# Existence Conditions for Bounded Solutions of Weakly Perturbed Linear Impulsive Systems

Authors: Alexander Boichuk; Martina Langerová; Jaroslava Škoríková
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

The weakly perturbed linear nonhomogeneous impulsive systems in the formẋ=A(t)x+εA1(t)x+f(t),t∈ℝ,t∉𝒯:={τi}ℤ,Δx|t=τi=γi+εA1ix(τi-),τi∈𝒯⊂ℝ,γi∈ℝn, and i∈ℤ are considered. Under the assumption that the generating system (for ε=0) does not have solutions bounded on the entire real axis for some nonhomogeneities and using the Vishik-Lyusternik method, we establish conditions for the existence of solutions of these systems bounded on the entire real axis in the form of a Laurent series in powers of small parameter ε with finitely many terms with negative powers of ε, and we suggest an algorithm of construction of these solutions.


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## Body

## 1. Introduction

In this contribution we study the problem of existence and construction of solutions of weakly perturbed linear differential systems with impulsive action bounded on the entire real axis. The application of the theory of differential systems with impulsive action (developed in [1–3]), the well-known results on the splitting index by Sacker [4] and by Palmer [5] on the Fredholm property of bounded solutions of linear systems of ordinary differential equations [6–9], the theory of pseudoinverse matrices [10] and results obtained in analyzing boundary-value problems for ordinary differential equations (see [10–12]), enables us to obtain existence conditions and to propose an algorithm for the construction of solutions bounded on the entire real axis of weakly perturbed linear impulsive differential systems.


## 2. Initial Problem

We consider the problem of existence and construction of solutions bounded on the entire real axis of linear systems of ordinary differential equations with impulsive action at fixed points of time(2.1)ẋ=A(t)x+f(t),t∈R∖T,Δx|t=τi=γi,τi∈T,i∈Z,
where A∈BC𝒯(ℝ) is an n×n matrix of functions, f∈BC𝒯(ℝ) is an n×1 vector function, BC𝒯(ℝ) is the Banach space of real vector functions bounded on ℝ and left-continuous for t∈ℝ with discontinuities of the first kind at t∈𝒯:={τi}ℤ with the norm: ∥x∥BC𝒯(ℝ):=supt∈ℝ∥x(t)∥, γi are n-dimensional column constant vectors: γi∈ℝn;   ⋯<τ-2<τ-1<τ0=0<τ1<τ2<⋯, and Δx|t=τi:=x(τi+)-x(τi-).The solutionx(t) of the system (2.1) is sought in the Banach space of n-dimensional bounded on ℝ and piecewise continuously differentiable vector functions with discontinuities of the first kind at t∈𝒯:x∈BC𝒯1(ℝ).Parallel with the nonhomogeneous impulsive system (2.1), we consider the corresponding homogeneous system(2.2)ẋ=A(t)x,Δx|t=τi=0,
which is the homogeneous system without impulses, and let X(t) be the fundamental matrix of (2.2) such that X(0)=I.Assume that the homogeneous system (2.2) is exponentially dichotomous (e-dichotomous) [5, 10] on semiaxes ℝ-=(-∞,0] and ℝ+=[0,∞), that is, there exist projectors P and Q(P2=P,Q2=Q) and constants Ki≥1, αi>0(i=1,2) such that the following inequalities are satisfied:(2.3)‖X(t)PX-1(s)‖≤K1e-α1(t-s),t≥s,‖X(t)(I-P)X-1(s)‖≤K1e-α1(s-t),s≥t,t,s∈R+,‖X(t)QX-1(s)‖≤K2e-α2(t-s),t≥s,‖X(t)(I-Q)X-1(s)‖≤K2e-α2(s-t),s≥t,t,s∈R-.For getting the solutionx∈BC𝒯1(ℝ) bounded on the entire axis, we assume that t=0∉𝒯, that is, x(0+)-x(0-)=γ0=0.We use the following notation:D=P-(I-Q); D+ is a Moore-Penrose pseudoinverse matrix to D; PD and PD* are n×n matrices (orthoprojectors) projecting ℝn onto N(D)=kerD and onto N(D*)=kerD*, respectively, that is, PD:ℝn→N(D), PD2=PD=PD*, and PD*:ℝn→N(D*), PD*2=PD*=PD**;  H(t)=[PD*Q]X-1(t);  d=rank[PD*Q]=rank[PD*(I-P)] and r=rank[PPD]=rank[(I-Q)PD].The existence conditions and the structure of solutions of system (2.1) bounded on the entire real axis was analyzed in [13]. Here the following theorem was formulated and proved.Theorem 2.1.
