Differential systems involving impulses.

*(English)*Zbl 0539.34001
Lecture Notes in Mathematics. 954. Berlin-Heidelberg-New York: Springer-Verlag. vii, 102 p. DM 19.80; $ 8.30 (1982).

From the authors’ preface: “An attempt is made to unify the results from several research papers published during the last fifteen years. Chapter 2 contains results on existence and uniqueness (for \(Dx=F(t,x)+G(t,x)Du\), \(Dx\), \(Du\) denote distributional derivatives, \(u\) is a function of bounded variation and \(Du\) can be viewed as a Stieltjes measure which has the effect of instantaneously changing the state). In Chapter 3 fixed point theorems and generalized integral inequalities are employed to derive results on stability and asymptotic equivalence. Chapter 4 provides a study for systems

\[ Dy=A(t)yDu+f(t,y)+g(t,y)Du\quad u(t)=t+\sum^{\infty}_{k=1}a_k H_k(t),\]

\(H_k(t)=0,\ t<t_k\), \(H_k(t)=1,\ t\geq t_ k\). Chapter 5 deals with extensions of Lyapunov’s second method to the study of impulse systems.”

\[ Dy=A(t)yDu+f(t,y)+g(t,y)Du\quad u(t)=t+\sum^{\infty}_{k=1}a_k H_k(t),\]

\(H_k(t)=0,\ t<t_k\), \(H_k(t)=1,\ t\geq t_ k\). Chapter 5 deals with extensions of Lyapunov’s second method to the study of impulse systems.”

Reviewer: Aristide Halanay (Bucureşti)

##### MSC:

34-02 | Research exposition (monographs, survey articles) pertaining to ordinary differential equations |

34A37 | Ordinary differential equations with impulses |

34D20 | Stability of solutions to ordinary differential equations |

34E99 | Asymptotic theory for ordinary differential equations |

34C10 | Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations |