1. Introduction
Impulsive dynamic systems play an important role in some theoretical and applied problems in the field of dynamic systems and the stabilization of control systems. Basically, impulses at certain time instants in a control law cause the solution trajectory to have finite jumps as a result, or equivalently, such a solution is piecewise continuous with discontinuities of the first kind. By appropriately monitoring the sizes and the signs of the impulses and their distribution through time, impulsive control becomes a useful tool for closed-loop stabilization either when combined with regular control designs or by injecting the necessary impulses as the only stabilizing actions. The control impulses might also be generated using feedback on the trajectory solution. In that context, impulsive control has often been used for the stabilization of time-delay systems. The related background literature is abundant. See, for instance, [
1,
2,
3,
4,
5], concerned with the impulsive stabilization of time-delay systems and some of the references therein as a reference sample of such a bibliography.
On the other hand, it is well-known that fixed point theory has been, and continues to be, widely used in the study of dynamic systems due to its applicability in analyzing the boundedness and convergence of sequences generated by various types of mappings, which play a fundamental role in the stability analysis of such systems. Very often, it is rigorously feasible to identify possible fixed points of the map, which generates the solution trajectory of a dynamic system from the initial conditions with the equilibrium points of such a system. As a direct result, eventual boundedness and/or convergence of the sequences generated by such a mapping to the fixed point is a useful suitable property to elucidate the dynamic system’s stability and asymptotic stability.
The related background literature is very exhaustive. We now mention some of it for the different kinds of contractions. For instance, the so-called weak contractive mappings in metric spaces do not ensure the existence of fixed points, but in the case they exist, they are unique. See, for instance, [
6,
7,
8,
9] and the references therein. The celebrated so-called (strict) contractions guarantee the existence of a unique fixed point and the convergence of all sequences to it if the considered metric space is complete. See, for instance, [
8,
9,
10,
11,
12] and some of the references therein. Non-expansive mappings are useful to investigate, in general, the boundedness of sequences without paying special attention to convergence [
8,
13]. It can be recalled that strict contractions are also weak contractions and that weak contractions are also non-expansive mappings so any result valid for the latter also applies to the former.
There are also other types of more general contractions like, for instance, quasi-contractions [
14] or those associated with the so-called controlled metric spaces, which basically admit weighting functions in the configuration of their associated triangle inequality [
15,
16]. Moreover, the so-called pseudocontractive mappings are formulated in Banach spaces, and they are more general than non-expansive mappings, while they are closely linked to accretive mappings [
8,
17]. The triangle inequality is sometimes generalized to the involvement of an extended version of its standard form, which consists of allowing a correcting scalar number that can exceed unity. The associated metric spaces are referred to as b-metric spaces [
18].
It can also be pointed out that fixed point theory has also been extended to “ad hoc” versions related to fractional differential equations, with the subsequent investigation of the existence of fixed points and the boundedness and convergence of the generated sequences. See, for instance, [
19,
20] and some of the references therein. The generalized contractions [
21] and the so-called large contractions (see, for instance, [
22,
23,
24,
25,
26,
27]) are of interest because they conform to a special case of weak contractions. Specifically, when the self-mappings generate distances that exceed a prescribed threshold, these large contractions become strict contractions. There are also weaker extended versions available in the background literature which, for distances exceeding such a prescribed threshold, the “extended large contractive” self-mapping is dominated by a c-comparison function [
28] rather than being a (strict) contraction. We pay some attention in this paper to mappings under weaker conditions compared to large contractions. For example, when distances exceed a maximum prescribed threshold, they are strict contractions, while for smaller distances, special constraints are not involved. This philosophy does not guarantee the existence of fixed points, but it guarantees the avoidance of large unsuitable distances between consecutive points of the generated sequences through the self-mapping in the metric space. In the event that there is a fixed point, then these mappings, when associated with a state trajectory solution of a differential system, guarantee global stability (that is, the global boundedness of the solution for any finite initial condition). In general, they do not guarantee global asymptotic stability, that is, the global boundedness for any finite initial condition, with the additional property of the convergence of sequences to the fixed point.
The main idea behind the stabilization of dynamic systems under the above design philosophy is to dynamically generate jumps in the self-mapping that generate the solution to reduce the potential large inconvenient distances at the next stages of the generated sequences. It can be noticed that jumps in the solution, which make it piecewise continuous rather than continuous are associated with impulsive actions in its derivative with respect to time. In the context of dynamic systems, the transients can become improved using non-periodic or adaptive sampling, which adapts itself to the signal transients. In this framework, each sampling action serves to capture the value of the signal at a particular moment, allowing for an adaptive discrete version of the dynamic system. As a result, the overshoot peaks become reduced. See [
29,
30,
31] and some of the references therein. Such a property can be reasonably linked to the idea that the above-mentioned jumps should typically be distributed through time in a non-uniform way rather than periodically.
