The present paper investigates the existence of integral manifolds for
impulsive differential equations with variable perturbations.
By means of piecewise continuous functions which are generalizations of the
classical Lyapunov’s functions, sufficient conditions for the existence of integral
manifolds of such equations are found

Stamov, G.

Bulgarian Digital Mathematics Library at IMI-BAS

Serdica Math. J. 22 (1996), 83-90APPLICATION OF LYAPUNOV’S DIRECT METHOD TO THEEXISTENCE OF INTEGRAL MANIFOLDS OF IMPULSIVEDIFFERENTIAL EQUATIONSG. T. StamovCommunicated by J.-P. Franc¸oiseAbstract. The present paper investigates the existence of integral manifolds forimpulsive differential equations with variable perturbations.By means of piecewise continuous functions which are generalizations of theclassical Lyapunov’s functions, sufficient conditions for the existence of integralmanifolds of such equations are found.1. Preliminary notes and definitions. In recent years, the impulsive differen-tial equations have been an object of numerous investigations (Bainov and Simeonov,1989; Dishliev and Bainov, 1989; Lakshmikantham, Bainov and Simeonov, 1989; Kulevand Bainov 1990; Simeonov and Bainov, 1991, 1993)In the present paper, some problems related to the existence of an integral ma-nifold are considered. The main results are obtained by means of piecewise continuousfunctions which are analogous to the classical Lyapunov’s functions.Let Rn be the n-dimensional Euclidean space with norm ‖·‖ and scalar product〈·, ·〉, and let I = [0,∞). We denote by PCk(J,Rn), where J ⊂ I, k = 1, 2, . . ., thespace of all piecewise continuous functions x : J → Rn such that:1991 Mathematics Subject Classification: 34A37Key words: integral manifold, impulsive differential equations84 G. T. Stamov1. The set D = {ti ∈ J, i = 1, 2, . . .} of all points of discontinuity of x has nofinite point of accumulation.2. For any ti ∈ D, x(ti − 0) = x(ti) and the limit x(ti + 0) is finite.3. x is a Ck-continuous function in J \D.Let Ω be a domain in Rn, f : I ×Ω→ Rn, Φi : Ω→ Rn, i = 1.2 . . ., τi : Ω→ R.Introduce the following assumptions:H1. f ∈ C1(I × Ω,Rn).H2. Φi ∈ C1(Ω, Rn), i = 1, 2, . . ..H3. If x ∈ Ω, then x+Φi(x) ∈ Ω, Li(x) = x+Φi(x) are invertible in Ω, and (Li(x))−1 ∈Ω for i = 1, 2, . . ..H4. τi(x) ∈ C1(Ω, I) and 0 < τ1(x) < τ2(x) < · · ·, limi→∞τi(x) =∞ uniformly on x ∈ Ω.H5. The following inequalities holdsup {‖f(t, x)‖ : (t, x) ∈ I ×Ω} ≤ A <∞,sup{∥∥∥∥∂τi(x)∂x∥∥∥∥ : x ∈ Ω, i = 1, 2, . . .}≤ B <∞, AB ≤ 1,sup{〈∂τi(x+ sΦi(x))∂x,Φi(x)〉: s ∈ [0, 1], x ∈ Ω, i = 1, 2, . . .}≤ 0.Let the conditions H1 – H5 be satisfied. Consider the system of impulsivedifferential equations(1) x˙(t) = f(t, x), t 6= τi(x),(2) ∆x = Φi(x), t = τi(x), i = 1, 2, . . .We recall (see Bainov and Simeonov, 1989) and (Lakshmikantham, Bainov andSimeonov, 1989) that for any point (t0, x0) ∈ I × Ω which is not in hypersurfacesσi = {(t, x) ∈ I × Ω : t = τi(x)}, the function x(t) = x(t; t0, x0) is called a solution ofthe system (1), (2) with initial condition x(t0 + 0) = x0, if:1. x(t; t0, x0) ∈ PC1(J,Rn) and, for any ti ∈ J ,x(ti + 0; t0, x0) = x(ti; t0, x0) + Φi(x(ti; t0, x0))2. x(t) satisfies in J \D equation (1).Application of Lyapunov’s direct method. . . 