A survey on impulsive dynamical systems

Electronic Journal of Qualitative Theory of Differential Equations Proc. 10th Coll. Qualitative Theory of Diff. Equ. (July 1–4, 2015, Szeged, Hungary) 2016, No. 7, 1–27; doi: 10.14232/ejqtde.2016.8.7 http://www.math.u-szeged.hu/ejqtde/ A survey on impulsive dynamical systems Everaldo Mello Bonotto1,Matheus C. Bortolan2, Tomás CaraballoB3and Rodolfo Collegari1 1Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Campus de São Carlos, Caixa Postal 668, São Carlos, SP, Brazil 2Departamento de Matemática, Universidade Federal de Santa Catarina, Campus Trindade, 88040-900, Florianópolis, Brazil 3Departamento de Ecuaciones Diferenciales y Análisis Numérico EDAN, Universidad de Sevilla, Sevilla, Spain Appeared 11 Agust 2016 Communicated by Tibor Krisztin Abstract. In this survey we provide an introduction to the theory of impulsive dynamical systems in both the autonomous and nonautonomous cases. In the former, we will show two different approaches which have been proposed to analyze such kind of dynamical systems which can experience some abrupt changes (impulses) in their evolution. But, unlike the autonomous framework, the nonautonomous one is being developed right now and some progress is being obtained over the recent years. We will provide some results on how the theory of autonomous impulsive dynamical systems can be extended to cover such nonautonomous situations, which are more often to occur in the real world. Keywords: impulsive dynamical systems, global attractors, nonautonomous dynamical systems, cocycle attractors, Navier–Stokes equation. 2010 Mathematics Subject Classification: 35B41, 34A37, 35R12. 1 Introduction The theory of impulsive differential equations (IDE, for short) describes the evolution of systems where the continuous development of a process is interrupted by abrupt changes of state. These systems are modeled by differential equations which describe the period of continuous variation of state and conditions which describe the discontinuities of first kind of the solution or of its derivatives at the moments of impulses. Many real world problems can experience abrupt external forces which can change completely their dynamics. For instance, an example of a real world problem that can be represented by an impulsive differential equation is a medicine intake, where the user must take regular doses of the medicine, which causes abrupt changes in the amount of medicine in their body, to control the disease or making it disappear. Examples that model real world problems in science and technology can be found BCorresponding author. Email: [email protected] 2E. M. Bonotto, M. C. Bortolan, T. Caraballo and R. Collegari in [1,13,19,20]. The reader is also referred to [2,3,26] to obtain more details about the theory of IDEs, for instance, results concerning existence and uniqueness of solutions, dependence of solutions on initial values, variation of parameters, oscillation and stability. As pointed out in [2,26] there exist different kinds of impulses, for instance, systems with impulses at fixed times and systems with impulses at variable times. Impulses that vary in time are more attractive due to their complexity, applicability in real world problems, and, moreover, the impulses may occur due to conditions on the phase space and not in time. As an example, we may cite the billiard-type system which can be modeled by differential systems with impulses acting on the first derivatives of the solutions. Indeed, the positions of the colliding balls do not change at the moments of impact (impulse), but their velocities gain finite increments (the velocity will change according to the position of the ball). Solutions of IDEs with impulses at variable time may generate “impulsive dynamical systems” (family of piecewise continuous functions that satisfy the identity and semigroup properties), for instance, when the differential equation is autonomous. As in the theory of IDEs, the case of impulsive dynamical systems with impulses that vary in time is more difficult to handle since we do not know previously the time of impulses. However, it provides us an effective tool to describe more types of discontinuous motions. The theory of impulsive dynamical systems is a new chapter of the theory of topological dynamical systems and it was started by Rozko in the papers [27,28], where he introduced several notions of impulsive systems with impulses at fixed times. In the early 90’s Kaul (see [24,25]) constructed the mathematical base for this theory with impulses at variable times, and has been followed by several authors in order to develop the theory which is known up to date. For instance, we would like to mention the papers by Ciesielski (see [16–18]), where it is analyzed the continuity of the function φ(see 2.2) that describes “the time of reaching impulse points”, and recently the works by Bonotto and his collaborators (see [6–10]) where the theory has been investigated. Throughout this work, an impulsive dynamical system is a dynamical system that possesses impulses depending on the state (and not on the time), that is, there is a set in the phase space which is responsible by the discontinuities of the solutions of the system. It is worth mentioning that the theory presented in this work provides a different approach from the theory presented in [21], where the author carries out a study of some types of discontinuous differential equations. Roughly speaking, Filippov considers in [21] the equation x0=f(t,x), where the right-hand side function is discontinuous and it is assumed to satisfy some Carathéodory conditions. Also, the solutions in this framework have to be absolutely