In addition, the stability properties of linear time-varying dynamic systems have been investigated exhaustively in the background literature. See, for instance, [
32,
33,
34] and some of the references therein, while the modeling aspects of differential systems, including the boundedness properties of the solutions and their related stability and convergence properties, can be found in [
35,
36,
37,
38,
39] and the references therein. Some known technical issues on the equivalences of norms are invoked in some mathematical proofs [
40]. It can be pointed out that fixed points are also relevant in certain algebraic and geometric studies. See, for instance, [
41,
42,
43,
44]. In particular, in [
41], two theorems are formulated that address the fixed points of automorphisms on the moduli space of the principal bundles on algebraic curves. Also, in [
44], Higgs pairs with group E6 are studied, and the fixed points of the sigma action are calculated.
The rest of the paper organizes its contribution as follows:
Section 2 formalizes some relevant properties of the boundedness of distances and that of their associated sequences and the related convergence properties of the introduced, so-called jumping self-mappings under given conditions. Such self-mappings have the characteristics that, at certain points of their domain, they can exhibit a finite jump between its left and right limits. In particular, conditions are given for such mappings to become contractions, weak contractions, or sub-contractions or bounded through the adequate distribution and size of the jumps. Also, the relevant properties of interest concerning the boundedness of distances between sequences and that of the generated sequences, as well as their related convergence and Cauchyness issues, are established and proved.
Section 3 focuses on the stabilization of a class of linear time-varying systems subject to impulsive actions. Two types of impulses might be allowed, namely, impulses in the derivative with respect to time of the system state and impulses in the matrix function of dynamics. These impulses can take place either separately or in a mixed way at the same or different time instants. The first type of impulse causes finite jumps in the solution trajectory at the impulsive time instants. The second kind of impulse generates finite jumps in the matrix of dynamics. Impulses are seen to be useful in the stabilization process via judicious choices of the time instants where impulses are generated at the entries of the matrix functions of impulsive gains. Also, the first mentioned type of impulses on the state first derivative with respect to time can be used to stabilize a system with several configurations (that is, with different alternative matrices of dynamics) using impulses to commute from one configuration to another one when the state exceeds a given, supposedly large, threshold. It is needed that at least one of the configurations be stable for global asymptotic stabilization and that this configuration be active along time intervals exceeding a minimum threshold. It can be pointed out that the above-mentioned second kind of impulse of the time derivative of the matrix of dynamics causes finite jumps in the first time derivative with respect to the time of the solution trajectory, which is then a piecewise continuous function under discontinuities of the first kind. The mathematical characterization of the impulses is addressed through the involvement of Dirac-type distributions in the system matrix. In the impulsive case, the evolution operator is seen to be, in general, non-unique. The formal study of the stabilization processes is included in the general formalism of
Section 2 on jumping self-mappings.
Section 4 discusses some illustrative examples that incorporate numerical simulations. Finally, some conclusions end the paper.
Nomenclature
;
is the n-th identity matrix;
is the n-th zero matrix;
; ;
; ;
where is the set of real numbers.
In the same way, we can define “mutatis-mutandis” as the respective subsets , , , and of the set of integer numbers.
An impulsive real function is that which has a nonzero set of Dirac distribution-type impulses on a finite set of impulsive points , such that there is an impulsive jump at of size . The same idea applies for a vector function in the sense that is impulsive if there is at least a component of , such that with . Thus, is an impulsive point of if is nonzero, that is, if there is at least one nonzero for . An abbreviated notation for this is . Again, a generalization for real matrix functions is direct in the sense that is an impulsive point of if with at least one entry of being nonzero. Note that if is non-impulsive, then the impulsive jump amplitude is null. To keep the notation less involved, the right limit of is simply denoted by so that stands as notation for .
denotes the infinity cardinal of a numerable set.
The entry-to-entry definition of the matrix is denoted by ; .
denotes the (finite or infinity) cardinal of a countable set . If such a cardinal is finite then . If the cardinal is (denumerable) infinity, then .
The acronym “iff” is the equivalent usual abbreviation of the claim “if and only if”.
If is symmetric, then and are the maximum and minimum eigenvalues of . If, in addition, , then and .
for (positive semidefinite real n-matrices) means that , i.e., is positive semidefinite, and (positive definite matrices) means that , i.e., is positive definite.
2. Main Results on Distances, Boundedness, and Convergence for Jumping Self-Mappings
This section relies on the main concepts and some mathematical results of the jumping self-mappings in metric spaces to be studied and the relevant related properties of boundedness and convergence under contractive conditions.
The self-mapping is being considered on a metric space which is defined by the composition of two piecewise continuous self-mappings , defined on the same metric space by , for any , and is the composed self-mapping on X. The notation for is useful to intuitively design the left limit of the point whose right limit is . Note that if , that is, there is not a finite jump of in , then is the identity, and is continuous in . In the same way, one may define the composed self-mapping on , such that , . Note from the above considerations that, even if have finite jump discontinuities, can be continuous.
