Stability conditions for impulsive dynamical systems

In this work, we consider impulsive dynamical systems evolving on an infinite-dimensional space and subjected to external perturbations. We look for stability conditions that guarantee the input-to-state stability for such systems. Our new dwell-time conditions allow the situation, where both continuous and discrete dynamics can be unstable simultaneously. Lyapunov like methods are developed for this purpose. Illustrative finite and infinite dimensional examples are provided to demonstrate the application of the main results. These examples cannot be treated by any other published approach and demonstrate the effectiveness of our results.


Introduction
Impulsive dynamical systems provide a mathematical modeling framework for practical processes where a combination of continuous and discrete dynamics takes place. Such a hybrid dynamics appears in many applications, for example, in case of mechanical collisions or in control systems involving a combination of analog and digital controllers. As well in pandemic systems, a mass vaccination can be modeled as an impulsive action meaning a (nearly instantaneous) transition of a large amount of susceptible individuals to become immune.  Vitalii Slynko vitalii.slynko@uni-wuerzburg.de 1 Institute for Mathematics, University of Würzburg, Emil-Fischer-Straße 40, 97074 Würzburg, Germany A combination of discrete and continuous dynamics leads to a higher complexity in the behavior of solutions compared with a purely discrete or purely continuous system. Such unexpected effects as instability [27] or chaos [14] can arise. In particular, such properties as stability and robustness are more difficult to investigate, especially in case of nonlinear systems.
Stability in the sense of Lyapunov of nonlinear impulsive systems has a long history of investigations, see [17,23]. Later, more general stability notions were developed for hybrid systems, which include impulsive ones as a particular case, see [12,28]. These notions use the generalized (hybrid) time concept, which allows to develop rather general results for a wide class of hybrid systems, including impulsive, switched and sampled data systems.
In case of systems having input signals, the notion of input-to-state stability (ISS) was introduced in [26] and was found very fruitful in many applications [16]. This framework was also successfully used for studying robust stability of impulsive systems, see [4,6,13,18,22]. In particular, [6] derives dwell-time conditions to establish the ISS property for nonlinear impulsive systems on infinite dimensional state spaces. This result is based on certain stability assumptions imposed either on the continuous or on the discrete dynamics. The ISS is assured by the stability property of either of both dynamics using a suitable dwell-time condition. Stability of interconnected impulsive systems is then studied in the case, when the ISS-Lyapunov functions are known for the subsystems, which leads to a combination of the dwell-time and small-gain conditions. The ISS property of impulsive systems where impulsive actions depend on time was studied in [4], where new and rather general dwell-time conditions were developed.
A small-gain theorem for n ≥ 2 interconnected hybrid systems was established in [22] for the case where not all subsystems are assumed to be ISS, which extends the results of [6]. Similar results for nonlinear interconnected impulsive systems were developed in [8] for the case of absent external perturbations. This work derives sufficient stability conditions for interconnected systems by means of vector Lyapunov functions, which leads to conditions similar to the small-gain ones.
In most of works (e.g., in those mentioned above) studying stability of impulsive systems by means of Lyapunov methods, it is assumed that the discrete and continuous dynamics share a common Lyapunov function V which decays either on jumps or along the continuous flow. A dwell-time condition allows to compensate the destabilizing effect of one type of dynamics by the stabilizing property of the other one. Certainly, if V increases in both cases, then the system is unstable. However, in general, it can happen that the whole system is asymptotically stable even in the case when both discrete and continuous dynamics are unstable. Identification of such systems needs a more refined consideration of the interaction between both dynamics types. It is expected that stability conditions become more involved in this case. It should be noted that there are not many stability results in the literature that cover the case of simultaneous instability of discrete and continuous dynamics: [1,[9][10][11]19,24], see also [3] and [2] for the linear case. However, these results cannot be extended directly to the case of nonlinear infinite-dimensional systems with inputs, also we note that only local stability was studied in the first group of these papers.
Vector Lyapunov functions were used in [19] to establish stability results, where second-order derivatives of Lyapunov functions along solutions enter to the stability conditions. These results were generalized in [9,10], where higher-order derivatives of Lyapunov functions are employed. This approach cannot be used in case of systems with inputs as one would need to require infinitely smooth disturbances, which is very restrictive in real applications.
Averaged dwell-time conditions were considered in [1,24] to establish stability of a linear impulsive system on a Banach space. Based on the identities from the commutator calculus new comparison theorems, constructions of Lyapunov functions and conditions for the local asymptotic stability were developed there. In this work, we are interested in global stability properties for nonlinear systems with inputs.
With an exception of [22], in the most of works devoted to investigations of the ISSlike properties of hybrid systems, it is assumed that either the continuous or discrete dynamics satisfies the ISS property. Hence, it is interesting to further develop the direct Lyapunov method for the case, when both types of dynamics fail to be ISS. Our paper contributes into this direction, providing a new approach and new stability conditions.
In this work, we improve the results of [9,10] and extend them to the case of ISS for nonlinear impulsive systems. We derive stability conditions by means of a series of Lyapunov-like functions. Instead of higher-order derivatives of Lyapunov functions employed in [9,10], we use an infinite sequence of auxiliary functions to provide estimates of the dwell-time in order to guarantee the ISS property. The obtained results are then applied to the studying of the global asymptotic stability of linear impulsive systems with continuous dynamics governed by a parabolic PDE. The ISS property is also studied for this type of systems. Moreover, we derive conditions for the ISS property of nonlinear locally homogeneous finite-dimensional impulsive systems.
The paper consists of six sections. Section 2 introduces the notation and several auxiliary inequalities used in the paper. The problem statement is described in Sect. 3. Section 4 contains the main results with their proofs. Application of the results to the investigation of GAS and ISS properties of linear impulsive systems in infinitedimensional spaces and of nonlinear finite-dimensional systems is provided in Sect. 5. A brief discussion and conclusions are collected in Sect. 6. Proofs of several technical results are placed in Appendix.
For M ⊂ R and a Banach space X by L ∞ (M, X ), we denote the space of For a linear bounded operator A acting on a Banach space, σ (A) denotes the spectrum of A and r σ (A) denotes its spectral radius.
R n×m is the space of n × m-matrices, for m = n the set R n×n is then a Banach algebra. We use the norm on R n×n induced by the Euclidean norm in R n : We will use the following well-known inequalities. For any a, b ∈ R + and ∈ K holds for