Assume that the linear nonhomogeneous impulsive differential system (2.1) has the corresponding homogeneous system (2.2) e-dichotomous on the semiaxes  ℝ-=(-∞,0] and  ℝ+=[0,∞) with projectors  P and  Q, respectively. Then the homogeneous system (2.2) has exactly  r linearly independent solutions bounded on the entire real axis. If nonhomogeneities  f∈BC𝒯(ℝ) and  γi∈ℝn satisfy  d linearly independent conditions
(2.4)∫-∞∞Hd(t)f(t)dt+∑i=-∞∞Hd(τi)γi=0,
then the nonhomogeneous system (2.1) possesses an r-parameter family of linearly independent solutions bounded on ℝ in the form
(2.5)x(t,cr)=Xr(t)cr+(G[fγi])(t),∀cr∈Rr.Here,Hd(t)=[PD*Q]dX-1(t) is a d×n matrix formed by a complete system of d linearly independent rows of matrix H(t), (2.6)Xr(t):=X(t)[PPD]r=X(t)[(I-Q)PD]ris ann×r matrix formed by a complete system of r linearly independent solutions bounded on ℝ of homogeneous system (2.2), and (G[fγi])(t) is the generalized Green operator of the problem of finding bounded solutions of the nonhomogeneous impulsive system (2.1), acting upon f∈BC𝒯(ℝ) and γi∈ℝn, defined by the formula (2.7)(G[fγi])(t)=X(t){∫0tPX-1(s)f(s)ds-∫t∞(I-P)X-1(s)f(s)ds+∑i=1jPX-1(τi)γi-∑i=j+1∞(I-P)X-1(τi)γi+PD+{∫-∞0QX-1(s)f(s)ds+∫0∞(I-P)X-1(s)f(s)ds+∑i=-∞-1QX-1(τi)γi+∑i=1∞(I-P)X-1(τi)γi},t≥0;∫-∞tQX-1(s)f(s)ds-∫t0(I-Q)X-1(s)f(s)ds+∑i=-∞-(j+1)QX-1(τi)γi-∑i=-j-1(I-Q)X-1(τi)γi+(I-Q)D+{∫-∞0QX-1(s)f(s)ds+∫0∞(I-P)X-1(s)f(s)ds+∑i=-∞-1QX-1(τi)γi+∑i=1∞(I-P)X-1(τi)γi},t≤0,
with the following property (2.8)(G[fγi])(0-)-(G[fγi])(0+)=∫-∞∞H(t)f(t)dt+∑i=-∞∞H(τi)γi.These results are required to establish new conditions for the existence of solutions of weakly perturbed linear impulsive systems bounded on the entire real axis.


