Abstract
Suppose we are given a system of ordinary differential equations (1) x′=f(x,t) (x,f are vectors), a sequence t0<t1<t2⋯→∞ and a vector-valued sequence Δix. The present paper considers discontinuous solutions of (1), that is, functions x(t), which, for ti<t<ti+1, satisfy (1), for t=ti, are discontinuous and x(ti+0)−x(ti−0)=Δix (i=1,2,⋯). Δix are called by the authors impulses.
The boundness and stability of such solutions of (1) are proved in several cases. Roughly speaking, this is obtained under the assumptions that the trivial solution of (1) is asymptotically stable in the usual sense (in case (1) is linear, it is assumed that the corresponding matrix has characteristic roots with real parts negative, and in case (1) is non-linear, the existence of an appropriate Liapunov function is assumed), the impulses Δix satisfy certain bounds, and ti+1−ti is bounded away from zero.
The boundness and stability of such solutions of (1) are proved in several cases. Roughly speaking, this is obtained under the assumptions that the trivial solution of (1) is asymptotically stable in the usual sense (in case (1) is linear, it is assumed that the corresponding matrix has characteristic roots with real parts negative, and in case (1) is non-linear, the existence of an appropriate Liapunov function is assumed), the impulses Δix satisfy certain bounds, and ti+1−ti is bounded away from zero.
| Original language | Russian |
|---|---|
| Pages (from-to) | 233-237 |
| Number of pages | 5 |
| Journal | Sibirsk. Mat. Ž. |
| Volume | 1 |
| State | Published - 1960 |