Problem statement and related stability notions
We consider dynamical systems with inputs defined similarly to [5,21,26] as follows Definition 1 Let X be the state space with the norm · X and U 1 ⊂ { f : R → U 1 } be the space of input signals normed by · U 1 with values in a nonempty subset U 1 of some linear normed space and invariant under the time shifts, that is, if 1 for some positive t 0 ,x,d 1 and satisfies the following axioms signals normed by · U 2 and taking values in a nonempty subset U 2 of some linear normed space. Let g : X × U 2 → X be a mapping defining impulsive actions and the mapping φ be defined for all (t, The We will denote for short The conditions (Σ c 1) and (Σ 2 ) imply and (Σ c 4), (Σ c 5), (Σ 2 ) imply that for t ≥ τ ≥ t 0 , (x, d 1 , d 2 ) ∈ X × U 1 × U 2 , the following holds: The system Σ c describes the continuous dynamics of the impulsive system Σ. One can also consider its discrete dynamics separately as a system Σ d defined next Definition 3 A discrete dynamical system with input Σ d = (X , g, φ d , U 2 ) is given by a normed state space (X , · X ); a space of input signals U 2 ⊂ { f : Z + → U 2 } with norm · U 2 and values in a nonempty subset U 2 of a linear normed space; a mapping g : X × U 2 → X ; and a mapping φ d : Assumption For any τ ≥ 0, there exist ξ, ξ τ ∈ K ∞ and χ τ , χ ∈ K ∞ such that for all and This assumption is not restrictive; for example, if g is continuous, then (6) is satisfied. We are interested in the stability properties of the system Σ and its robustness with respect to the input signals d 1 and d 2 . To this end, we use the notion of input-to-state stability (ISS). It was originally introduced in [25] for time invariant finite-dimensional systems. In our case, we adapt it as follows for Σ Definition 4 For a fixed time sequence E of impulsive actions, the system Σ is called ISS if there exist β t 0 ∈ KL, γ t 0 ∈ K ∞ , such that for any initial state x ∈ X , any t ≥ t 0 and any (d 1 , Next, we introduce a class of functions that we will use as Lyapunov functions to study the ISS property.
if it satisfies the following properties:

Main results
In this section, we provide sufficient conditions to guarantee the ISS property for Σ. Complementary stability conditions are given in Theorems 1 and 2, respectively. Application of these results are illustrated in Sect. 5.

