Stability analysis of large scale systems whose subsystems may be unstable.

*(English)*Zbl 0538.93048A method is proposed for studying the stability of nonlinear, stationary, interconnected systems. In particular the case is considered of two interconnected systems in the form \(\dot x_ i=f_ i(x_ i)+g_ i(x_ i,x_ 2),\quad i=1,2.\) Lyapunov functions are constructed for the individual systems (i.e. with \(g_ i=0)\), together with a function \(v_{12}(x_ 1,x_ 2)\) representing ”energy coupling” between the subsystems. A weighted sum of these functions is taken as a candidate for a Lyapunov function for the full system, and a sufficient condition is obtained for system stability. A method for constructing \(v_{12}\) is suggested and the technique is illustrated with two examples.

Reviewer: D.A.Wilson

##### MSC:

93D05 | Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory |

93A15 | Large-scale systems |

93C10 | Nonlinear systems in control theory |

93C15 | Control/observation systems governed by ordinary differential equations |

34D20 | Stability of solutions to ordinary differential equations |