Hindawi Publishing Corporation Advances in Difference Equations Volume 2010, Article ID 494379, 10 pages doi:10.1155/2010/494379 ResearchArticleSolutionsofLinearImpulsiveDifferentialSystemsBoundedontheEntireReal Axis Alexandr Boichuk, Martina Langerov ´ a, and Jaroslava ˇ Skor ´ ıkov ´ a Department of Mathematics, Faculty of Science, University of ˇ Zilina, 010 26 ˇ Zilina, Slovakia Correspondence should be addressed to Alexandr Boichuk, boichuk@imath.kiev.ua Received 21 January 2010; Accepted 12 May 2010 Academic Editor: Leonid Berezansky Copyright q 2010 Alexandr Boichuk et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We consider the problem of existence and structure ofsolutionsboundedontheentirereal axis of nonhomogeneous linearimpulsive differential systems. Under assumption that the corresponding homogeneous system is exponentially dichotomous onthe semiaxes R − and R and by using the theory of pseudoinverse matrices, we establish necessary and sufficient conditions for the indicated problem. Theresearch in the theory of differential systems with impulsive action was originated by Myshkis and Samoilenko 1, Samoilenko and Perestyuk 2, Halanay and Wexler 3,and Schwabik et al. 4. The ideas proposed in these works were developed and generalized in numerous other publications 5. The aim of this contribution is, using the theory ofimpulsive differential equations, using the well-known results onthe splitting index by Sacker 6 and by Palmer 7 onthe Fredholm property ofthe problem ofboundedsolutions and using the theory of pseudoinverse matrices 5, 8, to investigate, in a relevant space, the existence ofsolutionsboundedontheentirereal axis oflinear differential systems with impulsive action. We consider the problem of existence and construction ofsolutionsboundedontheentirereal axis oflinearsystemsof ordinary differential equations with impulsive action at fixed points of time ˙x A t x f t ,t / τ i , Δx| tτ i γ i ,i∈ Z,t,τ i ∈ R,γ i ∈ R n , 1 where At ∈ BCR \{τ i } I is an n × n matrix of functions; ft ∈ BCR \{τ i } I is an n × 1 vector function; BCR \{τ i } I is the Banach space ofreal vector functions continuous for t ∈ R 2 Advances in Difference Equations with discontinuities ofthe first kind at t τ i ; γ i are n-dimensional column constant vectors; ···<τ −2 <τ −1 <τ 0 0 <τ 1 <τ 2 < ···. The solution xt ofthe problem 1 is sought in the Banach space of n-dimensional piecewise continuously differentiable vector functions with discontinuities ofthe first kind at t τ i : xt ∈ BC 1 R \{τ i } I . Parallel with the nonhomogeneous impulsive system 1 we consider the homoge- neous system ˙x A t x, t ∈ R, 2 which is the homogeneous system without impulses. Assume that the homogeneous system 2 is exponentially dichotomous e-dichot- omous on semiaxes R − −∞, 0 and R 0, ∞; i.e. there exist projectors P and Q P 2 P, Q 2 Q and constants K i ≥ 1,α i > 0 i 1, 2 such that the following inequalities are satisfied: X t PX −1 s ≤ K 1 e −α 1 t−s ,t≥ s, X t I − P X −1 s ≤ K 1 e −α 1 s−t ,s≥ t, t, s ∈ R , X t QX −1 s ≤ K 2 e −α 2 t−s ,t≥ s, X t I − Q X −1 s ≤ K 2 e −α 2 s−t ,s≥ t, t, s ∈ R − , 3 where Xt is the normal fundamental matrix of system 2. By using the results developed in 5 for problems without impulses, the general solution ofthe problem 1 boundedonthe semiaxes has the form x t, ξ X t ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Pξ t 0 PX −1 s f s ds − ∞ t I − P X −1 s f s ds j i1 PX −1 τ i γ i − ∞ ij1 I − P X −1 τ i γ i ,t≥ 0; I − Q ξ t −∞ QX −1 s f s ds − 0 t I − Q X −1 s f s ds −j1 i−∞ QX −1 τ i γ i − −1 i−j I − Q X −1 τ i γ i ,t≤ 0. 4 For getting the solution xt ∈ BC 1 R \{τ i } I boundedontheentire axis, we assume that it has continuity in t 0: x 0,ξ − x 0−,ξ γ 0 0 5 Advances in Difference Equations 3 or Pξ − ∞ 0 I − P X −1 s f s ds − ∞ i1 I − P X −1 τ i γ i I − Q ξ 0 −∞ QX −1 s f s ds −1 i−∞ QX −1 τ i γ i . 