Stability of multi-dimensional switched systems with an application to open multi-agent systems

A multi-dimensional switched system or multi-mode ($M^3D$) extends the classic by allowing different subsystem dimensions. The stability problem of $M^3D$ system, whose state transitions at switching instants can be discontinuous due to dimension-varying feature, is studied. transition formulated an affine map that captures both dimension variations and impulses, with no extra constraint imposed. In presence unstable subsystems, general criteria featuring a series Lyapunov-like conditions for practical asymptotic properties are provided under proposed slow/fast transition-dependent average dwell time framework. Then, considering linear we propose class parametric multiple Lyapunov functions verify obtained explicitly reveal connection between practical/asymptotic property non-vanishing/vanishing impulsive effects in process. Further, results applied consensus open multi-agent (MAS), network topology size-varying migrations agents. It shows through proper transformation, seeking (practical) performance MAS disconnected digraphs boils down subsystems.