Remark 1
The inequality (12) in condition (4) of the theorem restricts the time intervals between jumps, which is a dwell-time condition. If both continuous and discrete dynamics are stable, the existence of W k and V p so that W k+1 (x) and are negative is guaranteed, so that (12) implies no restrictions on E. However if one of the dynamics is unstable, the dwell-time is restricted. For example, if the discrete dynamics is stable but the continuous one is not, we have W k+1 (x) < 0, but the second summand in (12) can be positive, and hence it needs to be small enough in order to satisfy (12). This implies that the time distances τ k+1 − τ k need to be small enough. Moreover, (12) allows the situation, where both dynamics types are not stable. This will be illustrated in our examples.
Let us fix any (t 0 , x 0 ) ∈ R × X as the initial data and assume without loss of generality that t 0 ≤ τ 0 . For any disturbances (d 1 , d 2 ) ∈ U 1 × U 2 , the corresponding solution to Σ will be denoted by Lemma 1 Under the conditions of Theorem 1 for any k ∈ Z + , n ≥ 2 the following inequality holds true: The proof can be found in Appendix.

Corollary 1
Under the conditions of Theorem 1, the following estimate is true: Proof The estimate follows after taking the limit for n → ∞ in the inequality (13), which is possible under the conditions of Theorem 1.
Proof (of Theorem 1) The combination of (11) and (14) implies that From this inequality together with condition (4) of Theorem 1, it follows that for all Let us show by contradiction that for some k * ∈ Z + , it holds that x(τ k * ) ∈ B r (0). Assume that this is not true, that is for all k ∈ Z + , we have x(τ k ) ≥ r , then from (15) follows: This means that the bounded from below sequence which leads to a contradiction. Hence, for some k * , it holds that x(τ k * ) < r . Let hence, by condition (1) of Theorem 1, we obtain Taking (6) into account, we obtain Hence, ) and from (15), it follows that The inequality (6) implies that It is easily seen that ϕ ∈ K ∞ , hence x(τ k ) ≥ ϕ −1 ( x(τ k + 0) ) and from (16) follows We denote , and conclude that for all k, 1 ≤ k ≤ N , the inequality (17) can be written as Let us define the sequence v k for k ∈ N by v k := v k for 1 ≤ k ≤ N and so that for Together with the inequality (19), we consider the comparison equation First, let us show that for all k ∈ Z + , the inequality v k ≤ w k is true. Indeed, if for some (20) can be written as Let δ 2 (s) := min{s, δ 2 (s)}, then We define By the mean value theorem for some s * ∈ (w k , w k−1 ) from (22), we obtain This means that for all k, 0 ≤ k ≤ N , we can estimate That is for all t ∈ (τ k , τ k+1 ], 0 ≤ k ≤ N − 1, the following inequality holds: Let It is easily seen that β ∈ KL. From (23), we get for all t ∈ (τ 0 , τ N ], the inequality Applying the estimates and the inequality (1), we obtain that for some functions ξ τ 0 −t 0 , Applying again (1), we derive From (24) using again (1), we get that for all t ∈ (τ 0 , τ N ], the inequality holds, where we have denoted It easy to check by definition that β t 0 ∈ KL, χ t 0 ∈ K ∞ . There exists a function Recall that for k ≥ N , we have x(τ k + 0) ≤ R. Hence from the estimate x(t) ≤ ξ θ ( x(τ k + 0) ) + χ θ (d), t ∈ (τ k , τ k+1 ] it follows that whereχ ∈ K ∞ by definition. Combining the inequalities (26) and (27), we see that for some γ t 0 ∈ K ∞ , the following estimate holds which proves the theorem. (5) of Theorem 1 assures the convergence of three last terms in (14) for n → ∞. However, if V n (t, x) ≤ 0 for some n ≥ 1, then we can set V p (t, x) ≡ 0 for all p > n and this condition (5) can be dropped. Stability investigation in this case is essentially easier because we deal with a finite number of auxiliary functions instead of an infinite sequence. The class of systems, where this simplification is possible becomes wider due to the next result (see the difference in the sign beforė V p in conditions (2) of the previous and the next theorem). Such simplification will be used in some of our examples later.