continuous, which is another relevant detail that makes Filippov’s theory different from the one presented in [2,3,26] and the theory presented here, where the solutions can be (and usually are) discontinuous. We aim to provide a survey on the theory of impulsive dynamical systems in both the autonomous and nonautonomous fields. We start with the autonomous framework which has being studied over the last years and, for the first part of this paper, we will recall some results established in the paper [5]. In this work the authors propose a new approach for the impulsive autonomous theory, by considering precompact attractors and pointing out several improvements that this precompact approach provides, when comparing with the previous theory in this framework. Examples to illustrate the impulsive autonomous theory are described in [5], one of them is reproduced in this survey, at the end of the section devoted to the autonomous case (see Example 2.22). To start off, in Section 2we include some basic definitions from the continuous au- A survey on impulsive dynamical systems 3 tonomous dynamical systems theory in order to introduce the definition of impulsive dynamical system. In the sequel we present some technical definitions and results, known as “tube conditions”, that is important in the development of this theory. Then, before presenting an impulsive autonomous example, we introduce the concept of omega limit sets, which is the key to construct the global attractor, as well as some results on the invariance and attraction in order to obtain an existence result for the global attractor. In Section 3, we analyze the nonautonomous case, taking into account that a complete description of the results and their proofs can be found in our paper [4], while in this survey we only intend to provide the main ideas of the new theory highlighting the difficulties that one can have in dealing with this much more complicated nonautonomous situation. Needless to say that most problems in the real world are, by their own nature, nonautonomous (or even stochastic) and, when we wish to mathematically analyze them, we usually approximate those problems by some autonomous models to simplify the study. However, even being the autonomous framework very useful, and providing a great amount of results, it does not take into account the whole richness of nonautonomous problems. In [11,12], one can find examples to illustrate how different the autonomous and nonautonomous settings can be. Mentioning again the medicine intake example, we could not expect that the action of the medicine in the user body depends only on the elapsed time but also the initial and final times must play their role in the evolution of the system. We follow the same structure than in the autonomous part, by starting with a brief introduction on the continuous nonautonomous dynamical systems in order to define the impulsive nonautonomous dynamical systems. We also state a result (see Theorem 3.9) that is important to transfer properties from the impulsive skew-product semiflow (autonomous) to the impulsive nonautonomous dynamical system. Next we present the nonautonomous version of the “tube conditions” and some convergence properties, which are more general than the first ones because take into account a second variable (the fibers). Then we define the notion of impulsive cocycle attractor and impulsive pullback omega limit, and also present some results about invariance and attraction. We would like to mention that the definition of impulsive pullback omega limit set introduced in [4] is a little different from the previous one, and this difference appears naturally when we start developing the impulsive nonautonomous theory, since in the impulsive scenario, the convergence results are obtained with some “correction times” (see Proposition 3.12). To conclude, we present, under suitable conditions, a result on the existence of impulsive cocycle attractor for an impulsive nonautonomous dynamical system and an example, borrowed from [4, Section 7], where a nonautonomous 2D-Navier–Stokes equation under impulses conditions is considered. Finally, some conclusions, comments and future lines of research are included in Section 4. 2 Impulsive dynamical systems To introduce the theory of impulsive dynamical system, we first recall, very briefly, the theory of continuous autonomous dynamical systems (or simply, semigroups). Let (X,d)be a metric space and R+be the set of nonnegative real numbers. A semigroup in Xis a family of mappings {π(t):t⩾0}, indexed on R+, satisfying (i) π(0)x=xfor all x∈X; (ii) π(t+s) = π(t)π(s)for all t,s⩾0; (iii) the map R+×X3(t,x)7→ π(t)xis continuous. 4E. M. Bonotto, M. C. Bortolan, T. Caraballo and R. Collegari A set A⊂Xis called π-invariant under {π(t):t⩾0}if π(t)A=Afor all t⩾0. Also A is π-positively (negatively) invariant if π(t)A⊆A(π(t)A⊇A), for all t⩾0. Given two subsets A,B⊆X, we say that Aπ-attracts Bif lim t→+∞dH(π(t)B,A) = 0, where dH(·,·)denotes the Hausdorff semidistance between two sets, i.e., dH(C,D) = sup x∈C inf y∈D d(x,y). A set A ⊂ Xis called a global attractor for the semigroup {π(t):t⩾0}if it is compact, π-invariant and π-attracts all bounded subsets of X. In this section, we present the definitions and basic properties of the impulsive dynamical systems theory (see [5–7,16,17] for more details). Let {π(t):t⩾0}be a semigroup in X. For each D⊆Xand J⊆R+we define F(D,J) = [ t∈J π(t)−1(D). A point x∈Xis called an initial point if F(x,t) = ∅for all t>0. Now we are able to define the impulsive dynamical systems. An impulsive dynamical system (IDS, for short) (X,π,M,I)consists of a semigroup {π(t):t⩾0}on a metric space (X,d), a nonempty closed subset M⊆Xsuch that for every x∈Mthere exists ex>0 such that F(x,(0, ex)) ∩M=∅and [ t∈(0,ex) {π(t)x} ∩ M=∅, (2.1) and a continuous function I:M→Xwhose action will be explained below in the description of the impulsive trajectory. Condition (2.1) is outlined in the next figure. Figure 2.1: The flow of the semigroup {π(t):t⩾0}is, in some sense, transversal to M. The set Mis called impulsive set and the function Iis called impulsive function. We also define M+(x) = [ t>0 π(t)x!