85We also note too that from H5 it follows that the phenomenon “beating” isabsent for the system (1), (2) i.e. the integral orbit of any solution of the system (1),(2) meets each hypersurface σi, i = 1, 2, . . . at most once (Dishliev and Bainov, 1989).We shall denote by J+ = J+(t0, x0) (resp. J− = J−(t0, x0)) the maximalforward interval (t0, ω) (resp. backward interval (α, t0)) of existence of x(t; t0, x0).Next, by θ+(t0, x0), θ−(t0, x0), θ(x0, t0) we shall denote the integral orbit of thesolution x(t; t0, x0) for t ∈ J+, t ∈ J− and t ∈ J respectively.Definition 1. We call an arbitrary manifold M in the extended phase space:a) r-integral manifold, if from (t0, x0) ∈M it follows that θ+(t0, x0) ⊂M ;b) l-integral manifold, if from (t0, x0) ∈M it follows that θ−(t0, x0) ⊂M ;c) integral manifold, if M is r-integral manifold and l-integral manifold.In this paper there are formulated and proved sufficient conditions for theexistence of an integral manifolds of the system (1), (2).The following is the main assumption in this paper:H6. J+(t0, x0) = [t0,∞).Definition 2. The impulsive system (1), (2) is said to be complete if for any(t0, x0) ∈ I × Ω such that (t0, x0) /∈ σi, i = 1, 2, . . ., it follows J(t0, x0) = I.It is easily verified that if the system (1), (2) is complete then conditions H5and H6 hold.Example 1. We consider the impulsive system (1), (2) with τi(x) = τi i. e. thefunctions τi(x), i = 1, 2, . . . are independent from x. If the sequence {τi}, i = 1, 2, . . .is a strictly increasing sequence, then the system (1), (2) is complete.In what follows we shall use the class VM of partially continuous auxiliaryfunctions V : I × Ω→ R which are analogue to Lyapunov’s functions.Let M be an arbitrary manifold on the extended phase space of (1), (2). Putτ0(x) = 0 for x ∈ Ω. Consider the setsGi = {(t, x) ∈ I × Ω : τi−1 < t < τi(x)} , i = 1, 2, . . . .Definition 3. We shall say that the function V : I × Ω → R belongs to theclass VM which kernel is the manifold M in the extend phase space of (1), (2), if thefollowing conditions hold:1. V (t, x) ∈ C1(∞∪i=1Gi,R).2. V (t, x) = 0, (t, x) ∈M ; V (t, x) > 0, (t, x) ∈ (I × Ω) \M,(3)86 G. T. Stamov3. For any i = 1, 2, . . . and each point (ξ, η) ∈ σi the following finitelimits exist V (ξ − 0, η)= lim(t,x)→(ξ,η)(t,x)∈GiV (t, x), V (ξ + 0, η)= lim(t,x)→(ξ,η)(t,x)∈Gi+1V (t, x)and moreover the equality V (ξ − 0, η) = V (ξ, η) holds.Note that if x = ϕ(t) is a solution of system (1), (2), then for (t, x) ∈ ∞∪i=1Gi(i.e. t 6= τi(x)), the equality V˙ (t, x) = D+V (t, x) is satisfied, whereD+V (t, x) = lim sup∆→0+∆−1 {V (t+∆, ϕ(t+∆))− V (t, ϕ(t))}is the upper right derivative of Dini of the function V (t, ϕ(t)).Finally denote by K the class of all continuous and strictly increasing functionsa : I → I such that a(0) = 0.2. Main results.Theorem 1. Assume that:1. The conditions H1 – H6 hold.2. For the system (1), (2) there exists a function V ∈ VM with kernel themanifold M , so that the following relations are satisfiedV˙ (t, x) ≤ 0 for (t, x) ∈ ∞∪i=1Gi,(4)(respectively 0 ≤ V˙ (t, x) for (t, x) ∈ ∞∪i=1Gi),V (t+ 0, x+Φi(x)) ≤ V (t, x) for (t, x) ∈ σi, i = 1, 2, . . .