6 Thus, the solution 4 will be boundedon R if and only if the constant vector ξ ∈ R n is the solution ofthe algebraic system: Dξ 0 −∞ QX −1 s f s ds ∞ 0 I − P X −1 s f s ds −1 i−∞ QX −1 τ i γ i ∞ i1 I − P X −1 τ i γ i , 7 where D is an n × n matrix, D : P − I − Q. The algebraic system 7 is solvable if and only if the condition P D ∗ 0 −∞ QX −1 s f s ds ∞ 0 I − P X −1 s f s ds −1 i−∞ QX −1 τ i γ i ∞ i1 I − P X −1 τ i γ i 0 8 is satisfied, where P D ∗ is the n × n matrix-orthoprojector; P D ∗ : R n → ND ∗ . Therefore, the constant ξ ∈ R n in the expression 4 has the form ξ D 0 −∞ QX −1 s f s ds ∞ 0 I − P X −1 s f s ds −1 i−∞ X t QX −1 τ i γ i ∞ i1 X t I − P X −1 τ i γ i P D c, ∀c ∈ R n , 9 where P D is the n × n matrix-orthoprojector; P D : R n → ND; D is a Moore-Penrose pseudoinverse matrix to D. Since P D ∗ D 0, we have P D ∗ Q P D ∗ I − P .Let d rank P D ∗ Q rank P D ∗ I − P ≤ n. 10 Then we denote by P D ∗ Q d a d × n matrix composed of a complete system of d linearly independent rows ofthe matrix P D ∗ Q and by H d tP D ∗ Q d X −1 t a d × n matrix. 4 Advances in Difference Equations Thus, the necessary and sufficient condition for the existence ofthe solution of problem 1 has the form ∞ −∞ H d t f t dt ∞ i−∞ H d τ i γ i 0 11 and consists of d linearly independent conditions. If we substitute the constant ξ ∈ R n given by relation 9 into 4, we get the general solution of problem 1 in the form x t, c X t ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ PP D c t 0 PX −1 s f s ds − ∞ t I − P X −1 s f s ds j i1 PX −1 τ i γi− ∞ ij1 I − P X −1 τ i γ i PD 0 −∞ QX −1 s f s ds ∞ 0 I − P X −1 s f s ds −1 i−∞ QX −1 τ i γ i ∞ i1 I − P X −1 τ i γ i ,t≥ 0; I − Q P D c t −∞ QX −1 s f s ds − 0 t I − Q X −1 s f s ds −j1 i−∞ QX −1 τ i γ i − −1 i−j I − Q X −1 τ i γ i I − Q D 0 −∞ QX −1 s f s ds ∞ 0 I − P X −1 s f s ds −1 i−∞ QX −1 τ i γ i ∞ i1 I − P X −1 τ i γ i ,t≤ 0. 12 Since DP D 0, we have PP D I − QP D .Let r rank PP D rank I − Q P D ≤ n. 13 Then we denote by PP D r an n × r matrix composed of a complete system of r linearly independent columns ofthe matrix PP D . Thus, we have proved the following statement. Theorem 1. Assume that thelinear nonhomogeneous impulsive differential system 1 has the corresponding homogeneous system 2 e-dichotomous onthe semiaxes R − −∞, 0 and R 0, ∞ with projectors P and Q, respectively. Then the homogeneous system 2 has exactly r r rank PP D rank I − QP D ,D P − I − Q linearly independent solutionsboundedontheentirereal axis. If nonhomogenities ft ∈ BCR \{τ i } I and γ i ∈ R n satisfy d d rank P D ∗ Q rank P D ∗ I − P linearly independent conditions 11, then the nonhomogeneous system 1 Advances in Difference Equations 5 possesses an r-parameter family of linearly independent solutionsboundedontheentirereal axis R in the form x t, c r X r t c r G f γ i t , ∀c r ∈ R r , 14 where X r t : X t PP D r X t I − QP D r 15 is an n × r matrix formed by a complete system of r linearly independent solutionsof homogeneous problem 2 and G f γ i t is the generalized Green operator ofthe problem of finding solutionsoftheimpulsive problem 1 boundedon R, acting upon ft ∈ BCR \{τ i } I and γ i ∈ R n , defined by the formula G f γ i t X t ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ t 0 PX −1 s f s ds − ∞ t I − P X −1 s f s ds j i1 PX −1 τ i γi− ∞ ij1 I − P X −1 τ i γ i PD 0 −∞ QX −1 s f s ds ∞ 0 I − P X −1 s f s ds −1 i−∞ QX −1 τ i γ i ∞ i1 I − P X −1 τ i γ i ,t≥ 0; t −∞ QX −1 s f s ds − 0 t I − Q X −1 s f s ds −j1 i−∞ QX −1 τ i γ i − −1 i−j I − Q X −1 τ i γ i I − Q D 0 −∞ QX −1 s f s ds ∞ 0 I − P X −1 s f s ds −1 i−∞ QX −1 τ i γ i ∞ i1 I − P X −1 τ i γ i ,t≤ 0. 