A
12
(
t
)
=
(
−
1.5
b
(
t
)
0
−
2
) with a
(
t
)
=
1
/
(
⌊
t
⌋
+
t
+
1
) and b
(
t
)
=
t
/
(
⌈
t
⌉
2
+
1
),
where ⌊
·
⌋ denotes the  largest integer not larger than
(·) and ⌈
·
⌉ denotes the smallest integer not smaller than
(·). Note that a
(
t
) and b
(
t
) are
discontinuous functions at integer values of time. A graphical representation
of these functions is shown in Figures 1 and 2.Figure 1
Graphical representation ofa
(
t
).Figure 2
Graphical representation ofb
(
t
).Initially,
it will be checked that Theorem2.16(ii) conditions
hold. Firstly, the switching instants between resetting systems have been
selected arbitrarily and defined by Figure 3.Figure 3
Sequence of switching instants between resetting systems.Secondly,
the time derivatives of the resetting systems defined above are given on each
real interval(
k
,
k
+
1
)
⊂
R with k
∈
N by (4.2)
A
˙
01
=
(
0
−
sin
t
4
0
0
)
,
A
˙
11
=
(
0
a
˙
(
t
)
0
0
)
,
A
˙
02
=
(
0
cos
t
4
0
0
)
,
A
˙
12
=
(
0
b
˙
(
t
)
0
0
) 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
(
ρ
0
/
K
0
)
2
>
ess
sup
t
∈
R
0
+
/
ST
(
σ
)
∥
∑
j
=
0
1
A
j
∥,
where λ
max
(
P
(
t
)
)
≤
K
0
/
2
ρ
0 for the existing
unique symmetric positive definite solution P
(
t
) of (
∑
j
=
0
1
A
j
T
)
P
(
t
)
+
P
(
t
)
(
∑
j
=
0
1
A
j
)
=
−
I
n. Numerical
computations lead to 10.33
=
2
(
ρ
0
/
K
0
)
2
>
ess
sup
t
∈
R
0
+
/
ST
(
σ
)
∥
∑
j
=
0
1
A
j
∥
=
1.25 which guarantees the fulfilment the second
item of the theorem. Finally, from Figures 1 and 2 above it becomes apparent
that the distributional derivative at integer time instants (where the ordinary
derivative does not exist) is negative since a
(
t
)
>
a
(
t
+
) and b
(
t
)
>
b
(
t
+
) for all t
∈
N.
Therefore, from Theorem 2.16(ii),  the
space-state trajectories of the solution asymptotically converge to zero as
time evolves as Figure 4 shows.Figure 4
Convergence to zero of the state-trajectories.The
phase plane is shown in Figure5, where it can be appreciated the convergence
of the state evolution to the origin. Note that delay-dependent stability is
achieved. As simulations show, the system becomes unstable as delay exceeds a
certain threshold. Global asymptotic stability is guaranteed within the delay
variation interval [
0
,
1
].Figure 5
Phase portrait of the evolution of the system.


## 4.2. Delay-Independent Stability

This
example is devoted to the delay independent stability ensured by
Theorem3.5(i).
Thus, consider the system x
˙
(
t
)
=
A
0
(
t
)
x
(
t
)
+
A
1
(
t
)
x
(
t
−
h
) and the resetting systems (4.3)
A
01
=
(
−
1
1
2
0
−
3
)
,
A
02
=
(
−
2.5
2
0
−
1.6
)
,
A
1
=
(
−
2.5
a
(
t
)
0
−
1.6
) with a
(
t
)
=
2
+
1
/
(
1
+
t
)
n
/
10 and n
=
⌊
t
⌋.
Again, this function possesses bounded discontinuities at integer values of
time as Figure 6 shows.Figure 6
Graphical representation ofa
(
t
).Furthermore,
the conditions of Theorem3.5(i) are especially easy to verify since the resetting
matrices A
0
i are time-invariant and hence its time-derivatives
are identically zero. The switching sequence is the same as depicted
in Figure 3. Figures
7,
8,
9,
and  10 show the convergence of the state trajectories of the
system to zero for different values of the delay showing the delay independence
property.Figure 7
State trajectories evolution for a delay of 1.75 seconds.Figure 8
State trajectories evolution for a delay of 7.25 seconds.Figure 9
State trajectories evolution for a delay of 15.8 seconds.Figure 10
State trajectories evolution for a delay of 35.8 seconds.


### 4.3. Potential Future Research

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


## 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.



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*Source: 231710-2008-12-31.xml*