Piecewise quadratic Lyapunov functions for linear control systems with First Order Reset Elements

In this paper we reconsider the piecewise quadratic Lyapunov construction preliminarily proposed in (Zaccarian et al., ACC 2005) and propose a different statement and proof which solves a subtle issue overlooked in that preliminary result. An example illustrates a situation where the result in (Zaccarian et al., ACC 2005) leads to erroneous conclusions. Another example is given to illustrate that the new statement leads to similar numerical bounds to the ones established in (Zaccarian et al., ACC 2005).