∩M A survey on impulsive dynamical systems 5 and the function φ:X→(0, +∞]by φ(x) = (s, if π(s)x∈Mand π(t)x/∈Mfor 0 <t<s, +∞, if M+(x) = ∅.(2.2) If M+(x)6=∅, the value φ(x)represents the first positive time such that the trajectory of x meets M. In this case, we say that the point π(φ(x))xis the impulsive point of x. Remark 2.1. The definition of the function φabove makes sense thanks to the following result. See [5,24]. Proposition 2.2. Let (X,π,M,I)be an IDS and x ∈X. If M+(x)6=∅then there exists s >0such that π(s)x∈M and π(t)x/∈M for 0<t<s. Now let us construct the impulsive trajectory of the IDS. Definition 2.3. The impulsive trajectory of x∈Xby the IDS (X,π,M,I)is a map ˜ π(·)x defined in an interval Jx⊆R+, 0 ∈Jx, taking values in Xwhich is given inductively by the following rule: if M+(x) = ∅, then ˜ π(t)x=π(t)xfor all t∈R+. However, if M+(x)6=∅ then we denote x=x+ 0and define ˜ π(·)xon [0, φ(x+ 0)] by ˜ π(t)x=(π(t)x+ 0, if 0 ⩽t<φ(x+ 0), I(π(φ(x+ 0))x+ 0), if t=φ(x+ 0). Now let s0=φ(x+ 0),x1=π(s0)x+ 0and x+ 1=I(π(s0)x+ 0). In this case s0<+∞and the process can go on, but now starting at x+ 1. If M+(x+ 1) = ∅, then we define ˜ π(t)x=π(t−s0)x+ 1for s0⩽t<+∞and in this case φ(x+ 1) = +∞. However, if M+(x+ 1)6=∅we define ˜ π(·)xon [s0,s0+φ(x+ 1)] by ˜ π(t)x=(π(t−s0)x+ 1, if s0⩽t<s0+φ(x+ 1), I(π(φ(x+ 1))x+ 1), if t=s0+φ(x+ 1). Now let s1=φ(x+ 1),x2=π(s1)x+ 1and x+ 2=I(π(s1)x+ 1). Assume now that ˜ π(·)xis defined on the interval [tn−1,tn]and that ˜ π(tn)x=x+ n, where t0=0 and tn=∑n−1 i=0sifor n∈N. If M+(x+ n) = ∅, then ˜ π(t)x=π(t−tn)x+ nfor tn⩽t<+∞and φ(x+ n) = +∞. However, if M+(x+ n)6=∅, then we define ˜ π(·)xon [tn,tn+φ(x+ n)] by ˜ π(t)x=(π(t−tn)x+ n, if tn⩽t<tn+φ(x+ n), I(π(φ(x+ n))x+ n), if t=tn+φ(x+ n). Now let sn=φ(x+ n),xn+1=π(sn)x+ nand x+ n+1=I(π(sn)x+ n). This process ends after a finite number of steps if M+(x+ n) = ∅for some n∈N, or it may proceed indefinitely, if M+(x+ n)6=∅for all n∈Nand in this case ˜ π(·)xis defined in the interval [0, T(x)), where T(x) = ∑+∞ i=0si. 6E. M. Bonotto, M. C. Bortolan, T. Caraballo and R. Collegari Figure 2.2: System (X,π)with Figure 2.3: Impulsive trajectory of x continuous trajectories. in the system (X,π,M,I). Remark 2.4. • We will always assume that all impulsive trajectories exist for all time t⩾0, i.e., T(x) = +∞for all x∈X, since we are interested in the asymptotic behavior of impulsive dynamical systems. • A simple consequence of the definition of impulsive trajectories is that if we assume that I(M)∩M=∅, then no point x∈Mis in any impulsive ˜ π-trajectory, except if the trajectory starts at x. The definitions of ˜ π-invariance and ˜ π-attraction are analogous to the notions of π-invariance and π-attraction, respectively, simply replacing πby ˜ π. 2.1 Tube conditions on impulsive dynamical systems In order to obtain some results in the impulsive theory of dynamical systems (for example, invariance and attraction results), we must ensure that the continuous semiflow possesses a nice behavior near the impulsive set Mand, for this purpose we introduce the so-called “tube conditions”. They are important to deduce a result ensuring the negative invariance of impulsive ω-limits. For more details and proofs see also [5,16,18]. Definition 2.5. Let {π(t):t⩾0}be a semigroup on X. A closed set Scontaining x∈Xis called a section through xif there exists λ>0 and a closed subset Lof Xsuch that: (a) F(L,λ) = S; (b) F(L,[0, 2λ]) contains a neighborhood of x; (c) F(L,ν)∩F(L,ζ) = ∅, if 0 ⩽ν<ζ⩽2λ. We say that the set F(L,[0, 2λ]) is a λ-tube (or simply a tube)and the set Lis a bar. A survey on impulsive dynamical systems 7 2λ λ π(x,λ) x LS q q Figure 2.4: Tube F(L,[0, 2λ]). Definition 2.6. Let (X,π,M,I)be an IDS. We say that a point x∈Msatisfies the strong tube condition (STC), if there exists a section Sthrough xsuch that S=F(L,[0, 2λ]) ∩M. Also, we say that a point x∈Msatisfies the special strong tube condition (SSTC)if it satisfies STC and the λ-tube F(L,[0, 2λ]) is such that F(L,[0, λ]) ∩I(M) = ∅. We finish this part presenting two proposition. The first one yields to a better understanding about the behavior of impulsive trajectories near the impulsive set Mand will be useful to obtain some results later. It states that the impulsive flow ˜ π(t)cannot reach the “right side” of the impulsive set Mfor large values of t. The second proposition summarizes some important convergence results that also will be useful to obtain further results. For details and proofs the reader may see [5]. Proposition 2.7 ([5]).Let (X,π,M,I)be an IDS such that I(M)∩M=∅and let y ∈M satisfy SSTC with λ-tube F(L,[0, 2λ]). Then ˜ π(t)X∩F(L,[0, λ]) = ∅for all t >λ. Proposition 2.8. Let (X,π,M,I)be an IDS. (i) Suppose that I(M)∩M=∅and each point of M satisfies STC. Let x ∈X\M and let {xn}n∈Nbe a sequence in X such that xnn→+∞ −→ x. Then, given t ⩾0, there exists a sequence {ηn}n∈N⊆[0, +∞)such that ηnn→+∞ −→ 0and ˜ π(t+ηn)xnn→+∞ −→ ˜ π(t)x. (ii) Suppose that each point in M satisfies STC. Let x ∈X\M and let {xn}n∈Nbe a sequence in X \M such that xnn→+∞ −→ x. Then if αnn→+∞ −→ 0and αn⩾0, for all n ∈N, we have ˜ π(αn)xnn→+∞ −→ x. (iii) Let z ∈M satisfy STC with λ-tube F(L,[0, 2λ]). Assume that there exists a sequence {zn}n∈N such that zn∈F(L,(λ, 2λ]) and znn→+∞ −→ z. Then there exist a subsequence {znk}k∈Nof {zn}n∈Nand a sequence {ek}k∈Nsuch that ek>0and ek→0as k →+∞, yk=π(ek)znk∈ M, φ(znk) = ekand yk k→+∞ −→ z. 2.2 Attractors We start with a first approach about attractors for the IDS. In [6], the authors propose the following definition of global attractor for an impulsive dynamical system. 