The metric
satisfies the following constraints for any
,
,
:
where
is a bounded continuous function. Note from the triangle inequality that
and assume that
and
satisfy the subsequent constraints:
where
if
under the constraints:
if
and
if
, where the functions
and
are control functions that modulate the allowed tolerances to jumps from the left to right limits when generating the sequences through the mapping
, and
if
if
, and
if
d(
x,
y) = 0.
Remark 1. Note that , and can be negative, zero, or positive according to the above constraints.
Define by Theorem 1. Assume that is a self-mapping on , which satisfies (1)–(14), and that the function , defined byis piecewise continuous bounded on . Then, the following properties hold: - (i)
If (X, d) is complete and for all and some real constant, then is a (strict) contraction, and it has a unique fixed point in to which all sequences in , which are Cauchy (then convergent and bounded), converge;
- (ii)
If for all , such that , then is a weak contraction, and it has a unique fixed point in if (X, d) is compact, that is, complete and totally bounded. All sequences in are Cauchy and converge to the fixed point;
- (iii)
If for all and for all , such that , then is a sub-contraction, and it has (at least) a fixed point in if (X, d) is compact. All sequences in are bounded and Cauchy and converge to one of the fixed points;
- (iv)
The self-mappings are continuous at the fixed points of , that is, , , in the three above properties.
Proof. It is obvious that if
for all
in (15) then, for all
,
so that
. Furthermore,
which implies that
as
, and one obtains:
so that
so that
, then
is bounded for any
, and there exists
. Since
is continuous and it is Lipschitz with constant
, then
so that
is convergent (and then Cauchy) to some
since
is complete.
On the other hand,
so that
for any given
, and
.
Taking
, it follows from the above results that:
Since
for all
, it follows that
is continuous, and the limit and distance can be interchanged. Then,
Then, is a Cauchy sequence, and since is complete, then is bounded, , and so that is a fixed point. Since as , where and for all , it follows that so that the fixed point of is unique, and all the sequences converge to it for any . Property (i) has been proved.
To prove property (ii), note that if for all if , then if . Thus, take . If for all , then any distance sequence converges so that . The following cases are considered:
Case a: for all and so that . Since is totally bounded, is bounded, and it as a Cauchy, and then convergent, subsequence , with since is compact. Furthermore, since , as and , then as so that . Thus, is a fixed point of in since is complete. Assume that there is some , such that and . Then, so that , a contradiction so that , and the fixed point of in is unique. Furthermore, one obtains for the whole sequence that , and then all ; that is, all the whole sequences also converge to the fixed point.
Case b: for all and so that with . Thus, is not totally bounded, contradicting the hypotheses. Thus, Case b is not possible.
Case c:
for all
and
so that the sequence
is bounded with
. Two sub-cases were evaluated: (1)
does not converge. This is impossible since from the weak contraction property,
. Concerning (2),
and
converge to distinct limits
. Then
which implies that
which implies that
which is a contradiction. Thus,
Case c is not possible.
Case d: for some , then for any ; thus, Case a applies.
Property (ii) has been proved. The proof of property (iii) is very similar to that of property (ii) and is omitted. Property (iv) follows directly from (3) to (6) from
if
, which leads for
to
so that
if
implies that
; thus,
. □
Remark 2. Note that since a sub-contraction is not necessarily (strictly or weakly) contractive for all pairs in but just non-expansive, then the uniqueness of the fixed point is not ensured even if is compact.
Corollary 1. Assume that in Theorem 1 (i), the condition for all is replaced with for a given , all and all .
Then, the composed self-mapping is a (strict) contraction, and it has a unique fixed point in . The fixed point is also the unique fixed point of .
Sketch of Proof. It is obvious that if
for all
and any
then, for all
,
. Define
as the composed map of
times on itself, that is,
. Then, the above contractive condition is fully equivalent to
which plays the role of (16) in the proof of Theorem 1 (i) after replacing
by
. Note that if
then
which is a contradiction so that
is a fixed point of
. Assume that there are two distinct fixed points
and
. Then,
, which is a contradiction, and then
, and the unique fixed point
of
is also a unique fixed point of
. □
Theorem 1 (i) can be extended to an asymptotic contraction as follows:
Corollary 2. Assume that the contractive hypothesis of Theorem 1 (i) is weakened as follows: for all and for all and some real constants and . Then, is an asymptotic contraction, and it has a unique fixed point in .