16 The generalized Green operator 16 has the following property: G f γ i 0 − 0 − G f γ i 0 0 ∞ −∞ H t f t dt ∞ i−∞ H τ i γ i , 17 where HtP D ∗ QX −1 t. 6 Advances in Difference Equations We can also formulate the following corollaries. Corollary 2. Assume that the homogeneous system 2 is e-dichotomous on R and R − with projec- tors P and Q, respectively, and such that PQ QP Q. In this case, the system 2 has r-parameter set ofsolutionsboundedon R in the form 14. The nonhomogeneous impulsive system 1 has for arbitrary ft ∈ BCR \{τ i } I and γ i ∈ R n an r-parameter set ofsolutionsboundedon R in the form x t, c r X r t c r G f γ i t , ∀c r ∈ R r , 18 where G f γ i t is the generalized Green operator 16 ofthe problem of finding boundedsolutionsoftheimpulsive system 1 with the property G f γ i 0 − 0 − G f γ i 0 0 0. 19 Proof. Since DP P − I − QP QP Q and P D ∗ D 0, we have P D ∗ Q P D ∗ DP 0. Thus condition 11 for the existence ofbounded solution of system 1 is satisfied for all ft ∈ BCR \{τ i } I and γ i ∈ R n . Corollary 3. Assume that the homogenous system 2 is e-dichotomous on R and R − with projectors P and Q, respectively, and such that PQ QP P . In this case, the system 2 has only trivial solution boundedon R. If condition 11 is satisfied, then the nonhomogeneous impulsive system 1 possesses a unique solution boundedon R in the form x t G f γ i t , 20 where G f γ i t is the generalized Green operator 16 ofthe problem of finding boundedsolutionsoftheimpulsive system 1. Proof. Since PD PP−I −Q PQ P and DP D 0, we have PP D PDP D 0. By virtue of Theorem 1, we have r 0 and thus the homogenous system 2 has only trivial solution boundedon R. Moreover, the nonhomogeneous impulsive system 1 possesses a unique solution boundedon R for ft ∈ BCR \{τ i } I and γ i ∈ R n satisfying the condition 11. Corollary 4. Assume that the homogenous system 2 is e-dichotomous on R and R − with projectors P and Q, respectively, and such that PQ QP P Q. Then the system 2 is e-dichotomous on R and has only trivial solution boundedon R. The nonhomogeneous impulsive system 1 has for arbitrary ft ∈ BCR \{τ i } I and γ i ∈ R n a unique solution boundedon R in the form x t G f γ i t , 21 where G f γ i t is the Green operator 16D D −1 ofthe problem of finding boundedsolutionsoftheimpulsive system 1. Advances in Difference Equations 7 Proof. Since PQ QP Q P and det D / 0, we have P D ∗ P D 0,D D −1 .Byvirtueof Theorem 1, we have r d 0 and thus the homogenous system 2 has only trivial solution boundedon R. Moreover, the nonhomogeneous impulsive system 1 possesses a unique solution boundedon R for all ft ∈ BCR \{τ i } I and γ i ∈ R n . Regularization ofLinear Problem The condition of solvability 11 ofimpulsive problem 1 for solutionsboundedon R enables us to analyze the problem of regularization oflinear problem that is not solvable everywhere by adding an impulsive action. Consider the problem of finding solutionsboundedontheentirereal axis ofthe system ˙x A t x f t ,A t ∈ BC R ,f t ∈ BC R , 22 the corresponding homogeneous problem of which is e-dichotomous onthe semiaxes R and R − . Assume that this problem has no solution boundedon R for some f 0 t ∈ BCR;i.e.the solvability condition of 22 is not satisfied. This means that ∞ −∞ H d t f 0 t dt / 0. 23 In this problem, we introduce an impulsive action for t τ 1 ∈ R as follows: Δx| tτ 1 γ 1 ,γ 1 ∈ R n , 24 and we consider the existence of solution oftheimpulsive problem 22-24 from the space BC 1 R \{τ 1 } I boundedontheentirereal axis. The parameter γ 1 is chosen from a condition similar to 11 guaranteeing that theimpulsive problem 22-24 is solvable for any f 0 t ∈ BCR and some γ 1 ∈ R n : ∞ −∞ H d t f 0 t dt H d τ 1 γ 1 0, 25 where H d τ 1 is a d × n matrix, H d τ 1 is an n × d matrix pseudoinverse to the matrix H d τ 1 , P NH ∗ d is a d × d matrix othoprojector, P NH ∗ d : R d → NH ∗ d ,andP NH d is an n × n matrix othoprojector, P NH d : R n → NH d . The algebraic system 25 is solvable if and only if the condition P NH ∗ d ∞ −∞ H d t f 0 t dt 0 26 is satisfied. Thus, Theorem 1 yields the following statement. 