8E. M. Bonotto, M. C. Bortolan, T. Caraballo and R. Collegari Definition 2.9. A compact subset Aof Xis a global attractor for an IDS (X,π,M,I)if the following conditions are fulfilled: (i) A ∩ M=∅; (ii) Ais ˜ π-invariant; (iii) A˜ π-attracts all bounded subsets of X. Remark 2.10. 1. This definition is consistent with the notion of a global attractor for semigroups, that is, when M=∅, both definitions coincide; and in fact, this notion of a global attractor is useful to describe the asymptotic dynamics of ˜ πin many cases. 2. Since Ais a compact set and Mis a closed set, condition (i) implies that there exists a positive distance between Aand M. Then the asymptotic behavior of the impulsive dynamical systems is qualitatively not different from the asymptotic behavior of the original dynamical system, thus, this notion does not consider some IDS. Let us see an example borrowed from [5] to illustrate these facts. Example 2.11. Consider the following continuous differential equation ˙ x=(1, if x<0, 1−x, if x≥0, (2.3) with the initial condition x(0) = x0∈Rand consider the action of the impulsive function I(0) = −1. The solutions of (2.3) without the action of Iare given by π(t)x0=       t+x0,x0<0, t∈[0, −x0), −e−t−x0+1, x0<0, t∈[−x0,+∞), (x0−1)e−t+1, x0⩾0, t∈[0, +∞). This problem has only one bounded invariant set; namely the asymptotically stable equilibrium solution {1}, and it is also the global attractor for (2.3). Now, the solutions of (2.3) with the action of I, are given by ˜ π(t)x0=       t+x0,x0<0, t∈[0, −x0), t+x0−n,x0<0, t∈[−x0+n−1, −x0+n),n∈N, (x0−1)e−t+1, x0≥0, t∈[0, +∞). (2.4) We can see that the dynamics is quite different, since there appeared the “impulsive periodic orbit” [−1, 0). Note that in this case there is no subset of Rsatisfying all the conditions of Definition 2.9. But we can distinguish some interesting sets: • The set A1= [−1, 0)∪ {1}is ˜ π-invariant and ˜ π-attracting bounded sets, A1∩M=∅, but A1is not compact. • The set A2= [−1, 0]∪ {1}˜ π-attracts bounded sets, A2is compact, but A2∩M6=∅and A2is neither ˜ π-positively nor ˜ π-negatively invariant. A survey on impulsive dynamical systems 9 • The set A3= [−1, 1]˜ π-attracts bounded sets, A3is compact, it is ˜ π-positively invariant, but it is not ˜ π-negatively invariant and A3∩M6=∅. Inspired by the ideas from this last example, in [5] the authors provide another definition of global attractor, in order to cover a larger class of impulsive dynamical systems. Their definition is the following. Definition 2.12. A subset A ⊂ Xwill be called a global attractor for the IDS (X,π,M,I)if it satisfies the following conditions: (i) Ais precompact and A=A \ M; (ii) Ais ˜ π-invariant; (iii) A˜ π-attracts bounded subsets of X. Remark 2.13. • The main difference between Definition 2.12 and Definition 2.9 is the compactness. In Definition 2.12, the global attractor does not need to be compact and now the attractor can “touch” the impulsive set M, while compact sets which do not intersect Mhave to be at a positive distance from M. • It is easy to see that, with Definition 2.12, if Aexists, it is unique. • We recall now that a function ψ:R→Xis a global solution of ˜ πif ˜ π(t)ψ(s) = ψ(t+s), for all t⩾0 and s∈R. Moreover, if ψ(0) = xwe say that ψis a global solution through x. Then, with Definition 2.12, if the IDS (X,π,M,I)possesses a global attractor Aand I(M)∩M=∅we have A={x∈X: there exists a bounded global solution of ˜ πthrough x}. Coming back to Example 2.11, we can see that set A1is the global attractor for the IDS, according to Definition 2.12. This example shows how different the continuous and the impulsive dynamics can be, as well as that a very large amount of impulsive dynamical systems, which do not fit the theory in [6], can now be considered. In what follows we will present some definitions and results to ensure the existence of a global attractor for an IDS (X,π,M,I)as defined in Definition 2.12. We will include a sketch of some proofs and for all the details the reader may see [5]. We start giving the definition of impulsive ω-limit. Definition 2.14. We represent the impulsive positive orbit of x∈Xstarting at s⩾0 by the set ˜ γ+ s(x) = {˜ π(t)x:t⩾s}. Also we set ˜ γ+(x) = γ+ 0(x). Given a subset B⊆Xwe define ˜ γ+ s(B) = Sx∈B˜ γ+ s(x)and we define the impulsive ω-limit of Bas the set ˜ ω(B) = \ t⩾0 ˜ γ+ t(B) 16 E. M. Bonotto, M. C. Bortolan, T. Caraballo and R. Collegari Proposition 3.12. Let [(ϕ,θ)(X,Σ),M,I]be an INDS. (i) Suppose that I(M)∩M=∅and each point of M satisfies ϕ-STC. Let also x ∈X\M and {xn}n∈Nbe a sequence in X such that xnn→+∞ −→ x. Then, given t ⩾0,σ∈Σand a sequence {σn}n∈N⊂Σwith σnn→+∞ −→ σ, there exists a sequence {ηn}n∈N⊆[0, +∞)such that ηnn→+∞ −→ 0and ˜ ϕ(t+ηn,σn)xnn→+∞ −→ ˜ ϕ(t,σ)x. (ii) Suppose that each point in M satisfies ϕ-STC, x ∈X\M, σ∈Σ,{xn}n∈Nbe a sequence in X\M such that xnn→+∞ −→ x and σnn→+∞ −→ σ. Then if αnn→+∞ −→ 0and αn⩾0, for all n ∈N, we have ˜ ϕ(αn,σn)xnn→+∞ −→ x. (iii) Assume that each x ∈M satisfies ϕ-SSTC and I(M)∩M=∅. Let ˆ B be a nonautonomous set, {tn}n∈N⊂R+,σ∈Σ,{ηn}n∈N⊂R+and {xn}n∈Nbe sequences such that ηnn→+∞ −→ 0, xn∈B(θ−tnσ)for each n ∈N. If {˜ ϕ(tn+ηn,θ−tnσ)xn}n∈Nis convergent with limit y∈M and {en}n∈N⊂R+is a sequence with enn→+∞ −→ 0, then there is a subsequence {˜ ϕ(tnk+ηnk,θ−tnkσ)xnk}k∈Nsuch that φ(˜ ϕ(tnk+ηnk,θ−tnkσ)xnk,θηnkσ)k→+∞ −→ 0and either ˜ ϕ(enk,θηkσ)˜ ϕ(tnk+ηnk,θ−tnkσ)xnk k→+∞ −→ y or ˜ ϕ(enk,θηkσ)˜ ϕ(tnk+ηnk,θ−tnkσ)xnk k→+∞ −→ I(y). In particular, ˜ ϕ(αk,θηkσ)˜ ϕ(tnk+ηnk,θ−tnkσ)xnk k→+∞ −→ I(y), where αk=φ(˜ ϕ(tnk+ηnk,θ−tnkσ)xnk,θηnkσ). 