Proof. The inequality (16) of the proof of Theorem 1 is modified as follows for any
:
which leads to
which implies that
as
since
. In addition,
is still continuous since it is globally Lipschitzian with Lipschitz constant M. Therefore, the proof of Theorem 1 (i) is extendable to this case. □
The following result is immediate from Theorem 1 [(i) and (ii)]:
Corollary 3. Consider the self-mapping of Theorem 1. The following properties hold:
- (i)
Assume that is complete, that for all and some real constant , and that the fixed point of the (strict contraction) is . Then, all sequences for any given are Cauchy and converge to ;
- (ii)
Assume that is compact and that for all , such that and that the self-mapping is a weak contraction with a unique fixed point . Thus, all the sequences are Cauchy and converge to . □
The following result relies on the inclusion of
within a special class of weak contractions, [
1,
2,
3,
4], which do not require the compactness of
but just its completeness to guarantee the existence and the uniqueness of a fixed point.
Corollary 4. Consider the self-mapping of Theorem 1. Let be a continuous non-decreasing function, such that iff , which satisfies the following constraint for all :subject to for all . Assume that (X, d) is complete. Then, is a weak contraction with a unique fixed point . Thus, all the sequences are Cauchy and converge to . Proof. It is direct since with the given constraints, it implies that if . Thus, is a type of weak contraction, and it has a unique fixed point if is complete. □
The following result considers not only the case when the iterations of the self-mapping are contractive but also that there exists a finite set of consecutive iterations where the mapping may be non-contractive (in particular, even expansive). In this case, one considers that there is another adjacent set where it is contractive with a sufficiently small contractive constant so as to neutralize the potential expansive effect of the former set. The above combination pattern is repeated as the iterations progress, while the sizes of the above respective sets can be, in general, variable. In particular, an appropriate and sufficiently drastic, contractive impulsive action can neutralize potential previous expansive effects. The second part of the result is related to stabilization by testing the values of the distances if they overpass a prescribed bound followed by an impulsive action of sufficient size to avoid unsuitable growing of the distance at the next iteration.
Theorem 2. Let be a self-mapping on , which satisfies (1)–(14), and the function , defined in (15), be piecewise continuous bounded on . Then, the following properties hold:
- (i)
Assume that is complete and that the subsequent boundedness conditions hold:where ; ; for all and some , where and , while the real constants and the real constants for are not necessarily less than one.
(C.3) , that is, the above difference of integers is finite for all (this implies from C.1–C.2 that ), and, for , (C.4) The sequence of self-mappings in is pointwise convergent in to the self-mapping , that is, for all .
Then, the following holds:
- (1)
as for all and the sequences of distances are bounded for all . Also, as for all and ;
- (2)
for any and any integers ;
- (3)
The sequences are bounded and Cauchy, then convergent in for all ;
- (4)
as and are bounded for all , , and is bounded;
- (5)
Assume pointwise convergence of to some limit mapping , that is, for and all . Then, one obtains:and is Lipschitz-continuous.
Then, and as for any for all . Equivalently, for any integer , as for any .
- (ii)
Assume that , take any and given constants , such that , where and assume that for some finite and all .
Assume also that, if for some given positive real constants, then one chooses Thus, for any , and all , one obtains , andAlso, all the distance sequences are bounded if is finite. If is a fixed point of , then , and then is bounded for any .
Proof. Then, since
,
as
for all
, and
On the other hand,
as
for all
implies that
as
. Proceed by complete induction by assuming that
and taking into account the already proved result that
as
for a given
and all
. Then, for all
,
And property (i.1) has been fully proved. On the other hand, note that
for any
so that
, and
is bounded. Furthermore,
. Then, either
(i.e.,
is a fixed point of
) so that
and the convergence is directly proved or, since
is bounded, then
if there exists a finite limit
, since
is Lipschitz-continuous in view of (25) and (26), which implies that
is convergent, and then Cauchy, to some
since
is complete.
Alternative Proof of Cauchyness (by contradiction). Assume that there is some , such that is not Cauchy. Then, for some given , there are some and some , such that , which contradicts property 1. Then, all the subsequences are Cauchy sequences convergent in since is complete. Properties (i.2) and (i.3) have been proved.
To prove property (i.4), note that
for
,
and, since
from C.3, then, for all
;
for
;
and, as
,
,
, which implies
as
for all
.
Then, property (i.4) follows from (34) and (35) and properties (i.1)–(i.3).
To prove property (i.5), note that, since
is continuous in view of (27) for
and all
(so that the operations of limit and distance can be permuted), and,
if as , as
, then if
, where
as
, one also obtains
On the other hand, from (28) and since
is Cauchy and then convergent, one obtains:
so that
, for any
, is a fixed point of
. Assume that
has another fixed point
. Thus, one obtains the subsequent contradiction if
:
so that
is the unique fixed point of
and independent of the initial
. Property (i.5) has been proved. To prove property (ii), note from (15) that
with
and then
if
, and
Thus, (30) holds and for any and any . If is a fixed point of , and since , then
for any
and any
. Thus,
is bounded since
is Lipschitz-continuous so that
□
Note that Theorem 2 (ii) guarantees the distances boundedness along the iteration procedure that generates the sequences using the appropriate impulses if the new distance exceeds an amount in order to reduce it at the next step while keeping it under the prefixed upper-bound .
