External stability and $H_{\infty }$ control of switching systems with delay and impulse

In this paper, we investigate the external stability and $H_{\infty }$ H ∞ control of switching systems with time-varying delay and impulse. First of all, a modified two-direction inequality (relation) between the switching numbers and the maximum, minimum dwell time is proposed. This new inequality is applied to proving the external stability of switching systems with delay and impulse consisting of subsystems with Hurwitz stable matrices of internal dynamics. By using this new inequality, a normal $L_{2}$ L 2 norm constraint is derived rather than weighted $L_{2}$ L 2 norm constraint. In addition, by a realizable switching law, the obtained result is extended to the switching systems comprised of subsystems with both Hurwitz stable and unstable matrices of internal dynamics. The results are finally applied to $H_{\infty }$ H ∞ control and illustrated by a numerical example.