3.2 Impulsive cocycle attractors Here we will present the notion of attractor for an INDS (impulsive cocycle attractor), define and establish some properties of the impulsive omega limit sets in order to obtain an existence result of impulsive cocycle attractors. We will see that this notion of attractor is not a natural generalization of the global attractor given in [5] (see Definition 2.12), since the results on the invariance in the impulsive case cannot be obtained as a natural generalization of the continuous case. A more complete analysis can be found in [4] and some of the proofs will be reproduced here to illustrate the techniques. Let us introduce the notion of attractor for an INDS (with respect to a universe). Definition 3.13. Given a universe D, a compact nonautonomous set ˆ Ais called a D-impulsive cocycle attractor for the INDS [(ϕ,θ)(X,Σ),M,I]if: (i) ˆ A\M={A(σ)\M}σ∈Σis ˜ ϕ-invariant; (ii) ˆ Ais (˜ ϕ,D)-pullback attracting; (iii) ˆ Ais minimal, that is, if ˆ Cis a closed nonautonomous set satisfying (ii), then A(σ)⊆C(σ) for each σ∈Σ. Remark 3.14. Note that, in the trivial case (i.e., Σ={σ}), the definition of the cocycle attractor reduces to a compact set Asuch that A\Mis invariant and attracts bounded sets of X A survey on impulsive dynamical systems 17 which is not the definition of a global attractor for the autonomous case, as given in Definition 2.12. We again emphasize that the nonautonomous framework is more challenging than the autonomous one, and so, it is reasonable that we find more restrictive conditions in the definition of impulsive cocycle attractors. Now we state the definition of the impulsive omega limit set along with its characterization. Definition 3.15. Given a nonautonomous set ˆ B. ={B(σ)}σ∈Σand σ∈Σwe define the impulsive pullback omega-limit of ˆ Bat the fiber σas the set ˜ ω(ˆ B,σ) = \ s≥0[ t≥s[ e∈[0,s−1) ˜ ϕ(t+e,θ−tσ)B(θ−tσ) and the impulsive pullback omega-limit of ˆ Bas the nonautonomous set ˜ ω(ˆ B). ={˜ ω(ˆ B,σ)}σ∈Σ. Lemma 3.16. It follows that ˜ ω(ˆ B,σ) = nx∈X:there exist sequences {tn}n∈N,{en}n∈N⊆R+and {xn}n∈N⊆B(θ−tnσ) with tnn→+∞ −→ +∞,enn→+∞ −→ 0such that ˜ ϕ(tn+en,θ−tnσ)xnn→+∞ −→ xo and ˜ ω(ˆ B,σ)is closed. Remark 3.17. Note that if M=∅, then ˜ ω(ˆ B,σ) = ω(ˆ B,σ), for each nonautonomous set ˆ B. Definition 3.18. An INDS [(ϕ,θ)(X,Σ),M,I]is said to be pullback D-asymptotically compact, if for each σ∈Σ,ˆ D∈Dand sequences {tn}n∈N⊆R+,{xn}n∈N⊂Xsuch that tnn→+∞ −→ +∞ and xn∈D(θ−tnσ), implies that the sequence {˜ ϕ(tn,θ−tnσ)xn}n∈Npossesses a convergent subsequence. Definition 3.19. A nonautonomous set ˆ Bis said to be pullback D-absorbing for the INDS [(ϕ,θ)(X,Σ),M,I], if for each σ∈Σand ˆ D∈D, there exists t0=t0(σ,ˆ D)⩾0 such that ˜ ϕ(t,θ−tσ)D(θ−tσ)⊆B(σ)for all t⩾t0. Following the same scheme as the autonomous case, we will present results on the impulsive pullback omega limit and finish this section giving a result on the existence of a impulsive cocycle attractor. The results can be found in [4, Section 4 and Section 5] and we will include some of the proofs in order to illustrate the techniques. Proposition 3.20 ([4, Proposition 4.8]).If the INDS [(ϕ,θ)(X,Σ),M,I]is pullback D-asymptotically compact, each point of M satisfies ϕ-SSTC, I(M)∩M=∅,ˆ B∈Dand σ∈Σ, then the nonautonomous set ˜ ω(ˆ B)is nonempty, compact and pullback attracts ˆ B, that is, for each σ∈Σ lim t→+∞dH(˜ ϕ(t,θ−tσ)B(θ−tσ),˜ ω(ˆ B,σ)) = 0. 18 E. M. Bonotto, M. C. Bortolan, T. Caraballo and R. Collegari Proof. Let σ∈Σand take sequences tnn→+∞ −→ +∞(tn≥0)and xn∈B(θ−tnσ),n∈N. By the asymptotic compactness, the sequence {˜ ϕ(tn,θ−tnσ)xn}n∈Nhas a convergent subsequence for a point x∈X. It is easy to verify that x∈˜ ω(ˆ B,σ), which proves that ˜ ω(ˆ B,σ)is nonempty. Now, since ˜ ω(ˆ B,σ)is closed, to show its compactness it is sufficient to prove that if {zn}n∈N⊆˜ ω(ˆ B,σ)is a sequence, then we can obtain a convergent subsequence. So, let {zn}n∈N⊆˜ ω(ˆ B,σ), then for each n∈N, one can obtain sequences {tn k}k∈N⊂R+,{en k}k∈N⊂ R+and {xn k}k∈N⊂B(θ−tn kσ)such that tn k k→+∞ −→ +∞,en k k→+∞ −→ 0 and ˜ ϕ(tn k+en k,θ−tn kσ)xn k k→+∞ −→ zn. Thus, there is a natural kn≥nsuch that d(˜ ϕ(tn kn+en kn,θ−tn knσ)xn kn,zn)≤1 n. Note that ˜ ϕ(tn kn+en kn,θ−tn knσ)xn kn=˜ ϕ(en kn,σ)˜ ϕ(tn kn,θ−tn knσ)xn kn, for n,k∈N. Since [(ϕ,θ)(X,Σ),M,I]is pullback D-asymptotically compact, we may assume without loss of generality that there is w∈Xsuch that ˜ ϕ(tn kn,θ−tn knσ)xn kn n→+∞ −→ w. If w/∈M, then using item (ii) of Proposition 3.12, we get ˜ ϕ(en kn,σ)˜ ϕ(tn kn,θ−tn knσ)xn kn n→+∞ −→ w, which shows that zkn n→+∞ −→ w. If w∈M, we may assume by item (iii) of Proposition 3.12 that either ˜ ϕ(en kn,σ)˜ ϕ(tn kn,θ−tn knσ)xn kn n→+∞ −→ w or ˜ ϕ(en kn,σ)˜ ϕ(tn kn,θ−tn knσ)xn kn n→+∞ −→ I(w), which shows that {zn}n∈Nadmits a convergent subsequence. Now, assume that the last statement does not hold, that is, there exist σ∈Σ,e0>0 and sequences tnn→+∞ −→ +∞and zn∈B(θ−tnσ)such that d(˜ ϕ(tn,θ−tnσ)zn,˜ ω(ˆ B,σ)) ⩾e0,n∈N. But ˜ ϕ(tn,θ−tnσ)znn→+∞ −→ xfor some x∈Xalong some subsequence. Clearly x∈˜ ω(ˆ B,σ) and 0=d(x,˜ ω(ˆ B,σ)) ⩾e0, which gives us a contradiction and proves the result. Proposition 3.21 ([4, Proposition 4.9]).Let [(ϕ,θ)(X,Σ),M,I]be an INDS such that I(M)∩M=∅ and each point of M satisfies ϕ-STC. Then for any nonempty nonautonomous set ˆ B, its impulsive ω-limit ˜ ω(ˆ B)\M. ={˜ ω(ˆ B,σ)\M}σ∈Σis positively ˜ ϕ-invariant. A survey on impulsive dynamical systems 19 Proof. Fix σ∈Σand t⩾0. Let x∈˜ ω(ˆ B,σ)\M. Then there exist {tn}n∈N,{en}n∈N⊂R+and {xn}n∈N⊆B(θ−tnσ)with tnn→+∞ −→ +∞,enn→+∞ −→ 0 such that ˜ ϕ(tn+en,θ−tnσ)xnn→+∞ −→ x. Since x/∈Mand Mis closed, we may assume that ˜ ϕ(tn+en,θ−tnσ)xn/∈Mfor all n∈N. Therefore, by item (i) of Proposition 3.12, there exists a sequence {ηn}n∈N⊂Rsuch that ηnn→+∞ −→ 0 and ˜ ϕ(tn+t+ηn+en,θ−(t+tn)θtσ)xn=˜ ϕ(t+ηn,θenσ)˜ ϕ(tn+en,θ−tnσ)xnn→+∞ −→ ˜ ϕ(t,σ)x. Hence, ˜ ϕ(t,σ)x∈˜ ω(ˆ B,θtσ). If t=0, there is nothing to do. If t>0 observe that ˜ ϕ(t,σ)x/∈M, since any impulsive trajectory starting at a point of X\Mnever reaches Min finite time (note that I(M)∩M=∅). This shows the positive ˜ ϕ-invariance of ˜ ω(ˆ B)\M. Before establishing the negative invariance for impulsive pullback ω-limit sets, we need an auxiliary result. Lemma 3.22 ([4, Lemma 4.10]).Let [(ϕ,θ)(X,Σ),M,I]be an INDS with I(M)∩M=∅. Assume that every point from M satisfies ϕ-SSTC and let ˆ B be a nonautonomous set. If y ∈˜ ω(ˆ B,σ)∩M then I(y)∈˜ ω(ˆ B,σ)\M. Proposition 3.23 ([4, Proposition 4.11]).Let [(ϕ,θ)(X,Σ),M,I]be an INDS with I(M)∩M=∅. Assume that every point from M satisfies ϕ-SSTC and let ˆ B be a nonautonomous set. If ˜ ω(ˆ B)is compact and pullback attracts ˆ B, then ˜ ω(ˆ B)\M is negatively ˜ ϕ-invariant. Proof. Let t⩾0, σ∈Σand x∈˜ ω(ˆ B,θtσ)\M. Then there exist sequences {tn}n∈N,{en}n∈N⊂ R+and {xn}n∈N⊆B(θ−tn+tσ)with tnn→+∞ −→ +∞and enn→+∞ −→ 0 such that ˜ ϕ(tn+en,θ−tn+tσ)xnn→+∞ −→ x. Now, since ˜ ω(ˆ B)is compact and pullback attracts ˆ Band we have item (ii) of Proposition 3.12, we can assume that {˜ ϕ(tn−t+en,θ−tn+tσ)xn}n∈Npossesses a convergent subsequence (which we denote by the same notation and we already assumed that tn>t, since tnn→+∞ −→ +∞and tis fixed). Thus yn. =˜ ϕ(tn−t+en,θ−tn+tσ)xnn→+∞ −→ y∈˜ ω(ˆ B,σ). Case 1: y∈X\M. By item (i) of Proposition 3.12, there exists a nonnegative sequence ηnn→+∞ −→ 0 such that ˜ ϕ(t+ηn,θenσ)ynn→+∞ −→ ˜ ϕ(t,σ)y. But ˜ ϕ(t+ηn,θenσ)yn=˜ ϕ(tn+en+ηn,θ−tn+tσ)xn, and using item (ii) of Proposition 3.12 we know that ˜ ϕ(t+ηn,θenσ)ynn→+∞ −→ x. Therefore, x=˜ ϕ(t,σ)y∈˜ ϕ(t,σ)( ˜ ω(ˆ B,σ)\M). Case 2: y∈M. In this case, using item (iii) of Proposition 3.12 and Lemma 3.22, we obtain a subsequence {ynk}k∈Nsuch that γk=φ(ynk,θenkσ)k→+∞ −→ 0 and z+ k . =˜ ϕ(γk,θenkσ)ynk k→+∞ −→ I(y). =z∈˜ ω(ˆ B,σ)\M. Now, by item (i) of Proposition 3.12, there exists a non-negative sequence αk k→+∞ −→ 0 such that ˜ ϕ(t+αk,θγk+enkσ)z+ k k→+∞ −→ ˜ ϕ(t,σ)z. 20 E. M. Bonotto, M. C. Bortolan, T. Caraballo and R. Collegari But ˜ ϕ(t+αk,θγk+ekσ)z+ k=˜ ϕ(tnk+enk+γk+αk,θ−tnk+tσ)xnkand again, using item (ii) of Proposition 3.12, we have ˜ ϕ(t+αk,θγk+enkσ)z+ nk k→+∞ −→ x. Therefore, x=˜ ϕ(t,σ)z∈˜ ϕ(t,σ)( ˜ ω(ˆ B,σ)\M). Now we present a result which guarantees the existence of an impulsive cocycle attractor. Theorem 3.24 ([4, Theorem 5.1]).Let [(ϕ,θ)(X,Σ),M,I]be an INDS pullback D-asymptotically compact such that I(M)∩M=∅and every point from M satisfies ϕ-SSTC. Assume that there exists a pullback (˜ ϕ,D)-absorbing nonautonomous set ˆ K∈D. Then, the nonautonomous set ˆ A defined by A(σ) = ˜ ω(ˆ K,σ) is a D-impulsive cocycle attractor for the INDS [(ϕ,θ)(X,Σ),M,I]. Proof. By Proposition 3.20 we have ˆ Ais nonempty, compact and pullback D-attracts ˆ K. The invariance of ˆ A\Mfollows from Proposition 3.21 and Proposition 3.23. Since ˆ Kis pullback (˜ ϕ,D)-absorbing then we deduce that ˆ Kis (˜ ϕ,D)-pullback attracting. Suppose there exists a nonautonomous closed set ˆ Cthat pullback D-attracts every nonautonomous set ˆ B∈D. Since ˜ ω(ˆ B)\Mis ˜ ϕ-invariant, we have dH(˜ ω(ˆ B,σ)\M,C(σ)) = dH(˜ ϕ(t,θ−tσ)˜ ω(ˆ B,θ−tσ)\M,C(σ)) t→+∞ −→ 0, that is, ˜ ω(ˆ B,σ)\M⊆C(σ), for every ˆ B∈Dand σ∈Σ. Now, let x∈˜ ω(ˆ B,σ)∩M. Then there exist sequences {tn}n∈N,{en}n∈N⊂R+and xn∈ B(θ−tnσ)with tnn→+∞ −→ +∞,enn→+∞ −→ 0 such that ˜ ϕ(tn+en,θ−tnσ)xnn→+∞ −→ x. Let zn. = ˜ ϕ(tn,θ−tnσ)xn,n∈N. We may assume that znn→+∞ −→ z∈˜ ω(ˆ B,σ). By items (ii) and (iii) of Proposition 3.12, we have (possibly, taking subsequences) either (1) ˜ ϕ(en,σ)znn→+∞ −→ zor (2) ˜ ϕ(en,σ)znn→+∞ −→ I(z), and since ˜ ϕ(en,σ)zn=˜ ϕ(tn+en,θ−tσ)xnn→+∞ −→ xand I(M)∩M=∅, item (2) cannot happen and we must have z=x∈Mand ˜ ϕ(tn,θ−tnσ)xnn→+∞ −→ x. Since ˆ Cpullback D-attracts nonautonomous sets, we have x∈C(σ). Then ˜ ω(ˆ B,σ)⊆C(σ)for every ˆ B∈Dand σ∈Σ, which implies in particular that ˜ ω(ˆ K,σ)⊂C(σ)and therefore A(σ)⊆C(σ)and ends the proof. Remark 3.25. With Definition 3.13, if ˆ Aexists, it is uniquely determined. To finish this section, we state an important characterization of the impulsive cocycle attractor. Definition 3.26. We say that a function ψ:R→Xis a global solution of ˜ ϕat σif ˜ ϕ(t−s,θsσ)ψ(s) = ψ(t)for all t⩾s,s∈R. Moreover, if ψ(0) = xwe say that ψis a global solution through x. We say that a global solution is bounded if ψ(R)is a bounded subset of X. A survey on impulsive dynamical systems 21 Proposition 3.27 ([4, Proposition 5.5]).At light of Definition 3.13, if the INDS [(ϕ,θ)(X,Σ),M,I] has an impulsive cocycle attractor ˆ A∈Dwith universe Dconsisting of all nonautonomous sets ˆ B such that Sσ∈ΣB(σ)is bounded in X and I(M)∩M=∅, then A(σ)\M={x∈X:ψis a bounded global solution of ˜ ϕat σthrough x}. Proof. If ψ(·)is a bounded global solution of ˜ ϕat σthrough xthen ψ(R)∩M=∅, since if ψ(t0)∈Mfor some t0∈Rthen ˜ ϕ(t0−s,θsσ)ψ(s) = ψ(t0)∈Mfor each ssuch that t0−s>0 which cannot happen (the impulsive cocycle from xcannot reach Min positive time for any x∈X, because I(M)∩M=∅). Hence, ψ(R)∩M=∅. By its invariance we can see that x∈A(σ)and therefore, x∈A(σ)\M. For the reverse inclusion, if x∈A(σ)\Mthen x∈˜ ϕ(1, θ−1σ)A(θ−1σ)and there exists x−1∈A(θ−1σ)such that ˜ ϕ(1, θ−1σ)x−1=x. Again, since x−1∈A(θ−1σ)there exists x−2∈ A(θ−2σ)such that ˜ ϕ(1, θ−2σ)x−2=x−1. Inductively, we can construct a sequence {x−n}n∈N such that ˜ ϕ(1, θ−n−1σ)x−n−1=x−nfor all n⩾0, with x0=x. Then we can define ψ(t) = (˜ ϕ(t+n,θ−nσ)x−n, if t∈[−n,−n+1],n∈N, ˜ ϕ(t,σ)x0, if t⩾0. Since ˆ A∈D, it is clear that this global solution is bounded and completes the proof. 