8 Advances in Difference Equations Corollary 5. By adding an impulsive action, the problem of finding solutionsboundedon R oflinear system 22, that is solvable not everywhere, can be made solvable for any f 0 t ∈ BCR if and only if P NH ∗ d 0 or rank H d τ 1 d. 27 The indicated additional (regularizing) impulse γ 1 should be chosen as follows: γ 1 −H d τ 1 ∞ −∞ H d t f 0 t dt P NH d c, ∀c ∈ R n . 28 So theimpulsive action can be regarded as a control parameter which guarantees the solvability of not everywhere solvable problems. Example 6. In this example we illustrate the assertions proved above. Consider theimpulsive system ˙x A t x f t ,t / τ i , Δx| tτ i γ i ⎛ ⎜ ⎜ ⎜ ⎝ γ 1 i γ 2 i γ 3 i ⎞ ⎟ ⎟ ⎟ ⎠ ∈ R 3 ,t,τ i ∈ R,i∈ Z, 29 where Atdiag{− tanh t, − tanh t, tanh t}, ftcolf 1 t,f 2 t,f 3 t ∈ BCR. The normal fundamental matrix ofthe corresponding homogenous system ˙x A t x, t / τ i , Δx| tτ i 0 30 is X t diag 2 e t e −t , 2 e t e −t , e t e −t 2 , 31 and this system is e-dichotomous as shown in 9 onthe semiaxes R and R − with projectors P diag{1, 1, 0} and Q diag{0, 0, 1}, respectively. Thus, we have D 0,D 0,P ND P ND ∗ I 3 , r rank PP ND 2,d rank P ND ∗ Q 1, X r t ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 2 e t e −t 0 0 2 e t e −t 00 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , H d t 0, 0, 2 e t e −t . 32 Advances in Difference Equations 9 In order that theimpulsive system 29 with the matrix At specified above has solutionsboundedontheentirereal axis, the nonhomogenities ftcol f 1 t,f 2 t,f 3 t ∈ BCR and γ i ∈ R 3 must satisfy condition 11. In the analyzed impulsive problem, this condition takes the following form: ∞ −∞ 2f 3 t e t e −t dt ∞ i−∞ 2 e τ i e −τ i γ 3 i 0, ∀f 1 t ,f 2 t ∈ BC R , ∀γ 1 i ,γ 2 i ∈ R. 33 If we consider the system 29 only with one point of discontinuity ofthe first kind t τ 1 ∈ R with impulse Δx| tτ 1 γ 1 ∈ R 3 , 34 then we rewrite the condition 33 in the form ∞ −∞ 2f 3 t e t e −t dt 2 e τ 1 e −τ 1 γ 3 1 0. 35 It is easy to see that 35 is always solvable and, according to Corollary 5, the analyzed impulsive problem has bounded solution for arbitrary f 0 t ∈ BCR if the pulse parameter γ 1 should be chosen as follows: γ 3 1 − e τ 1 e −τ 1 ∞ −∞ f 3 t e t e −t dt, ∀γ 1 1 ,γ 2 1 ∈ R. 36 Remark 7. It seems that a possible generalization to systems with delay will be possible. In a particular case when the matrix oflinear terms is constant, a representation ofthe fundamental matrix given by a special matrix function so-called delayed matrix exponential, etc., for example, in 10, 11for a continuous case and in 12, 13for a discrete case, can give concrete formulas expressing solution ofthe considered problem in analytical form. Acknowledgments This research was supported by the Grants 1/0771/08 and 1/0090/09 ofthe Grant Agency of Slovak Republic VEGA and project APVV-0700-07 of Slovak Research and Development Agency. 10 Advances in Difference Equations References 1 A. D. Myshkis and A. M. Samoilenko, “Systems with impulses at given instants of time,” Mathematics Sbornik, vol. 74, no. 2, pp. 202–208, 1967 Russian. 2 A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations, Vyshcha Shkola, Kiev, Russia, 1974. 3 A. Halanay and D. Wexler, Qualitative Theory ofImpulsive Systems, vol. 309, Mir, Moscow, Russia, 1971. 4 ˇ S. Schwabik, M. Tvrdy, and O. 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We consider the problem of existence and construction of solutions bounded on the entire. analyze the problem of regularization of linear problem that is not solvable everywhere by adding an impulsive action. Consider the problem of finding solutions bounded on the entire real axis of the. Corporation Advances in Difference Equations Volume 2010, Article ID 494379, 10 pages doi:10.1155/2010/494379 Research Article Solutions of Linear Impulsive Differential Systems Bounded on the Entire Real