# Attractors for Multivalued Impulsive Systems: Existence and Applications to Reaction-Diffusion System

Authors: S. Dashkovskiy; O. A. Kapustian; O. V. Kapustyan; N. V. Gorban
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this paper, we develop a general approach to investigate limit dynamics of infinite-dimensional dissipative impulsive systems whose initial conditions do not uniquely determine their long time behavior. Based on the notion of an uniform attractor, we show how to describe limit behavior of such complex systems with the help of properties of their components. More precisely, we prove the existence of the uniform attractor for an impulsive multivalued system in terms of properties of nonimpulsive semiflow and impulsive parameters. We also give an application of these abstract results to the impulsive reaction-diffusion system without uniqueness.


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## Body

## 1. Introduction

The global attractors’ theory of dissipative dynamical systems provides information about the long-term behavior of the system which is rather valuable for the design of practical systems in the framework of mathematical systems and control theory. Such information is of interest for applications, but its derivation leads to challenges from the mathematical viewpoint problems, especially in the case of nonlinear infinite-dimensional systems. It is known that, in case of a finite dimensional phase space, a dissipative continuous dynamical system possesses a global attractor, that is, an invariant and globally, uniformly attracting compact set. To prove existence of such a set for an infinite-dimensional continuous dynamical system, we have to impose additional assumption, asymptotic compactness, which can be verified in different ways for rather general classes of evolutionary processes [1–3]. Moreover, using the set-valued analysis approach [4], theory of global attractors can be applied to systems with nonunique solutions of the Cauchy problem [5–18].However, in some practical situations, the dynamics is not always continuous due to effects such as elastic collisions, sampling of signals, triggered events, and other actions causing impulsive affects [19]. In mathematical modeling of the corresponding real-world applications in mechanics engineering, biology, etc., one has to deal with hybrid systems [20], in which discrete and continuous dynamics are combined in one model. An important subclass of such systems is impulsive dynamical systems. Their continuous dynamics is described by means of a continuous semigroup as long as the solution does not touch a given set in the state space. Having reached this set, the state changes instantaneously due to an impulsive action. Then, the dynamics is governed again by the continuous semigroup. Such systems were actively studied in view of their applications in mechanics [21–23]. Recently, this framework was used due to application in biology and medicine [19]. Qualitative analysis of such systems in finite dimensional case was carried out in [24–30].The main problem we face when we try to expand the global attractor theory to impulsive DS is the lack of continuous dependence on the initial data. To overcome this difficulty, two approaches were proposed. One approach has been proposed and developed in [31–34]. The key idea of those papers is to keep the invariance property in the definition of the attractor. This approach allows to construct theory of attractors for impulsive DS similar to the classical one, but it requires additional information about impulsive trajectories in a neighborhood of the impulsive set and causes significant restrictions on their behavior.Another approach was developed in [35–38] and uses the notion of the uniform attractor. An advantage of this approach is that we work with a compact uniformly attracting set without any restrictive assumptions on the impulsive semiflow. In the present paper, we develop this approach and prove the existence of the uniform attractor for the impulsive multivalued system in terms of properties of nonimpulsive semiflow and impulsive parameters. We also give an application of these abstract results to the impulsive reaction-diffusion system without uniqueness of solutions.Let us note that this paper uses a common approach applied to study attracting sets of impulsive infinite-dimensional systems, as, for example, in [31–39]. In these and many other works, the existence of a uniform attractor is proved by means of asymptotic compactness verification for the impulsive semiflow. This verification is rather difficult already in case of relatively simple problems because the times of impulsive actions are usually not known in advance. The novelty of our results is that they provide new sufficient conditions for the asymptotic compactness property which can be verified easily (see Lemmas 2 and 3). This allows us to consider a more general class of systems with more general impulsive action classes. In particular, we were able to establish the existence of a uniform attractor for an impulsive system of N∈ℕ reaction-diffusion equations in a general form.


## 2. Setting of the Problem

Assume that we have an evolution continuous system, i.e., the familyK of continuous maps φ:0,+∞⟶X:(A1)∀x∈X∃φ∈K such that φ0=x(A2)∀φ∈K∀τ≥0φ⋅+τ∈KIn applications, the familyK is generated by autonomous evolution equation which is globally resolvable in the phase space X. Uniqueness of solution is not assumed.Impulsive systemK,M,I is an evolution system such that a phase point xt moves along trajectories of K until it reaches a fixed subset M⊂X which is called an impulsive set. At that moment τ (which is not fixed and depends on a particular trajectory), the phase point “jumps” into a new position x+τ∈Ixτ, where I:M⟶X is called the impulsive map. After that, the phase point x continues its motion along a trajectory of K up to the next meeting with M. The trivial example is a bouncing ball described by phase vector xx˙, where “jumps” means instantaneous change of x˙ when x reaches a certain surface M. Under suitable assumptions (see Definition 1), such an impulsive system can be considered as a multivalued (discontinuous) dynamical system. If this system is dissipative, its limit behavior can be described by the investigation of properties of uniform attractor, minimal compact uniformly attracting set in the phase space. The main aim of the present paper is to give effective sufficient conditions for investigation of such sets in terms of K, M, and I. Under obtained conditions, we apply to study qualitative behavior of impulsive systems generated by reaction-diffusion evolution problems.