3.3 Nonautonomous 2D-Navier–Stokes equations with impulses Here we present an example to illustrate the impulsive nonautonomous theory. A more detailed description can be found in [4, Section 6]. The Navier–Stokes equations model fluid flow and are obtained used the conservation of linear momentum, which is ut−ν∆u+ (u· ∇)u+∇p=g(t), (3.2) together with an incompressibility condition ∇ · u=0, where u(t,x)denotes the vector velocity, ν>0 is the kinematic viscosity, gis a body force and pis a scalar pressure. Now, we will consider this model with impulses in the state space, that can be imagined as forced changes on the vector velocity of the fluid, in order to prevent problems that may occur when the fluid reaches certain speeds. One can modify this model a little and add a component v(t,x)for the position of the fluid, and we could imagine impulses as a way to avoid barriers and obstacles along the fluid trajectory. We use here the approached adopted in [15], and we treat this problem in Ω= [0, 2π]2(a periodic domain) and we require zero total momentum, that is, if RΩu0=0 and RΩg(t) = 0 for all t⩾0, then RΩu(t) = 0 for all t⩾0. Writing ˙ Z2=Z2\ {0, 0}, let ˙ Hsbe the subspace of the Sobolev space Hswhich consists of all divergence-free, zero average, periodic real functions ˙ Hs. =(u=∑ k∈˙ Z2 ˆ ukeik·x:ˆ uk=ˆ uk,∑ k∈˙ Z2 |k|2s|ˆ uk|2<∞,k·ˆ uk=0), 22 E. M. Bonotto, M. C. Bortolan, T. Caraballo and R. Collegari with the norm kuk2 s=∑ k∈˙ Z2 |k|2s|ˆ uk|2. We have that H. =˙ H0is the natural phase space for this problem, and we write k · k for the norm in H(the usual L2-norm). Remember also that the space of divergence-free functions is perpendicular (in L2(Ω)) to the space of gradients, since integrating by parts, we have ZΩu· ∇p=−ZΩ(∇ · u)p=0, and so we use the Leray projector P, which is the orthogonal projection of L2(Ω)into the space os divergence-free fields. Applying this projector to (3.2) we obtain du dt +νAu +B(u,u) = f(t), (3.3) where A=−P∆is the Stokes operator, B(u,u) = P[(u· ∇)u]and f(t) = Pg(t). In the periodic case, we have that Au =−∆Pu and so Au =−∆u, for u∈˙ Hs. We define the fractional power As/2 of Aby D(As/2) = ˙ Hsand As/2 ∑ k∈˙ Z2 ˆ ukeik·x!=∑ k∈˙ Z2 |k|sˆ ukeik·x. We note that the norms k · k1and the norm kA1/2 · k are equivalent, and also that ˙ H1is compactly embedded in H. Also, we denote the dual space of ˙ H1by H−1. A simple integration by parts leads to the following antisymmetric identity (B(u,v),w) = −(B(u,w),v), which implies in particular that (B(u,v),v) = 0. (3.4) Also, with a little more effort and using the incompressibility condition, one can prove that in the two-dimensional periodic case, we have (B(u,u),Au) = 0. Then, we can summarize the results of [15, Section 11.1] in the next proposition. Proposition 3.28. Assume that kf(t)k⩽αfor all t ⩾0, then we have: (i) equation (3.3)defines a nonautonomous dynamical system (ϕ,θ)(H,R), where θts=t+s for all t⩾0and s ∈Rand ϕ(t,s)u0=u(t+s,s,u0) is the unique solution in H of (3.3), with u(s,s,u0) = u0∈H; (ii) ϕ(·,s)u0∈L∞(0, T;H)∩L2(0, T;D(A1/2)) and ϕt(·,s)u0∈L2(0, T;D(A−1/2)) for every T>0; (iii) for u0∈H and s ∈R kϕ(t,s)u0k2⩽e−νλ1tku0k2+α2 ν2λ2 1 ,for all t ⩾0, where λ1is the first eigenvalue of A. A survey on impulsive dynamical systems 23 Now we assume that Mis an impulsive set in Hfor (ϕ,θ)(H,R), and assume that every point of Msatisfies ϕ-SSTC. Also, let I:M→Hbe an impulsive function such that (H1) I(M)∩M=∅; (H2) kI(v)k2⩽µ, for all v∈M. (H3) Assume that there exists ξ>0 such that φ(v,s)⩾2ξ, for all v∈I(M)and s∈R. Let ˜ ϕ(t,s)u0be the associated impulsive solution of        du dt +νAu +B(u,u) = f(t), u(0) = u0∈H, I:M→H. (3.5) We assume that kf(t)k ≤ αfor all t≥0. Now we summarize some results (see [4, Section 6]) which are useful to obtain an existence result of impulsive cocycle attractor for this example. Proposition 3.29. (i) ([4, Lemma 6.2])For each t >0and s ∈R, the map ϕ(t,s):H→H is compact. (ii) ([4, Lemma 6.3])We have k˜ ϕ(t,s)u0k2⩽µ+α2 ν2λ2 1 , for all u0∈I(M), t ⩾0and s ∈R. (iii) ([4, Proposition 6.4])If B ⊂H is a bounded subset then there exists t0=t0(B)⩾0such that k˜ ϕ(t,s)u0k2⩽µ+α2 ν2λ2 1 , if t ⩾t0, for all u0∈B and s ∈R. (iv) ([4, Lemma 6.5])If G is a precompact subset of H and τ∈[0, ξ), then ˜ ϕ(τ,s)G is precompact in H for each s ∈R. Using the results in Proposition 3.29, we can construct a compact nonautonomous set ˆ K={K(s)}s∈Rwhich ˜ ϕ-pullback absorbs all bounded subsets of H. We will reproduce its proof here. Theorem 3.30 ([4, Theorem 6.6]).There exists a compact nonautonomous set ˆ K={K(s)}s∈Rwhich ˜ ϕ-pullback absorbs all nonautonomous sets ˆ D with Ss∈RD(s)bounded in H, and such that Ss∈RK(s) is bounded in H. Proof. Let B0=u∈H:kuk2⩽µ+α2 ν2λ2 1. Firstly, we fix τ∈(ξ, 2ξ). We claim that G(s) = ˜ ϕ(τ,θ−τs)B0is precompact for each s∈R. Indeed, we can write B0=C1∪C2∪C3where C1={u∈B0:φ(u,θ−τs)⩾2ξ},C2={u∈B0:ξ<φ(u,θ−τs)⩽2ξ}and C3={u∈B0:φ(u,θ−τs)⩽ξ}. Then we have G(s) = ϕ(τ,θ−τs)C1∪˜ ϕτ−ξ,θ−τ+ξsϕ(ξ,θ−τs)C2∪ϕτ−ξ,θ−τ+ξs˜ ϕ(ξ,θ−τs)C3, since φ(v,s)⩾2ξfor all v∈I(M)and s∈R, and τ−ξ∈(0, ξ). By Proposition 3.29, since C1and ˜ ϕ(ξ,θ−τs)C3are bounded (see item (ii)), it follows that sets ϕ(τ,θ−τs)C1and ϕτ−ξ,θ−τ+ξs˜ ϕ(ξ,θ−τs)C3are precompact in H(see item (i)). Also, 24 E. M. Bonotto, M. C. Bortolan, T. Caraballo and R. Collegari since ϕ(ξ,θ−τs)C2is precompact in H, it follows that ˜ ϕτ−ξ,θ−τ+ξsϕ(ξ,θ−τs)C2is also precompact in H(item (iv)). Therefore, K(s). =G(s)is compact in H, for each s∈R. Clearly, we have that sup v∈K(s) kvk2⩽β+α2 ν2λ2 1 , where β=max{µ,L0}and L0=supu∈B0kuk2. Now it remains to prove that ˆ K˜ ϕ-pullback absorbs nonautonomous bounded sets ˆ Dwith Ss∈RD(s)bounded in H. To this end, let ˆ Da nonautonomous set in Hwith B. =Ss∈RD(s) bounded in Hand fix s∈R. We know, by item (iii) of Proposition 3.29, that there exists t0=t0(B)>0 such that ˜ ϕ(t,θ−t−τs)B⊂B0, for all t⩾t0. Thus ˜ ϕ(t+τ,θ−t−τs)B=˜ ϕ(τ,θ−τs)˜ ϕ(t,θ−t−τs)B⊂˜ ϕ(τ,θ−τs)B0⊂K(s), which shows that if t⩾t0+τ ˜ ϕ(t,θ−ts)D(θ−ts)⊂˜ ϕ(t,θ−ts)B⊂K(s), and proves that ˆ Kis a ˜ ϕ-pullback absorbs ˆ D. As a consequence of this last theorem we obtain that the INDS [(ϕ,θ)(H,R),M,I]defined by (3.5) has an impulsive cocycle attractor (see [4, Corollary 6.7]). 