## 3. Perturbed Problems

Consider a weakly perturbed nonhomogeneous linear impulsive system in the form(3.1)ẋ=A(t)x+ɛA1(t)x+f(t),t∈R∖T,Δx|t=τi=γi+ɛA1ix(τi-),τi∈T,γi∈Rn,i∈Z,
where A1∈BC𝒯(ℝ) is an n×n matrix of functions, A1i are n×n constant matrices.Assume that the condition of solvability (2.4) of the generating system (2.1) (obtained from system (3.1) for ɛ=0) is not satisfied for all nonhomogeneities f∈BC𝒯(ℝ) and γi∈ℝn, that is, system (2.1) does not have solutions bounded on the entire real axis. Therefore, we analyze whether the system (2.1) can be made solvable by introducing linear perturbations to the differential system and to the pulsed conditions. Also it is important to determine perturbations A1(t) and A1i required to make the problem (3.1) solvable in the space of functions bounded on the entire real axis, that is, it is necessary to specify pertubations for which the corresponding homogeneous system(3.2)ẋ=A(t)x+ɛA1(t)x,t∈R∖T,Δx|t=τi=ɛA1ix(τi-),τi∈T,i∈Z,
turns into a system e-trichotomous or e-dichotomous on the entire real axis [10].We show that this problem can be solved using thed×r matrix (3.3)B0=∫-∞∞Hd(t)A1(t)Xr(t)dt+∑i=-∞∞Hd(τi)A1iXr(τi-),constructed with the coefficients of the system (3.1). The Vishik-Lyusternik method developed in [14] enables us to establish conditions under which a solution of impulsive system (3.1) can be represented by a function bounded on the entire real axis in the form of a Laurent series in powers of the small fixed parameter ɛ with finitely many terms with negative powers of ɛ.We use the following notation:B0+ is the unique matrix pseudoinverse to B0 in the Moore-Penrose sense, PB0 is the r×r matrix (orthoprojector) projecting the space Rr to the null space N(B0) of the d×r matrix B0, that is, PB0:Rr→N(B0), and PB0* is the d×d matrix (orthoprojector) projecting the space ℝd to the null space N(B0*) of the r×d matrix B0* (B0*=BT), that is, PB0*:ℝd→N(B0*).Now we formulate and prove a theorem that enables us to solve indicated problem.Theorem 3.1.
Suppose that the system (3.1) satisfies the conditions imposed above, and the homogeneous system (2.2) is e-dichotomous on  ℝ+ and  ℝ- with projectors P and Q, respectively. Let nonhomogeneities  f∈BC𝒯(ℝ) and  γi∈ℝn be given such that the condition (2.4) is not satisfied and the generating system (2.1) does not have solutions bounded on the entire real axis. If
(3.4)PB0*=0,
then the system (3.2) is e-trichotomous on ℝ and, for all nonhomogeneities f∈BC𝒯(ℝ) and γi∈ℝn, the system (3.1) possesses at least one solution bounded on ℝ in the form of a series
(3.5)x(t,ɛ)=∑k=-1∞ɛkxk(t),
uniformly convergent for sufficiently small fixed ɛ∈(0,ɛ*].Here,ɛ* is a proper constant characterizing the range of convergence of the series (3.5) and the coefficients xk(t) of the series (3.5) are determined from the corresponding impulsive systems as (3.6)xk(t)=xk(t,ck)=Xr(t)ck+(G[A1(⋅)xk-1(⋅,ck-1)A1ix(τi-,ck-1)])(t)fork=1,2,…,ck=-B0+[∫-∞∞Hd(t)A1(t)(G[A1(⋅)xk-1(⋅,ck-1)A1ixk-1(τi-,ck-1)])(t)dt+∑i=-∞∞Hd(τi)A1i(G[A1(⋅)xk-1(⋅,ck-1)A1ixk-1(⋅,ck-1)])(τi-)],x-1(t)=x-1(t,c-1)=Xr(t)c-1,c-1=B0+{∫-∞∞Hd(t)f(t)dt+∑i=-∞∞Hd(τi-)γi},x0(t)=x0(t,c0)=Xr(t)c0+(G[A1(⋅)Xr(t)c-1+f(⋅)γi+A1iXr(τi-)c-1])(t),c0=-B0+[∫-∞∞Hd(t)A1(t)(G[A1(⋅)x-1(⋅,c-1)+f(⋅)A1ix-1(τi-,c-1)+γi])(t)dt+∑i=-∞∞Hd(τi)A1i(G[A1(⋅)x-1(⋅,c-1)+f(⋅)A1ix-1(⋅,c-1)+γi])(τi-)].Proof.
We suppose that the problem (3.1) has a solution in the form of a Laurent series (3.5). We substitute this solution into the system (3.1) and equate the coefficients at the same powers of ɛ. The problem of determination of the coefficient x-1(t) of the term with ɛ-1 in series (3.5) is reduced to the problem of finding solutions of homogeneous system without impulses
(3.7)ẋ-1=A(t)x-1,t∉T,Δx-1|t=τi=0,i∈Z,
bounded on the entire real axis. According to the Theorem 2.1, the homogeneous system (3.7) possesses r-parameter family of solutions
(3.8)x-1(t,c-1)=Xr(t)c-1bounded on the entire real axis, where c-1 is an r-dimensional vector column c-1∈ℝr and is determined from the condition of solvability of the problem used for determining the coefficient x0 of the series (3.5).