## 3. Attracting Sets for Multivalued Impulsive Systems

LetX,⋅ be a Banach space; PXβX be a set of all nonempty (nonempty bounded) subsets of space X. Let K be an evolution continuous system on X:(1)Kx=φ∈K:φ0=x.LetM⊂X be an impulsive set and I:M⟶PX be an impulsive map. For x∈M, we denote by x+ some element of Ix. For φ∈K, we denote M+φ=∪t>0φt∩M. At the points of “jumps,” we assume impulsive trajectories to be right continuous. For “well-posedness” of our impulsive problem, we assume the following conditions:(A3)M⊂X is closed and I:M⟶PX is closed-valued(A4)M∩IM=∅(A5)∀x∈M∀φ∈Kx∃τ=τφ>0 such that ∀t∈0,τφt∉MThese assumptions allow us to claim [39] that, for every φ∈K, either M+φ=∅ or ∃s=sφ>0 such that φs∈M and φt∉M∀t∈0,s. This function s:K⟶0,+∞ (we put sφ=+∞ if M+φ=∅) will play an important role in the sequel. In particular, it allows to define impulsive trajectory φ˜.So, letx0∈X and φ0∈Kx0 be arbitrary. If sφ0=+∞, then φ˜t=φ0t∀t≥0. Otherwise, we set s0=sφ0 and put(2)φ˜t=φ0t,t∈0,s0,x1+∈Iφ0s0,t=s0.Letφ1∈Kx1+. If sφ1=+∞, then φ˜t=φ1t−s0∀t≥s0. Otherwise, we set s1=sφ1 and put(3)φ˜t=φ1t−s0,t∈s0,s0+s1,x2+∈Iφ1s1,t=s0+s1.Continuing this process, we obtain an impulsive trajectoryφ˜ with finite or infinite number of impulsive points xn+n≥1⊂IM, corresponding time intervals between jumps snn≥0, and functions φnn≥0⊂K such that ∀n≥0and∀t≥0:(4)φ˜t=φnt−tn,t∈tn,tn+1,xn+1+∈Iφnsn,t=tn+1,where t0=0 and tn+1=∑k=0nsk.Denote byK˜ (Kx˜) the set of all (starting from x) impulsive trajectories. We assume the following condition.(A6)∀x∈X and ∀φ˜∈K˜x: φ˜ is defined on 0,+∞, i.e., either the number of jumps is no more than finite (including nonimpulsive case) or ∑n=0∞sn=∞Remark 1.
According to our construction, an impulsive trajectory may start fromM but cannot belong to M at any positive time moment, i.e.,(5)∀x∈X∀φ˜∈K˜x∀t>0:φ˜t∉M.Definition 1.
A set-valued mapG˜:0,+∞×X⟶PX given by(6)G˜t,x=φ˜t:φ˜∈K˜xis called impulsive (or discontinuous) multivalued dynamical system.
It can be shown [38] that G˜ is a multivalued semiflow (m-semiflow), i.e.,(7)∀x∈XG˜0,x=x,∀t,s≥0G˜t+s,x⊆G˜t,G˜s,x.
In the last embedding, equality takes place if∀φ,ψ∈K such that φ0=ψs; the concatenation function θp=ψp,p∈0,sφp−s,p≥s belongs to K.Definition 2.
A compact setΘ⊂X is called a uniform attractor of m-semiflow G˜ if Θ is a uniformly attracting set, i.e., ∀B∈βX:(8)distG˜t,B,Θ⟶0ast⟶∞,where Θ is the minimal set among all closed uniformly attracting sets.Remark 2.
In the classical dynamical systems theory (whereG˜ is single-valued and continuous map), the uniform attractor coincides with the global attractor, i.e., compact, invariant, and uniformly attracting set. As it was shown in [37], we cannot expect the invariance of the attractor in impulsive case.
The following lemma provides criteria for a dissipative semiflow to have a uniform attractor.Lemma 1 (see [2]).
Assume that m-semiflowG˜ is dissipative, i.e., there exists B0∈βX such that, for every subset B∈βX exists, the time moment T=TB>0:(9)∀t≥TBG˜t,B⊂B0.Then,G˜ has a uniform attractor if and only if G˜ is asymptotically compact, i.e., ∀tn↗∞, and for all bounded sequences xn∈βX, every sequence ξn=G˜tn,xn has a convergent subsequence.In applications, dissipativity can be verified (only technical difficulties can appear) with the help of a priori estimates for trajectories ofK. However, verification of asymptotically compactness for the infinite-dimensional impulsive problem is much more delicate problem. This property can be false in the simplest cases.Example 1.
Let us consider the following parabolic problem in bounded domainΩ⊂ℝn:(10)∂φ∂t=Δφ,φ|∂Ω=0.
In the phase spaceX=L2Ω with L2-norm ⋅, this problem is uniquely, globally resolvable and globally exponentially stable, i.e.,(11)∀t≥0φt≤e−λ1tφ0,where λ1>0 is the first eigenvalue of the operator −Δ in H01.
Let us consider impulsive parameters:(12)M=φ∈X:φ=ε,ε>0,I:M⟶X,Iφ=1+μφ,μ>0.
