Solutions of Linear Impulsive Differential Systems Bounded on the Entire Real Axis
© Alexandr Boichuk et al. 2010
Received: 21 January 2010
Accepted: 12 May 2010
Published: 9 June 2010
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 , Samoilenko and Perestyuk , Halanay and Wexler , and Schwabik et al. . The ideas proposed in these works were developed and generalized in numerous other publications . The aim of this contribution is, using the theory of impulsive differential equations, using the well-known results on the splitting index by Sacker  and by Palmer  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; .
which is the homogeneous system without impulses.
Thus, we have proved the following statement.
We can also formulate the following corollaries.
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).
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.
So the impulsive action can be regarded as a control parameter which guarantees the solvability of not everywhere solvable problems.
In this example we illustrate the assertions proved above.
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.
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.
- 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 pressGoogle Scholar
- 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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