Forɛ0, the problem of determination of the coefficient x0(t) of series (3.5) reduces to the problem of finding solutions of the following nonhomogeneous system:
(3.9)ẋ0=A(t)x0+A1(t)x-1+f(t),t∉T,Δx0|t=τi=A1ix-1(τi-)+γi,i∈Z,
bounded on the entire real axis. According to the Theorem 2.1, the condition of solvability of this problem takes the form
(3.10)∫-∞∞Hd(t)[A1(t)Xr(t)c-1+f(t)]dt+∑i=-∞∞Hd(τi)[A1iXr(τi-)c-1+γi]=0.Using the matrix B0, we get the following algebraic system for c-1∈ℝr:
(3.11)B0c-1=-∫-∞∞Hd(t)f(t)dt+∑i=-∞∞Hd(τi-)γi,which is solvable if and only if the condition
(3.12)PB0*{∫-∞∞Hd(t)f(t)dt+∑i=-∞∞Hd(τi-)γi}=0
is satisfied, that is, if
(3.13)PB0*=0.In this case, this algebraic system is solvable with respect to c-1∈ℝr within an arbitrary vector constant PB0c(∀c∈ℝr) from the null space of the matrix B0, and one of its solutions has the form
(3.14)c-1=B0+{∫-∞∞Hd(t)f(t)dt+∑i=-∞∞Hd(τi-)γi}.
Therefore, under condition (3.4), the nonhomogeneous system (3.9) possesses an r-parameter set of solution bounded on ℝ in the form
(3.15)x0(t,c0)=Xr(t)c0+(G[A1(⋅)x-1(⋅,c-1)+f(⋅)γi+A1ix-1(τi-,c-1)])(t),where (G[**])(t) is the generalized Green operator (2.7) of the problem of finding bounded solutions of system (3.9), and c0 is an r-dimensional constant vector determined in the next step of the process from the condition of solvability of the impulsive problem for coefficient x1(t).
We continue this process by problem of determination of the coefficientx1(t) of the term with ɛ1 in the series (3.5). It reduces to the problem of finding solutions of the system(3.16)ẋ1=A(t)x1+A1(t)x0,t∉T,Δx1|t=τi=A1ix0(τi-),i∈Z,
bounded on the entire real axis. If the condition (3.4) is satisfied and by using the condition of solvability of this problem, that is,
(3.17)∫-∞∞Hd(t)A1(t)[Xr(t)c0+(G[A1(⋅)x-1(⋅,c-1)+f(⋅)A1ix-1(τi-,c-1)+γi])(t)]dt+∑i=-∞∞Hd(τi-)A1i[Xr(τi-)c0+(G[A1(⋅)x-1(⋅,c-1)+f(⋅)A1ix-1(⋅,c-1)+γi])(τi-)]=0,we determine the vector c0∈ℝr (within an arbitrary vector constant PB0c, ∀c∈ℝr) as
(3.18)c0=-B0+[∫-∞∞Hd(t)A1(t)(G[A1(⋅)x-1(⋅,c-1)+f(⋅)A1ix-1(τi-,c-1)+γi])(t)dt+∑i=-∞∞Hd(τi)A1i(G[A1(⋅)x-1(⋅,c-1)+f(⋅)A1ix-1(⋅,c-1)+γi])(τi-)].Thus, under the condition (3.4), system (3.16) possesses an r-parameter set of solutions bounded on ℝ in the form
(3.19)x1(t,c1)=Xr(t)c1+(G[A1(⋅)x0(⋅,c0)A1ix(τi-,c0)])(t),where (G[**])(t) is the generalized Green operator (2.7) of the problem of finding bounded solutions of system (3.16), and c1 is an r-dimensional constant vector determined in the next stage of the process from the condition of solvability of the problem for x2(t).
If we continue this process, we prove (by induction) that the problem of determination of the coefficientxk(t) in the series (3.5) is reduced to the problem of finding solutions of the system
(3.20)ẋk=A(t)xk+A1(t)xk-1,t∉T,Δxk|t=τi=A1ixk-1(τi-),i∈Z,k=1,2,…,
bounded on the entire real axis. If the condition (3.4) is satisfied, then a solution of this problem bounded on ℝ has the form
(3.21)xk(t)=xk(t,ck)=Xr(t)ck+(G[A1(⋅)xk-1(⋅,ck-1)A1kxk-1(τi-,ck-1)])(t),
where (G[**])(t) is the generalized Green operator of the problem of finding bounded solutions of impulsive system (3.20) and the constant vector ck∈Rr is given by the formula
(3.22)ck=-B0+[∫-∞∞Hd(t)A1(t)(G[A1(⋅)xk-1(⋅,ck-1)A1ixk-1(τi-,ck-1)])(t)dt+∑i=-∞∞Hd(τi)A1i(G[A1(⋅)xk-1(⋅,ck-1)A1ixk-1(⋅,ck-1)])(τi-)]
(within an arbitrary vector constant PB0c, c∈Rr).
The fact that the series (3.5) is convergent can be proved by using the procedure of majorization.In the case where the numberr=rankPPD=rank(I-Q)PD of linear independent solutions of system (2.2) bounded on ℝ is equal to the number d=rank[PD*Q]=rank[PD*(I-P)], Theorem 3.1 yields the following assertion.Corollary 3.2.
Suppose that the system (3.1) satisfies the conditions imposed above, and the homogeneous system (2.2) is e-dichotomous on ℝ+ and ℝ- with projectors P and Q, respectively. Let nonhomogeneities f∈BC𝒯(ℝ) and γi∈ℝn be given such that the condition (2.4) is not satisfied, and the generating system (2.1) does not have solutions bounded on the entire real axis. If condition
(3.23)det⁡B0≠0(r=d),
is satisfied, then the system (3.1) possesses a unique solution bounded on ℝ in the form of series (3.5) uniformly convergent for sufficiently small fixed ɛ∈(0,ɛ*].Proof.
Ifr=d, then B0 is a square matrix. Therefore, it follows from condition (3.4) that PB0=PB0*=0, which is equivalent to the condition (3.23). In this case, the constant vectors ck∈ℝr are uniquely determined from (3.22). The coefficients of the series (3.5) are also uniquely determined by (3.21), and, for all f∈BC𝒯(ℝ) and γi∈ℝn, the system (3.1) possesses a unique solution bounded on ℝ, which means that system (3.2) is e-dichotomous.We now illustrate the assertions proved above.Example 3.3.
Consider the impulsive system(3.24)ẋ=A(t)x+ɛA1(t)x+f(t),t∈R∖T,Δx|t=τi=γi+ɛA1ix(τi-),γi={γi(1)γi(2)γi(3)}∈R3,i∈Z,
where
(3.25)A(t)=diag⁡{-tanh⁡t,-tanh⁡t,tanh⁡t},f(t)=col{f1(t),f2(t),f3(t)}∈BCT(R),A1(t)={aij(t)}i,j=13∈BCT(R),A1i={ãij}i,j=13.
The generating homogenous system (for ɛ=0) has the form
(3.26)ẋ=A(t)x,Δx|t=τi=0and is e-dichotomous (as shown in [6]) on the semiaxes ℝ+ and ℝ- with projectors P=diag{1,1,0} and Q=diag{0,0,1}. The normal fundamental matrix of this system is
(3.27)X(t)=diag⁡{2et+e-t,2et+e-t,et+e-t2}.Thus, we have
(3.28)D=0,D+=0,PD=PD*=I3,r=rank⁡PPD=2,d=rank⁡PD*Q=1,Xr(t)=(2et+e-t002et+e-t00),(3.29)Hd(t)=(0,0,2et+e-t).
In order that the generating impulsive system (2.1) with the matrix A(t) specified above has solutions bounded on the entire real axis, the nonhomogeneities f(t)=col{f1(t),f2(t),f3(t)}∈BC𝒯(ℝ) and γi=col{γi(1),γi(2),γi(3)}∈ℝ3 must satisfy condition (2.4). In this analyzed impulsive problem, this condition takes the form
(3.30)∫-∞∞2f3(t)et+e-tdt+∑i=-∞∞2eτi+e-τiγi(3)=0,∀f1(t),f2(t)∈BCT(R),∀γi(1),γi(2)∈R.