It can be proved [35] that ∀ε>0 and ∀μ>0 such that the impulsive system satisfies assumptions (A1)–(A6), and it is dissipative. However, it is not asymptotically compact. Therefore, it does not possess the uniform attractor.
The reason is that every trajectory meets the impulsive setM too often, so the set of impulsive points is not compact.
For further consideration, we need additional assumptions which take place for the most evolution systemsK.(A7) For any sequencexn, xn⟶x as n⟶∞, and for any φn∈Kxn, there exists φ∈Kx such that, on some subsequence, φnt⟶φt∀t≥0(A8)∀r>0∃c1r such that ∀x,x≤r, ∀φ∈Kx, and ∀t≥0: φt≤c1rLemma 2.
Assume that impulsive systemK,M,I satisfies conditions (A1)–(A8), the corresponding m-semiflow G˜ is dissipative, and there exists compactly embedded space Y⋐X such that(A9)∀t>0 and ∀r>0 exist c2t,r such that ∀x, x≤r and ∀φ∈Kx; it is true that(13)φtY≤c2t,r.(A10)∀r>0∃c3r such that ∀x∈M∩Y, xY≤r, and ∀x+∈Ix; it is true that(14)x+Y≤c3r.(A11)∃s¯>0 such that ∀x∈IM∩Y and ∀φ∈Kx; it is true that sφ≥s¯.Then, the impulsive m-semiflowG˜ has a uniform attractor.Remark 3.
Y⋐X implies that ∃α>0 such that ∀x∈Yx≤αxY, and if xnY≤c, then, up to subsequence, xn⟶x in X.Remark 4.
Assumption (A9) means thatK generates a compact semigroup.Proof.
Dissipativity of m-semiflowG˜ implies that ∀r>0∃Tr such that ∀x, x≤r, ∀φ˜∈K˜x, and ∀t≥Tr: φ˜t≤R0. Moreover, combining (A8), (A10), and (A11), we get that ∃cr such that ∀t∈0,Trφ˜t≤cr. So, we can deduce that(15)∀r>0∃crsuch that∀x,x≤r,∀φ˜∈K˜x,∀t≥0φ˜t≤cr+R0≕Dr.
Due to Lemma1, we have to prove that the set,(16)ξm=φ˜mtm:φ˜m∈K˜xm,xm≤r,tm↗∞,is precompact in X.
So, we consider an impulsive trajectoryφ˜m which consists of impulsive points xnm+n≥1⊂IM∩Y, time intervals snmn≥0, and functions φnmn≥0⊂K. We have the following alternatives.(a)
Iftm<s0m=sφ0m≤∞, then(17)ξm=φ˜mtm=φ0mtm=φ0m1+tm−1,φ0m⋅+tm−1∈Kφ0mtm−1,φ0mtm−1≤Dr.Hence, due to (A9), we have that(18)ξmY≤c21,Dr.So,ξm is precompact in X.(b)
Iftm=sφ0m, then ξm=φ˜mtm∈Iφ0mtm. So, due to (A10), ξmY≤c3c21,Dr, and ξm is precompact in X.(c)
Iftm>sφ0m, then, for some n=nm≥1,(19)ξm=φ˜mtm=φnmtm−∑k=0mskm,φnm∈Kxnm+,or ξm=xn+1m+, and due to (15) and (A10), ∀n≥1,(20)xnm+Y≤c3Dr.
Let us denoteτm=tm−∑k=0mskm, ηm=xnm+. Then, up to subsequence, ηm⟶η in X.
Ifτm⟶τ≥0, then, due to (A7), for some φ∈Kη, up to subsequence,(21)φnmτm⟶φτasm⟶∞.
So,ξm is precompact.
Otherwise,τm⟶∞. Then, arguing analogously as at to the beginning of the proof, we obtain(22)ξmY≤c21,Dr.Lemma 3.
Assume that impulsive systemK,M,I satisfies conditions (A1)–(A8), the corresponding m-semiflow G˜ is dissipative, and there exists a compactly embedded space Y⋐X.(A9 ′)∀x∈X, ∀φ∈Kx, and ∀t≥0,(23)φt=φ1t+φ2t,where(24)φ1tY≤c4x,c4⋅is continuous on0,+∞,φ2t≤c5xe−δt,whereδ>0,andc5≥0are constants.(A10 ′)∃x¯∈X: ∀r>0∃c6r such that ∀x∈M, x≤r, and ∀x+∈Ix; it is true that(25)x+−x¯Y≤c6r.(A11 ′)∃s¯>0 such that ∀x∈IM and ∀φ∈Kx; it is true that sφ≥s¯.Then, the impulsive m-semiflowG˜ has a uniform attractor.Remark 5.
Assumption (A9′) means thatK generates an asymptotically compact semigroup.Proof.
The alternatives (a)–(c) of the proof are the same as in the proof of Lemma2. We consider an impulsive trajectory φ˜m with components xnm+n≥1⊂IM, snmn≥0, φnmn≥0⊂K, and φ˜m0≤r, and (15) takes place.(a)
Iftm<s0m=sφ0m≤∞, then, due to (A9’),(26)ξm=φ˜mtm=φ0mtm=φ0m1tm+φ0m2tm,whereφ0m1tmY≤c4r and φ0m2tm≤c5re−δtm.So, the first summand has a convergent subsequence, and the second one tends to zero asm⟶∞. Therefore, ξm is precompact.(b)
Iftm=s0m, then(27)ξm=φ˜tm∈Iφ0mtm.Due to case (a), we have that(28)φ0mtm≤αc4r+c5r.So, from (A11’),(29)ξm−x¯Y≤c6αc4r+c5r.It means precompactness ofξn in X.(c)
Iftm>sφ0m, then, for some n=nm≥1,(30)ξm=φ˜tm=φnmτm,φnm∈Kηm,where τm=tm−∑k=0mskm and ηm=xnm+.
Due to (A10’),(31)ηm−x¯Y≤c6Dr.
So, up to subsequence,ηm⟶η in X. After that, we can repeat arguments from Lemma 2 and finish the proof.