(5) (respectively V (t, x) ≤ V (t+ 0, x+Φi(x)) for (t, x) ∈ σi, i = 1, 2, . . .).Then M is an r-integral manifold, (respectively an l-integral manifold) of thesystem (1), (2).P r o o f. We shall prove Theorem 1 for the case of an r-integral manifold. Forl-integral manifold the proof is similar. Suppose that M is not an r-integral manifold.Therefore there exists t′, t′ > t0 such that, if (t0, x0) ∈M then (t, x(t; t0, x0)) ∈M for t0 ≤ t ≤ t′ but (t, x(t; t0, x0)) /∈ M for t > t′. Then V (t′, x′) = 0, wherex′ = x(t′; t0, x0). Moreover, x(t) ∈ PC1(J+(t0, x0),Rn). Then for t′ the following twocases are possible:Application of Lyapunov’s direct method. . . 87a) If t′ ∈ σi, i = l, l + 1, . . ., then (t′, x(t′ + 0; t0, x0)) /∈ M and from Definition3 it follows that V (t′ + 0, x(t′ + 0; t0, x0)) > 0.Consequently 0 = V (t′, x′) < V (t′ + 0, x(t′ + 0; t0, x0)) which contradicts (4).b) If t′ /∈ σi, i = l, l + 1, . . . then there exists t′′ ∈ J+(t′, x′), t′′ > t′ such that(t′′, x(t′′; t′, x′)) /∈M . From (4) and (5) it follows that the function is not increasing in(t0,∞).From (3) it follows that V (t′′, x(t′′; t′, x′)) > 0 so V (t′′, x(t′′; t′, x′)) > V (t′, x′)for t′′ > t′ which contradicts the fact that the function is not increasing in (t0,∞).From a) and b), it follows that M is an r-manifold. Theorem 2. Assume that:1. The conditions H1 – H6 hold.2. There exist a function V ∈ VM and a function c ∈ K such that the followingrelations are satisfiedV˙ (t, x) ≤ −c(‖x‖) for (t, x) ∈ ∞∪i=0Gi(resp V˙ (t, x) ≥ c(‖x‖) for (t, x) ∈ ∞∪i=0Gi),V (t+ 0, x+Φi(x)) ≤ V (t, x) for (t, x) ∈ σi, i = 1, 2, . . .(resp V (t, x) ≤ V (t+ 0, x+Φi(x)) for (t, x) ∈ σi, i = 1, 2, . . .).Then M is an r-integral manifold, (l-integral manifold) of the system (1), (2).P r o o f. The proof of Theorem 2 is analogous to the proof of Theorem 1. Theorem 3. Assume the following:1. The conditions H1 – H6 hold.2. There exist functions V ∈ VM and W ∈ VM such that the following relationsare satisfied:V˙ (t, x) ≤ 0 for (t, x) ∈ ∞∪i=0Gi,W˙ (t, x) ≥ 0 for (t, x) ∈ ∞∪i=0Gi,V (t+ 0, x+Φi(x)) ≤ V (t, x) for (t, x) ∈ σi, i = 1, 2, . . . ,W (t, x) ≤W (t+ 0, x+Φi(x)) for (t, x) ∈ σi, i = 1, 2, . . . .Then M is an integral manifold of (1), (2).P r o o f. The proof of Theorem 3 follows from Theorem 1. 88 G. T. StamovExample 2. We consider the following system of impulsive differential equa-tions(6)dydt= −y − t2√yz2, dzdt= t2y−2(z − 2), t 6= i,∆y = −12∆z = 0, t = i, i = 1, 2, . . .where t ∈ I, y ∈ I, z ∈ I.Take the manifold(7) M ={(t, y, z) ∈ I3 : z = 2, t > 0, y > 0}and the functionsV (t, y, z) =(34)iexp{−(ty)2}(z − 2)2andW (t, y, z) = (z − 2)2.Then(8)V˙ (t, y, z) =(34)i (−2ty−2 exp{−(ty)2}(z − 2)2)++2(34)iexp{−(ty)2}(z − 2)t2y−2 +(34)i2t2y−3 exp{−(ty)2}(z − 2)2××(−y − t2√yz2) = 2(34)ity−2 exp{−(ty)2}(z − 2)2(−1− t3y− 12 z2)≤ 0,i < t < i+ 1, i = 1, 2, . . . , y > 0, z > 0.On the other handW˙ (t, y, z) = 2(z − 2)2t2y−2 ≥ 0, i < t < i+ 1, i = 1, 2, . . . , y > 0, z > 0,(9)V(i+ 0, y − 12, z)≤ V (i, y, z), y > 0, z > 0, i = 1, 2, . . . ,(10)W(i+ 0, y − 12, z)=W (i, y, z), y > 0, z > 0, i = 1, 2, . . . .