Let f3 and γi(3) be given such that the condition (3.30) is not satisfied and the corresponding generating system (2.1) does not have solutions bounded on the entire real axis. The system (3.24) will be an e-trichotomous on ℝ if the coefficients a31(t),a32(t)∈BC𝒯(ℝ) of the perturbing matrix A1(t) and the coefficients ã31,ã32∈ℝ of the perturbing matrix A1i satisfy condition (3.4), that is, PB0*=0, where the matrix B0 has the form
(3.31)B0=∫-∞∞[a31(t)(et+e-t)2,a32(t)(et+e-t)2]dt+∑i=-∞∞[ã31(eτi-+e-τi-)2,ã32(eτi-+e-τi-)2].
Therefore, if a31(t),a32(t)∈BC𝒯(ℝ) and ã31,ã32∈ℝ are such that at least one of the following inequalities
(3.32)∫-∞∞a31(t)(et+e-t)2dt+∑i=-∞∞ã31(eτi-+e-τi-)2≠0,∫-∞∞a32(t)(et+e-t)2dt+∑i=-∞∞ã32(eτi-+e-τi-)2≠0
is satisfied, then either the condition (3.4) or the equivalent condition rankB0=d=1 from Theorem 3.1 is satisfied and the system (3.2) is e-trichotomous on ℝ. In this case, the coefficients a11(t), a12(t), a13(t), a21(t),a22(t),a23(t),a33(t) are arbitrary functions from the space BC𝒯(ℝ), and ã11, ã12, ã13,ã21,ã22,ã23,ã33 are arbitrary constants from ℝ. Moreover, for any
(3.33)f(t)=col{f1(t),f2(t),f3(t)}∈BCT(R)a solution of the system (3.24) bounded on ℝ is given by the series (3.5) (within a constant from the null space N(B0), dimN(B0)=r-rankB0=1).Another Perturbed Problem
In this part, we show that the problem of finding bounded solutions of nonhomogeneous system (2.1), in the case if the condition (2.4) is not satisfied, can be made solvable by introducing linear perturbations only to the pulsed conditions.
Therefore, we consider the weakly perturbed nonhomogeneous linear impulsive system in the form(3.34)ẋ=A(t)x+f(t),t∈R∖T,A,f∈BCT(R),Δx|t=τi=γi+ɛA1ix(τi-),γi∈Rn,i∈Z,
where A1i are n×n constant matrices. For ɛ=0, we obtain the generating system (2.1). We assume that this generating system does not have solutions bounded on the entire real axis, which means that the condition of solvability (2.4) is not satisfied (for some nonhomogeneities f∈BC𝒯(ℝ) and γi∈ℝn). Let us show that it is possible to make this problem solvable by adding linear perturbation only to the pulsed conditions. In the case, if this is possible, it is necessary to determine perturbations A1i for which the corresponding homogeneous system
(3.35)ẋ=A(t)x,t∈R∖T,Δx|t=τi=ɛA1ix(τi-),i∈Z,
turns into the system e-trichotomous or e-dichotomous on the entire real axis.
This problem can be solved with help of thed×r matrix
(3.36)B0=∑i=-∞∞Hd(τi)A1iXr(τi-)
constructed with the coefficients from the impulsive system (3.34).
By using Theorem3.1, we seek a solution in the form of the series (3.5). Thus, we have the following corollary.Corollary 3.4.
Suppose that the system (3.34) satisfies the conditions imposed above and the generating homogeneous system (2.2) is e-dichotomous on ℝ+ and ℝ- with projectors P and Q, respectively. Let nonhomogeneities f∈BC𝒯(ℝ) and γi∈ℝn be given such that the condition (2.4) is not satisfied, and the generating system (2.1) does not have solutions bounded on the entire real axis. If the condition (3.4) is satisfied, then the system (3.35) is e-trichotomous on ℝ, and the system (3.34) possesses at least one solution bounded on ℝ in the form of series (3.5) uniformly convergent for sufficiently small fixed ɛ∈(0,ɛ*].



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