Corollary 5 below extends Theorem 2 (ii) by allowing the distance to grow over several consecutive steps of the iteration before performing the large distance text for distance reduction. Furthermore, the violation of the lower-bound distance before applying the reduction might be optionally allowed on a finite number of (step-dependent) iteration steps. The proof of Corollary 5 is very close to that of Theorem 2 (ii), and it is omitted.
Corollary 5. Assume that in Theorem 2 (ii) the large distance text for distance reduction is performed after several consecutive iteration steps for and some for each any , then the constraint (29) is applied while the remaining constraints of Theorem 2 (ii) are kept.
Thus, for any , and all , one obtains , andso that all the distance sequences are bounded if is finite. If is a fixed point of then , and then is bounded for any . □
Remark 3. (1) Note that Theorem 2 (i) covers the particular cases when all iterations are contractive, with identical different contractive constants, by fixing for all ;
(2) Note that the proof of Theorem 2 can be re-arranged dually, with the necessary direct “ad hoc changes” by considering the compositions of the mapping on the subsequences instead of on the subsequences ;
(3) Note that in view of (25) and (26), the compositions of the mapping on itself are not commutative, in general, but this fact is not relevant for the validity of the results and proof.
Note that the sequence of self-mappings in is pointwise convergent in to the composed self-mapping times on itself of , that is, ; for all, if .
Remark 4. Note that it is direct to define the self-mapping from as follows: In this way, if the element of the sequence generated by has no jump, then , while if it has a jump, then , where is generated by with and . Thus, describes equivalently the sequences generated by by keeping constant consecutive values of them when jumps do not take place.
3. Applications to Stability of Time-Varying Linear Dynamic Systems Under Eventually Impulsive Parameterizations
The above results are now linked to the investigation of the stability of a class of time-varying dynamic systems. The following result is of usefulness in the practical use of Theorem 2 (ii) in the case of linear time-varying dynamic systems eventually subject to impulsive actions in their dynamic configurations. In fact, under the assumption that the matrix of dynamics is bounded everywhere, although not necessarily stable for all time, it might be achieved that the maximum increase in the solution norm along finite time intervals is bounded with a prescribed bound (in terms of norms) depending on the interval length. In the impulsive case, the above consideration remains valid with prescribed increases of the solution norm along the inter-impulsive time intervals. This concern, together with the existence of some potential configuration of the time-varying dynamics being stable, may allow the achievement of the solution trajectory boundedness in light of Theorem 2 (ii).
Theorem 3. In general, consider the following linear time-variant differential system of the n-th order:where is bounded with , and it has piecewise continuous entries. The following properties hold: If has bounded piecewise continuous entries in with , with , being an impulsive set of zero Lebesgue measures and , then the following properties hold: In particular, .
Proof. Since
, then
which implies that
if
. Property (i) is proved if
. Now, assume that
and
for some
. Then,
for
, leading to the contradiction
so that
for some,
is impossible if
, and then,
for all
. On the other hand, if
, one obtains from (51) the following contradiction:
which implies that
for all
. As a result, property (i) has been proved for any
irrespective of
being zero or not for some
. From property (i) and (45), one obtains:
then, property (ii) follows directly. Properties (ii)–(iv) follow along the inter-impulsive switching intervals under the same arguments used in the proof of properties (i)–(ii) since the essential supremum on the union of inter-impulsive intervals coincides with the finite supremum
used in the proof of Theorem 3 [(i) and (ii)], and
is of a zero Lebesgue measure, even with
, namely, if there are countably many impulsive time instants. □
Remark 5. Note that the proof of Theorem 3 does not directly apply, as given, for other vector and vector-induced matrix norms other than -norms since the auxiliary inequality:arising from (50) does not hold for all time instants in general. However, one can invoke the equivalence of norms to find its validity under very close forms for other vector norms and corresponding matrix vector-induced norms. In this way, one easily obtains the simple subsequent result for alternative norms, which implies that Theorem 3 is valid for any tandem of vector norms with corresponding vector-induced matrix norms applied to (45):
Lemma 1. The following properties hold for :
- (i)
; ; ;
- (ii)
For any vector norm , there exists , such that : ;
- (iii)
Equation (49) of Theorem 3 (iv) becomes modified as follows:for some norm-dependent where for the -norm, and for the -norm.