Lemma 2 Under the conditions of Theorem 2, we have for all k
Corollary 2 Under the conditions of Theorem 2, the following holds: Proof Condition (6) of Theorem 2 assures the possibility to take the limit for n → ∞ in (33) which implies the assertion.
Proof (of Theorem 2) From (30) and (34) follows Taking (31) into account, we obtain Obviously η ∈ K ∞ and from (32) and (36) follows (recall that d = Let ∈ (0, 1), r = δ −1 ( η(d) 1− ). First we show by contradiction that there exists k * ∈ Z + such that x(τ k * ) < r . Indeed, otherwise, for all k ∈ Z + x(τ k ) ≥ r and hence from (37), it follows that This means that the sequence {V 0 (τ k , x(τ k ))} k∈Z + is strictly decreasing and is bounded from below, hence it possesses a nonnegative limit m * = lim k→∞ V 0 (τ k , x(τ k )). From (37) follows which leads to a contradiction. We denote R := max{(α −1 where 1 ≤ j(m) ≤ ∞, then from (37), we have that for i ≥ 2 and hence from (5) and by the condition (1) of Theorem 2, it follows that Taking (6) into account, we obtain Similarly, for i = 1 we get Let S r := {k ∈ Z + : ∀l, 0 ≤ l ≤ k x(τ l ) ≥ r } and N = max S r , for S r = ∅, 0 for S r = ∅ It is enough to consider the case N ≥ 1. Let k be such that 0 ≤ k ≤ N , then η(d) ≤ (1 − )δ( x(τ k ) ), and from (37), the next inequality follows: By condition (1) of Theorem 2, we obtain x(τ k ) ≥ α −1 2 (V 0 (τ k , x(τ k ))) and hence (38) can be written as Hence, Δ v 0 ( v k ) ≥ k for all k ∈ Z + . From its definition, it follows that Δ v 0 (s) is strictly growing with respect to v 0 for a fixed s and strictly decreasing in s for a fixed v 0 with Δ v 0 (s) → ∞ for s → 0+. This implies that The desired ISS estimate follows similarly as in the proof of Theorem 1, which proves Theorem 2.

Remark 3
If for some n ≥ 1 we have V n (t, x) ≤ 0, then we can take V p (t, x) ≡ 0 for all p > n, and in this case, condition (5) of Theorem 2 can be dropped.

Heat equation with variable coefficients
Let X be the normed linear vector space defined by with l > 0, 0 < 2δ < l and the norm · X := · L 2 [0,l] . Consider the following linear system with impulsive actions with the following initial and boundary conditions u(0, t) = u(l, t) = 0, where ( and u satisfies the initial and boundary conditions (41).
Since c ∈ C 2 [0, l], it follows that the state space X is invariant under jumps, that is, f ∈ X implies (1 + c) f ∈ X . It is also clear that the jump operator is consistent with the boundary conditions. Theorem 1 from [15] implies the existence and uniqueness of solutions to (40)-(41) so that u(·, t) ∈ X for t ≥ 0.
We consider the question of asymptotic stability of solutions to (40)-(41) with respect to the norm in L 2 [0, l]. The next proposition verifies the estimates (5)-(6).

Proposition 1 Solutions of (40)-(41) satisfy the estimates (5)-(6) with the following choice of functions
The proof can be found in Appendix.
To study stability properties, we introduce the following Lyapunov function: for which we havė u(·, t)).

