Vector -gain and stability of feedback switched systems

A concept of vector L 2 -gain is presented for switched systems. Each subsystem does not necessarily have L 2 -gain in the classic sense but is assumed to have individual L 2 -gain during any time interval when the subsystem is active. Stability is derived from this vector L 2 -gain under some constraints on inactive storage functions. Asymptotic stability is also achieved if in addition a small-time norm-observability property is imposed. A small-gain theorem for feedback switched systems with vector L 2 -gain assured for each subsystem is established. The small-gain condition is given in terms of the L 2 -gains of the coupled active subsystems and the changes of inactive storage functions. A switching law design method is also given to achieve vector L 2 -gain.