Proof. Thus, property (i) follows from Theorem 2 (ii) and (55) since
Property (ii) follows, since from the equivalence of norms, there exist constants
, such that any vector norm
of
satisfies
. Then, from Theorem 2 [(i)–(ii)], one obtains:
with
. Property (iii) is direct from Theorem 3 (iv) and properties (i) and (ii). □
In particular, Lemma 1 is useful to evaluate conditions (25) and (26) of Theorem 2, via Theorem 3, in the case of a linear time-varying differential system under the use of norm-induced distances other than the -vector norms. Basically, one concludes that Theorem 3 is still valid for any vector norms by changing “ad hoc” the involved real constants, which are norm-dependent in general. Note that Theorem 3 (ii) guarantees that the Constraint C.1 of Theorem 2, Equation (25), holds by identifying the vector norm with a distance from a current value of the generated solution of (45) to a concrete value that could be a suited equilibrium value, if any. Lemma 1 allows the above idea to be extended to any alternative vector norm.
The subsequent result refers to exponential stability of linear time-varying dynamic systems.
Theorem 4. Assume that has bounded entries in . The following properties hold:
- (i)
If there is a positive definite symmetric constant matrix , such that for some , the subsequent matrix inequality holds:then (45) is globally exponentially stable for any given finite initial condition; - (ii)
Let be a constant stability matrix and be defined by . Assume that for some , the subsequent matrix inequality holds for the positive definite symmetric constant matrix and some : Then, (45) is globally exponentially stable for any given finite initial condition.
Proof. Consider the Lyapunov function candidate
, which implies that
where
and
are the minimum and maximum eigenvalues of
. One obtains:
Then,
so that
leading to
Property (i) has been proved. Property (ii) follows with
, where
is the unique real symmetric
n-matrix, which satisfies the relation
where the integral converges since
is a stability matrix so that there exist real constants
and
, such that
for all
. Then,
and
under the necessary condition
. Property (ii) is proved. □
Remark 6. Note that (59) implies that is a stability matrix for each , and this fact is considered in Theorem 4 (ii).
The subsequent result refers to the exponential stability of linear time-varying dynamic systems. It addresses the stabilization of a time-varying linear system in the case of when the matrix of dynamics can have discontinuities that translate into impulsive jumps in its derivative with respect to time and are eventually combined with discontinuities of the first kind in the solution trajectory, which translate, equivalently, into the mentioned impulses in its first time derivative. Both impulsive phenomena take place at isolated time instants belonging to the respective sets and , which can be either disjointed or intersect. This result covers, in a unified fashion, the case when those impulsive sets are disjointed, the case when they are not disjointed, and the cases when one of them is empty, or none of them is empty.
Theorem 5. Consider the following linear time-variant differential impulsive system of the n-th order:where the following conditions are assumed to hold: - (1)
is bounded with , and it has piecewise continuous entries, and is bounded, and its support has zero Lebesgue measure with iff ;
- (2)
for all , where the impulsive set of is that of the discontinuities of , that is, and iff so that is bounded, and its support has zero Lebesgue measure with iff ;
- (3)
is a stability matrix for all (i.e., for some , ; for ) so that there exist real constants , such that - (4)
One of the conditions below holds for some , and all and some :
Then, the following propositions hold:
- (i)
Assume the following particular constraints on the above conditions (1)–(4):
- (a)
The entries of are bounded continuous time-differentiable functions in ;
- (b)
, that is, both and its time derivative, are nowhere impulsive in .
Thus, the following properties hold:
- (i.1)
The following Lyapunov matrix equation:has a unique positive definite bounded solution (i.e., ) for all ,which follows directly by taking time derivatives in the Lyapunov matrix equation, such that - (i.2)
; and the particular differential impulsive-free system of (68) and (69) and Equation (45) is globally uniformly exponentially stable for any given finite initial condition; - (i.3)
is an exponentially decreasing Lyapunov function which satisfies: ; ; and and as exponentially fast, where:
- (a)
under the conditions (a), Equation (71) or (c), Equation (73), one obtains:and are such that with and are the minimum and the maximum eigenvalues of , and - (b)
under condition (b), Equation (72), one obtains:
- (ii)
Under conditions (1)–(4), the impulsive differential system of (68) and (69) is globally uniformly exponentially stable for any given finite initial condition if the following additional conditions hold:for all and some and all consecutive pairs of impulsive time instants , with , where: ; if and for and ; if and .
Thus, is bounded and at an exponential rate as ;
- (iii)
Property (ii) still holds if the condition for in (76) is replaced with for some finite ;
- (iv)
Property (iii) also holds if the condition for in (76) is replaced with for some sequence of bounded positive integers .
Proof. Property (i) is proved in [
34].
To prove property (ii), note that (68) and (69) lead to their subsequent integral forms:
Now, the bounded stability matrices for all satisfy , and .