(42)
Let us calculate the full time derivative of V 1 (u(·, t)) along solutions to (40)-(41): Hence, we can apply Theorem 1 taking V 2 = 0. From (40) for t ∈ E, we obtain With help of Theorem 1, we arrive to the following conditions for the global asymptotic stability for some ε 0 > 0, where T k = τ k+1 −τ k denotes the dwell-time between two impulsive actions. By means of the Friedrich's inequality, we arrive to the following condition guaranteeing the GAS property of the system (40)-(41) For illustration, we apply this condition to the following particular case of (40)-(41): where ε > 0, δ < π/2, c ∈ C 2 , −2ε ≤ c(z) ≤ ε . We will see that for some choice of parameters ε, δ both continuous and discrete dynamics have unstable behavior.
In this case, the dwell-time condition (45) reduced to the following two inequalities: or equivalently to the following explicit condition applied on the dwell-times Let us consider continuous and discrete dynamics of (40)-(41) separately. First, we consider the stability properties of the differential equation with the following boundary and initial conditions The corresponding self-adjoint spectral problem is We can show that at least for ε small enough there exists some critical δ * , (δ * ≈ 0.651331) such that for all δ < δ * the linear system (47)-(48) is not stable, and for δ > δ * , this system is asymptotically stable. This follows immediately from the next proposition proved in Appendix:
This mens that for sufficiently small ε, the parameter δ can be chosen so that the partial differential equation (47)-(48), which describes the continuous dynamics of the original impulsive system (40)-(41), is unstable.
We consider the difference equation in the state space X , that describes the discrete dynamics of the original impulsive system (40)-(41), and demonstrate its unstable behavior. Let Cψ := (1 + c(z))ψ for ψ ∈ X , then C ∈ L(X ) and showing the instability property. Remark 4 Let us note that in case of an unstable scalar ODE of the first order subjected to destabilizing impulsive actions, the overall dynamics of the whole impulsive system is always unstable. In contrary to this, our example shows that for the considered unstable PDE with destabilizing impulses the overall dynamics is stable under certain condition on dwell times.

Nonlinear impulsive ODE
Consider the following nonlinear ODE system with impulsive actionṡ where x ∈ R n , f , g : R n → R n are homogeneous functions of the order m > 1 with odd m, that is, Let there exist a positive definite matrix P such that for all x ∈ R n such that the function f , g satisfies the inequalities for some positive constants b 0 , c 0 , ν 0 and ν 1 . For short, we denote

Proposition 3 Solutions of the system (50) satisfy the estimates (5)-(6) with functions
where ω i , i = 1, 2 are certain positive functions (provided explicitly in the proof).
The proof can be found in Appendix. Stability properties will be studied with help of the following Lyapunov function V 0 (x) = x T Px, P 0. Its time derivative along solutions to (50) for t = kθ iṡ By means of the Young's inequality (2), we have for any τ > 0. Hence from (52), we obtaiṅ We chose On the jumps due to impulsive actions from the definition of V 0 , we calculate Again by the Young's inequality (2), we obtain From (51), it follows that |g T (x)Pg(x)| ≤ P b 2 0 x 2m . Hence from (55), the following estimate follows: By means of the Cauchy inequality and from (51), it follows that Hence, Now, we apply the Young's inequality with the parameter τ > 0 and obtaiṅ Denoting we can writė .

Proposition 4 If
then for τ > 0 small enough and for all s ∈ R + , the next inequality is true The proof can be found in Appendix.

Proposition 5
Let system (50) satisfy the conditions of (51) as well as the inequality
To illustrate this result, we take It is easy to see that conditions of the last proposition are satisfied. Hence, the nonlinear impulsive system (50) is ISS. Note that in this case, both continuous and discrete dynamics are unstable.

Discussion and conclusions
The main results of this paper are given by Theorems 1 and 2. They allow us to study the ISS property of nonlinear impulsive systems with different assumptions imposed on the discrete and continuous dynamics. These theorems can establish the ISS property even for the case when neither discrete nor continuous dynamics are ISS. Our approach enables usage of a wider class of ISS-Lyapunov functions to study the ISS property of nonlinear impulsive systems. One advantage of our approach, demonstrated in the examples, is that a rather simple (energetic) Lyapunov function equipped with a sequence of auxiliary functions allows to derive desired stability condition to assure the ISS or GAS property.
An interesting direction for future research would be to develop an approach of stability investigation of nonlinear impulsive systems by means of a combination of a Lyapunov and Chetaev functions, which was used for studying local stability of finite-dimensional systems without inputs [11]. This can provide stability conditions alternative to the ones developed in the current paper. First steps in this direction can be found in [7].
Another interesting direction of research is to explore an extension possibility of our results to the strong ISS notion in the context of time varying impulsive systems (see [20]).
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Proof of Lemma 1
The proof is by induction as follows. First, we check that the statements is true for n = 2 from the obvious equality and the first inequality of (10) for s ∈ (τ k , τ k+1 ]: From the next two obvious relations we obtain the desired inequality Now, let the statement be true for n = q, that is Similarly to the previous step, we write and from the inequalities (10), we obtain We substitute this estimate into (59) and obtain the desired inequality which proves the Lemma.
We use the following two basic facts from analysis that enable us to write for ε → 0+ the next equalities.