(11)From (8), (9), (10) and (11) it follows that the hypotheses of Theorem 3 aresatisfied. Then (7) is an integral manifold of the system (6).Application of Lyapunov’s direct method. . . 89Now we consider the function(12) L(t, s, x) =V (t, x), t > s, (t, x) ∈ I × Ω, (s, x) ∈ I × Ω,W (t, x), t < s, (t, x) ∈ I × Ω, (s, x) ∈ I × Ω,max{V (t, x),W (t, x)}, (t, x) ∈ I × Ω,max{V (t+ 0, x+Φi(x)),W (t + 0, x+Φ(x))},(t, x) ∈ σi, i = 1, 2, . . . ,where V (t, x) and W (t, x) are as in Theorem 3.Theorem 4. Let the conditions H1 – H6 hold. Then a manifold M from theextended phase space of (1), (2) is an integral manifold of (1), (2), if and only if thereexists a function L(t, s, x) of form (12) such that:L˙(t, s, x) ≤ 0 for t > s, (t, x) ∈ ∞∪i=1Gi, (s, x) ∈∞∪i=1Gi,L(ti + 0, s+ 0, x+Φi(x)) ≤ L(ti, s, x) for t > s, (t, x) ∈ σi, i = 1, 2, . . . ,0 ≤ L˙(t, s, x) for t < s, (t, x) ∈ ∞∪i=1Gi, (s, x) ∈∞∪i=1Gi,L(ti, s, x) ≤ L(ti + 0, s + 0, x+Φi(x)) for t < s, (t, x) ∈ σi, i = 1, 2, . . . .P r o o f. The proof of Theorem 4 follows from (12) and Theorem 3. REFERENCES[1] D. D. Bainov and P. S. Simeonov. Systems with Impulsive Effect: Stability,Theory and Applications. Ellis Harwood Limited, 1989.[2] D. D. Bainov, S. I. Kostadinov, Nguyen Van Min, Nguyen Hong Thaiand P. S. Zabreiko. Integral manifolds of impulsive differential equation. J.Appl. Math. Stochastic Anal. 5, 2 (1992), 99-110.[3] D. D. Bainov, S. I. Kostadinov, Nguyen Hong Thai, P. P. Zabreiko.Existence of integral manifolds for impulsive differential equations in a BanachSpace. Internat. J. Theoret. Phys. 28, 7 (1989) 815-833.[4] A. B. Dishliev and D. D. Bainov. Continuous dependence of the solution of asystem of differential equations with impulses on the initial condition. Zeitschriftfu¨r Analysis und ihre Anwendungen 8, 2 (1989), 183-196.90 G. T. Stamov[5] G. K. Kulev and D. D. Bainov. Second method of Lyapunov and comparisonprinciple for systems with impulse effect. Bull. Inst. Math. Sinica 18, 2 (1990),155-178.[6] V. Lakshmikantham, D. D. Bainov and P. S. Simeonov. Theory of Impul-sive Differential Equations, World Scientific, Singapore, 1989.[7] A. M. Samoilenko, N. A. Perestyuk. Differential Equations with ImpulseEffect. Visa Skola, Kiev, 1987, (in Russian).[8] P. S. Simeonov and D. D. Bainov. Application of the method of the two-sidedapproximations to the solution of the periodic problem for impulsive differentialequations. Tamkang J. Math. 22, 3 (1991), 275-284.[9] P. S. Simeonov and D. D. Bainov. Criteria for dichotomy of linear impulsivedifferential equations. J. Korean Math. Soc. 30, 2, 345-359.Gani StamovTechnical UniversitySlivenBulgariaReceived July 27, 1995Revised March 4, 1996

Application of Lyapunov's Direct Method to the Existence of Integral Manifolds of Impulsive Differential Equations

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Application of Lyapunov's Direct Method to the Existence of Integral Manifolds of Impulsive Differential Equations

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