# Solutions of Linear Impulsive Differential Systems Bounded on the Entire Real Axis

- Alexandr Boichuk
^{1}Email author, - Martina Langerová
^{1}and - Jaroslava Škoríková
^{1}

**2010**:494379

**DOI: **10.1155/2010/494379

© Alexandr Boichuk et al. 2010

**Received: **21 January 2010

**Accepted: **12 May 2010

**Published: **9 June 2010

## Abstract

We consider the problem of existence and structure of solutions bounded on the entire real axis of nonhomogeneous linear impulsive differential systems. Under assumption that the corresponding homogeneous system is exponentially dichotomous on the semiaxes and and by using the theory of pseudoinverse matrices, we establish necessary and sufficient conditions for the indicated problem.

The research 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 of impulsive differential equations, using the well-known results on the splitting index by Sacker [6] and by Palmer [7] on the Fredholm property of the problem of bounded solutions and using the theory of pseudoinverse matrices [5, 8], to investigate, in a relevant space, the existence of solutions bounded on the entire real axis of linear differential systems with impulsive action.

where is an matrix of functions; is an vector function; is the Banach space of real vector functions continuous for with discontinuities of the first kind at ; are -dimensional column constant vectors; .

The solution of the problem (1) is sought in the Banach space of -dimensional piecewise continuously differentiable vector functions with discontinuities of the first kind at : .

which is the homogeneous system without impulses.

where is the normal fundamental matrix of system (2).

is satisfied, where is the matrix-orthoprojector; .

Then we denote by a matrix composed of a complete system of linearly independent rows of the matrix and by a matrix.

and consists of linearly independent conditions.

Then we denote by an matrix composed of a complete system of linearly independent columns of the matrix .

Thus, we have proved the following statement.

Theorem 1.

where .

We can also formulate the following corollaries.

Corollary 2.

Proof.

Since and , we have . Thus condition (11) for the existence of bounded solution of system (1) is satisfied for all and .

Corollary 3.

where is the generalized Green operator (16) of the problem of finding bounded solutions of the impulsive system (1).

Proof.

Since and , we have . By virtue of Theorem 1, we have and thus the homogenous system (2) has only trivial solution bounded on . Moreover, the nonhomogeneous impulsive system (1) possesses a unique solution bounded on for and satisfying the condition (11).

Corollary 4.

where is the Green operator (16) of the problem of finding bounded solutions of the impulsive system (1).

Proof.

Since and , we have . By virtue of Theorem 1, we have and thus the homogenous system (2) has only trivial solution bounded on . Moreover, the nonhomogeneous impulsive system (1) possesses a unique solution bounded on for all and .

Regularization of Linear Problem

The condition of solvability (11) of impulsive problem (1) for solutions bounded on enables us to analyze the problem of regularization of linear problem that is not solvable everywhere by adding an impulsive action.

is satisfied. Thus, Theorem 1 yields the following statement.

Corollary 5.

So the impulsive 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.

Remark 7.

It seems that a possible generalization to systems with delay will be possible. In a particular case when the matrix of linear terms is constant, a representation of the fundamental matrix given by a special matrix function (so-called delayed matrix exponential, etc.), for example, in [10, 11] (for a continuous case) and in [12, 13] (for a discrete case), can give concrete formulas expressing solution of the considered problem in analytical form.

## Declarations

### Acknowledgments

This research was supported by the Grants 1/0771/08 and 1/0090/09 of the Grant Agency of Slovak Republic (VEGA) and project APVV-0700-07 of Slovak Research and Development Agency.

## Authors’ Affiliations

## References

- Myshkis AD, Samoilenko AM:
**Systems with impulses at given instants of time.***Mathematics Sbornik*1967,**74**(2):202-208.Google Scholar - Samoilenko AM, Perestyuk NA:
*Impulsive Differential Equations*. Vyshcha Shkola, Kiev, Russia; 1974.Google Scholar - Halanay A, Wexler D:
*Qualitative Theory of Impulsive Systems*.*Volume 309*. Mir, Moscow, Russia; 1971.Google Scholar - Schwabik Š, Tvrdy M, Vejvoda O:
*Differential and Integral Equations, Boundary Value Problems and Adjoints*. , Academia, Prague; 1979.MATHGoogle Scholar - Boichuk AA, Samoilenko AM:
*Generalized Inverse Operators and Fredholm Boundary-Value Problems*. Koninklijke Brill NV, Utrecht, The Netherlands; 2004.View ArticleMATHGoogle Scholar - Sacker RJ:
**The splitting index for linear differential systems.***Journal of Differential Equations*1979,**33**(3):368-405. 10.1016/0022-0396(79)90072-XMathSciNetView ArticleMATHGoogle Scholar - Palmer KJ:
**Exponential dichotomies and transversal homoclinic points.***Journal of Differential Equations*1984,**55**(2):225-256. 10.1016/0022-0396(84)90082-2MathSciNetView ArticleMATHGoogle Scholar - Boichuk AA:
**Solutions of weakly nonlinear differential equations bounded on the whole line.***Nonlinear Oscillations*1999,**2**(1):3-10.MathSciNetMATHGoogle Scholar - Samoilenko AM, Boichuk AA, Boichuk AnA:
**Solutions, bounded on the whole axis, of linear weakly perturbed systems.***Ukrainian Mathematical Zhurnal*2002,**54**(11):1517-1530.MathSciNetMATHGoogle Scholar - Diblík J, Khusainov DYa, Lukáčová J, Růžičková M:
**Control of oscillating systems with a single delay.***Advances in Difference Equations*2010,**2010:**-15.Google Scholar - Boichuk A, Diblík J, Khusainov DYa, Růžičková M:
**Boundary-value problems for delay differential systems.***Advances in Difference Equations*. In press - Diblík J, Khusainov DYa:
**Representation of solutions of linear discrete systems with constant coefficients and pure delay.***Advances in Difference Equations*2006,**2006:**-13.Google Scholar - Diblík J, Khusainov DYa, Růžičková M:
**Controllability of linear discrete systems with constant coefficients and pure delay.***SIAM Journal on Control and Optimization*2008,**47**(3):1140-1149. 10.1137/070689085MathSciNetView ArticleMATHGoogle Scholar

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