For
, consider the non-negative real function:
with a unique
, which is almost time-differentiable everywhere, being time-differentiable for
, which is the solution for the Lyapunov equation from Theorem 4, with
in (55) since (70) holds, then:
If
, then some entry (entries) of
have finite jump(s) at the time instant
according to
. If
, then
and
so that
is continuous at the time instant
since
is continuous at
, and (81) is valid at the right limit of
, that is, one obtains from (82) and (83) that
By taking time derivatives at
in (81) yields:
The time derivative of
becomes at non-impulsive time instants:
for all
, such that
if
. Note that
so that
Then, since
one obtains from (85)–(87) that
and, using (69) for
, one obtains:
so that
and, from (86) and (90),
Also, from (84) and (79), one obtains:
then,
Let
be two consecutive time instants of
, where
. Then, it follows that:
with
so that
and, under (71) or (73), one obtains:
Under (72), note from the Schwartz inequality that
and then,
Now, let
be two consecutive time instants of
. If
, then one obtains from (79)–(81) and (92) and (93), since
and
is symmetric,
where
is the inter-impulsive time interval for two consecutive impulsive time instants
, with
, which depends on the particular invoked condition (72), (73), or (74), such that:
Thus, taking into account the “ad hoc” use of the expression
from property (i.3), on an inter-impulsive time period, one can derive the subsequent expression in-between consecutive impulsive time instants:
Then, one obtains from (93)–(96) that
for all
, provided that (76)–(78) hold. To avoid division by zero in (97), it is requested that for some given positive real constant
and, since
, the above conditions leads to:
equivalently since
and equivalently, after fixing
according to the constraint
,
is selected according to the following constraint for (97) to be well-posed:
which is the first condition of (76) on the impulses of
. Note that
;
is the only equilibrium point of (79) and (80) since the matrix of dynamics is non-singular as it is a stability matrix, then
in (68) for all
so that
is the only point leading to
. As a result,
, and it is also the only fixed point of the self-mapping on
, which defines the solution trajectory from any finite initial conditions. Then, it follows from (97), by defining the distance as the
(or Euclidean) norm and fixing
, one obtains from (97) for any finite
and
that:
for all
, where
. Since the self-mapping
on
is a (strict) contraction of constant
, then
as
,
as
, and
at an exponential rate for
as
, irrespective of
or not, and
is bounded if
is bounded. Since
is continuous in
, then it cannot be unbounded in
since
for
are consecutive impulsive time instants for
irrespective of
being in
or not. Therefore, one obtains the following:
- (1)
If , then for all and as at an exponential rate;
- (2)
If
, that is, if there is a finite number of impulsive time instants, then
is bounded by the same above reasoning, while
is bounded too and
as since it follows from (77) that
with
;
if
, and
, and then
. Property (ii) has been proved. The proof of property (iii) is omitted since it is quite close to that of property (ii) based on modifying (98) as follows:
where
. The proof of property (iv) is also direct from the “ad hoc” modification of (99) by replacing
for
. □
The subsequent result addresses the way of globally stabilizing, for any given finite initial conditions, a particular case of the differential system (68) and (69) through the injection of impulses of appropriate distribution and sign on the state solution trajectory. The impulses were monitored with certain adjustable gains of appropriate signs and amplitudes and appropriate lengths of the inter-impulsive time intervals in the event that the dynamics in-between impulsive time instants was unstable, with in (77) being non-exponentially decaying but of an exponential positive order. In that context, the stability condition (3) of Theorem 5 along the inter-impulsive time intervals was replaced by either a critical stability one or an instability one, and condition (4) was removed. For exposition simplicity, it is assumed that is continuous everywhere so that and , that is, ; .
Theorem 6. Consider the time-variant differential impulsive system (68) and (69) with , that is, . Assume also that:
- (1)
is continuous in and bounded with ;
- (2)
is not a stability matrix for almost any (i.e., for some , ; for ), while it fulfils that there exist real constants , such that: - (3)
Given any , with , then if the impulsive gain matrix function is selected as diagonal and the next both fulfil the subsequent conditions for some given real constants and some constant : Then, the following properties hold:
- (i)
The differential impulsive system has a bounded solution trajectory, and then it is globally stable for any give finite initial condition if (i.e., if there are infinite many impulsive time instants). If (i.e., there is a finite number of impulses), then the differential impulsive system is not guaranteed to be globally stable unless (i.e., if the matrix function is critically stable), and the constraints (101) are not applied;
- (ii)
If and , then the differential impulsive system is globally asymptotically stable.
Proof. First note that (80) is modified as
for all
since
, and
is continuous everywhere and differentiable with respect to time. From condition (3), Equation (77) is replaced with:
for
if
and
.
Also (since
), Equation (78) is replaced with:
for and ; if and .
Note from (102) and (103) that
if
is selected as some
, which fulfils for some prescribed real constant
that:
equivalently if
with
and
. Note that
which leads to
, that is, and provided that
, to
provided that
, which guarantees that
so that (106) is well-posed. The above conditions may be rewritten as (101).
