Basics of Linear Systems

Chapter

Abstract

We start discussion of linear impulsive systems with the following differential equation:
$$\begin{array}{rcl} & & x^\prime = A(t)x, \\ & & \Delta x{\vert }_{t={\theta }_{i}} ={B}_{i}x,\end{array}$$
(4.1)
where $$(t,x) \in \mathbb{R} \times {\mathbb{R}}^{n},{\theta }_{i},i \in \mathbb{Z},$$ is a B-sequence, such that | θ i | → as | i | → . We suppose that the entries of n ×n matrix A(t) are from $$\mathcal{P}C(\mathbb{R},\theta ),$$ real valued n ×n matrices $${B}_{i},i \in \mathbb{Z},$$ satisfy
$$\det (\mathcal{I} + {B}_{i})\not =0,$$
(4.2)
where $$\mathcal{I}$$ is the identical n ×n matrix.

References

1. 33.
M.U. Akhmetov, N.A. Perestyuk, The comparison method for differential equations with impulse action, Differ. Equ., 26 (1990) 1079–1086.
2. 59.
E.A. Coddington, N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955.
3. 60.
C. Corduneanu, Principles of differential and integral equations, Chelsea Publishing Co., Bronx, NJ, 1977.Google Scholar
4. 75.
A. Halanay, D. Wexler, Qualitative theory of impulsive systems (romanian), Edit. Acad. RPR, Bucuresti, 1968.Google Scholar
5. 78.
C.S. Hcu, W.H. Cheng, Applications of the theory of impulsive parametric excitation and new treatment of general parametric excitations problems, Trans. ASME, 40 (1973) 2174–2181.Google Scholar
6. 111.
A.D. Myshkis, A.M. Samoilenko, Systems with impulses at fixed moments of time (russian), Math. Sb., 74 (1967) 202–208.Google Scholar
7. 138.
A.M. Samoilenko, N.A. Perestyuk, Stability of solutions of impulsive differential equations (russian), Differentsial’nye uravneniya, 13 (1977) 1981–1992.Google Scholar
8. 139.
A.M. Samoilenko, N.A. Perestyuk, Periodic solutions of weakly nonlinear impulsive differential equations (russian), Differentsial’nye uravneniya, 14 (1978) 1034–1045.Google Scholar
9. 140.
A.M. Samoilenko, N.A. Perestyuk, On stability of solutions of impulsive systems (russian), Differentsial’nye uravneniya, 17 (1981) 1995–2002.
10. 141.
A.M. Samoilenko, N.A. Perestyuk, Differential Equations with impulsive actions (russian), Vishcha Shkola, Kiev, 1987.Google Scholar
11. 142.
A.M. Samoilenko, N.A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995.