## 4. Application to Reaction-Diffusion Impulsive System

We consider the reaction-diffusion type system:(32)∂u∂t=AΔu+εfu,t>0,x∈Ω,u|∂Ω=0,where Ω⊂ℝd is a bounded domain, ε>0 is a small parameter, u=u1,…,uNT is unknown vector-function, A is N×N matrix with positive symmetric part A+AT/2≥βI,β>0, f=f1,…,fNT is continuous nonlinear interaction vector-function, and(33)supu∈ℝNfuℝN≤C.It is known [2] that, under such assumptions, for every u0∈X=L2ΩN, there exists (maybe nonunique) global weak solution of problem (32), and every weak solution belongs to ℂ0,+∞,X.We consider the following impulsive problem:(34)M=u∈X:∑i=1Nui,ψ2=1,where ψ is chosen as one of solutions of the spectral problem:(35)Δψ=−λψ,ψ∈H01Ω.We putL=c¯ψ:c¯∈ℝN. Then, X=L⊕L⊥, and we put, for u∈M, u=c¯ψ+u⊥, u⊥∈L⊥, ∑i=1Nc¯i2=1, and(36)Iu=v=d¯ψ+u⊥:∑i=1Nd¯i2=1+μ.Remark 6.
In the case,N=1, the described impulsive problem consists of increasing 1+μ times the value u,ψ2.
LetK be the set of global weak solutions of problem (32).Theorem 1.
The impulsive systemK,M,I has the uniform attractor in the phase space X.Proof.
Let us verify assumptions from Lemma2. Within this proof constant, C means a constant which depends on parameters of (32).
It is known thatK satisfies (A1), (A2), (A7), and (A8). Moreover, for every u∈K,(37)∀t≥0utX2≤u0X2e−βt+1.
Additionally, forY=H01ΩN⋐X, assumption (A9) takes place. Let us take u∈M∩Y. Then,(38)uY2=c¯ψY2+u⊥Y2.
And, for arbitraryu+∈Iu,(39)u+Y2=d¯ψY2+u⊥Y2.
So, ifuY2≤r2, then(40)u+Y2≤r2+ψY2∑i=1Nd¯i2−∑i=1Nc¯i2≤r2+μψY2.
Hence, (A10) takes place.
Also, it is easy to see thatM⊂X is closed, ∀u∈XIu is closed, and M∩IM=∅. So, (A3) and (A4) hold.
For verifying other assumptions, we need more detailed analysis of our system. Let us introduce denotations:(41)v¯t=u1t,ψ…9uNt,ψ,f¯ut=f1u,ψ…fNu,ψ.
Then, from (32), we obtain(42)ddtv¯t+λAv¯t=εf¯ut.
And, after multiplying byv¯t in ℝN, we obtain(43)12ddtv¯tℝN2+λAv¯t,v¯tℝN=εf¯ut,v¯tℝN.
Hence,(44)ddtv¯tℝN2+λA+ATv¯t,v¯tℝN=2εf¯ut,v¯tℝN.
So,∀t≥0:(45)∑i=1Nuit,ψ2≤∑i=1Nui0,ψ2e−βλt+ε2C.
Now, let us consider(46)gεt=v¯tℝN2=∑i=1Nuit,ψ2.
Here, indexε means that ut=u1t,…,uNtT is a solution of problem (32) with ε>0.
Assume thatu0∈M. Then, gε0=1, and due to (44), we have that(47)gε′0≤−2βλ+2εC.
Inequality (47) means that, for sufficiently small ε>0, it is true that gε′0≤−βλ. So, for such ε>0∃τ=τu0,ε: ∀t∈0,τgεt<1. Hence, (A5) takes place.
Now, let us takeu0∈IM. Assume that sε>0 is the first meeting of M by u∈Ku0. Then, gεsε=1, i.e., due to (44),(48)1+λ∫0sεA+ATv¯t,v¯tℝN=1+μ+2ε∫0sεf¯ut,v¯tℝN.
Due to (45), ∀t∈0,sε:(49)v¯tℝN2≤1+μ+ε2C≤2+μ,for sufficiently small ε.
So,(50)A+ATv¯t,v¯tℝℕ≤2A2+μand(51)2λA2+μsε≥μ−2εC2+μsε.
Finally,(52)sε≥μ2λA2+μ+2C2+μ≕s¯.
Therefore, (A11) holds.
It is left to prove dissipativity property. Let us considerφ˜∈K˜u0 with u0≤r.
Ifφ˜ has not impulsive perturbation, then, from (37),(53)∀t≥T=2βlnr:φ˜tX≤2.
Otherwise, without loss of generality, we can assume thatu0∈IM, u0≤r, and φ˜ has jumps at the moments s0,s0+s1,… with impulsive points yi+i≥1. Then, using (52) and equality u+=u2+μ∀u∈M, we have that, for k≥0,(54)uk+1+2≤u02e−βk+1s¯+1+μ∑i=0ke−βis¯.
Combining (53) and (54), we obtain required dissipativity. The theorem is proved.



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