Thus, the impulsive time instants and their gains are such that (101) holds guarantee that if . Note that is the unique equilibrium point, which is the only fixed point of the mapping and defines the solution trajectory. That is, the mapping on that generates the sequence of solution of the differential system is non-expansive. Since is continuous in the disjointed union of finite intervals and is bounded if is finite, then is bounded in , which equalizes in the case of infinitely many impulses, that is, when . If , , and , then the solution can be unbounded as time tends to infinity since , and then, it is not guaranteed for a finite guaranteed that . However, the global boundedness of the solution is guaranteed if (critical stability of almost everywhere), and there is a finite number of impulses . Property (i) has been proved. Property (ii) follows since if , then ; . The self-mapping on , which constructs the solution trajectory sequence from any finite initial condition at the impulsive time instants, is then a weak contraction, which has a unique fixed point of . Then, since . Since the sequence of inter-impulsive time instants is bounded from (101), then the solution of the differential system is piecewise continuous bounded within each inter-impulsive interval and then in and as since from (100) and . □
Note that, since the fixed point is an unstable or critically stable equilibrium point of the differential system since (see Equation (100)), the global asymptotic stabilization addressed in Theorem 6 (ii) based on stabilizing through the use of infinitely many appropriate impulses is not robust in the sense that small perturbations in the parameterization may cause the stability property to become lost. The subsequent result weakens the condition of the impulsive gain matrix to be diagonal in Theorem 6.
Corollary 6. Assume that the conditions of Theorem 6 hold except that on its support set is extended to be non-diagonal of the form for λ R under the constraints:for some given real constant ; . Then, Theorem 6 holds if from some prescribed minimum inter-impulsive time interval , given , then the next is selected as follows: Proof. Note that the second constraint in (107) is equivalent to
Note by inspection that
for any
. Using the relation of norms
, Equation (104) is now replaced with
that is, for
selected subject to (109),
which is guaranteed if for all
, the next
is fixed with an inter-impulsive time interval (108). □
Note that Theorem 6 is directly extendable to the case of non-diagonal impulsive gain matrix function with non-positive entries, such that there is at least a non-zero entry per row at the impulsive time instants.
The subsequent result, based on Theorem 4 (iii), Theorem 5, and Theorem 2, addresses the stabilization of a time-varying linear system, which switches in-between a finite or infinite set of possible configurations under the assumption that at least one of such configurations is stable, and the matrix of dynamics can have discontinuities, which translate into impulsive jumps in its time derivative. As a result, the whole impulsive set of time instants might affect the matrix of dynamics and its time derivative. The next additional notation is used in the next result:
denotes the set of impulsive time instants, while denotes the impulsive subset of switches between configurations. Impulsive time instants not related to switches between the configurations are denoted by ; if they are related to a switch to the configuration , then they are denoted by ; and if they are denoted by , then they refer to switching time instants to an unspecified configuration.
Theorem 7. In general, consider the following linear time-variant differential impulsive system of the n-th order (68) and (69) subject to conditions (1) and (2) of Theorem 5 and, furthermore, to the additional conditions below:
(3) , and that , with , is a piecewise constant switching law in-between the set of parameterizations (or configurations) of , such that:
- (a)
for , where are two consecutive parameterization switching time instants with an inter-impulsive interval of , is the parameterization switching set of time instants, with and being a minimum threshold inter-impulsive interval and being the maximum inter-impulsive time interval;
- (b)
There is a nonempty (stable) configuration subset:namely, there is at least a , , which is a stability matrix for all so that there exist real constants , such that ; . It is also assumed that if for , then ; for some real constants ; - (c)
Any fulfils that any one of the conditions below holds for all and some , and , such that for some and some for .
Remark on the theorem statement: Note that the boundedness constraints:imply that there can exist either a countable finite number of discontinuities in on or infinitely many ones of vanishing jumps between the right and left limits compatible with such constraints). Then, the following properties hold:
- (i)
There exist non-unique switching rules between the configurations with both a finite or infinite number of switches, such that the state trajectory solution of (68) and (69) is bounded for any given finite initial condition, and then (68) and (69) is globally stable;
- (ii)
Assume that , , and card . Then, there exist non-unique switching rules between each proper or improper subset of the configurations with both a finite or infinite number of switches, such that the state trajectory solution of (68) and (69) is bounded for any given finite initial condition, and then (68) and (69) is globally asymptotically stable.
Proof. Since the set of stable configurations
is nonempty, it suffices for global stabilization to have, through distributed tested time instants, a residence time interval of a sufficiently large duration at a stable configuration. Given the prescribed constants
and
, if
for any
then choose the next
at a stable
i-th configuration, that is, in
, so that
for
and
according to:
Thus, if is finite, then it follows from Theorem 5 that ; , where , , , , and . It is clear that the above property holds for and . Property (i) has been proved. Property (ii) follows directly if the number of configurations is finite by choosing after a finite time to switch to a stable configuration under the assumption that the whole impulsive set of time instants is finite. □