4 Conclusion, comments and future directions In this survey paper we described the theories of impulsive dynamical systems in both autonomous and nonautonomous frameworks. In the first part of this survey we presented two different approaches to study the asymptotic dynamical behavior of autonomous systems, proposed by Bonotto and Demuner (see [6,7]) and Bonotto et al. (see [5]), respectively. In [6,7], the definition of global attractors for impulsive autonomous dynamical systems was first introduced, where the attractor is invariant, consists of a compact set which does not intersect the impulsive set Mand attracts bounded sets. This definition is consistent with the notion of global attractors for semigroups (they coincide when M=∅) and describes the asymptotic behavior of many impulsive dynamical systems. However, it is not suitable for a large class of impulsive dynamical systems. For example, when the global attractor is compact and is disjoint with the closed set M, the compactness of the global attractor implies a separation between them and hence the asymptotic behavior of the impulsive dynamical system is not qualitatively different from the asymptotic behavior of the original system without impulse (see, e.g., Example 2.11). Later in [5] the notion of precompact global attractors was introduced, where the global attractor can “touch” the impulsive set M,i.e., the boundary of the global attractor can have points which belong to M. The simplicity of autonomous framework allows us to study various types of impulsive dynamical systems, along with many interesting new applications. In this survey we illustrated one of the three interesting applications presented in [5]. A survey on impulsive dynamical systems 25 In the second part of this survey we described the recently developed theories of nonautonomous impulsive dynamical systems, with multiple lines of prospective research. In particular, we recalled the main results of our recent work [4], where we proposed the first approach in the nonautonomous theory to study impulsive dynamical systems. This is done by defining the notion of impulsive nonautonomous dynamical systems, in which the trajectories have to be defined in a careful manner to obtain their relationship with the associated impulsive skew-product semiflow (see Theorem 3.9). The main goal is to construct a proper notion of impulsive cocycle attractors and develop their existence. To this end, we introduced a different notion of omega limit set (see Definition 3.15), to overcome the difficulties encountered in proving the usual properties such as invariance and pullback attraction in the nonautonomous theory. It is worth mentioning again that the theory of impulsive dynamical systems is still in the early stage of investigation and has many interesting topics to be discovered, especially in the nonautonomous framework. We have made an initial step toward establishing the modern theory of impulsive dynamical systems, by developing a definition of impulsive nonautonomous dynamical systems and presenting an existence result of impulsive cocycle attractor. Yet there are many other interesting and important problems along this direction to be investigated, even in the autonomous framework. For example, on the one hand, there are no studies to date on the semi-continuity and geometrical structures of attractors for impulsive dynamical systems, and on the other hand, there are not many examples from applications analyzed in a detailed way. The main reasons are the difficulties in order to check some of the hypotheses ensuring the generation of an impulsive system, as well as the conditions required for the existence of attractors. Therefore, this is a field to be explored in a more detailed way in the future and we plan to work on this direction. Another major research direction would be developing a set of analog theories for impulsive dynamical systems where the nonautonomous character involves uncertainty, i.e., noise. This leads to a framework for random impulsive dynamical systems, a brand new area of research. Acknowledgements E. M. B. is partially supported by FAPESP grant 2014/25970-5 and CNPq grant 307317/2013-7. M. C. B. and T. C. are partially supported by FEDER and Ministerio de Economía y Competitividad (Spain) under grant MTM2015-63723-P, and Consejería de Innovación, Ciencia y Empresa (Junta de Andalucía) under Proyecto de Excelencia P12-FQM-1492. R. C. is supported by FAPESP grants 2013/23933-2 and 2014/20691-0. The authors also would like to thank the referee for the helpful comments and suggestions which allowed us to improve the presentation of this survey. References [1] N. U. Ahmed, Existence of optimal controls for a general class of impulsive systems on Banach spaces, SIAM J. Control Optim. 42(2003), No. 2, 669-685. MR1982287;url [2] D. D. Bainov, P. S. Simeonov,Systems with impulsive effect. Stability, theory and applications, Wiley, New York, 1989. MR1010418