Autonomy, forward non-Zenoness and quadratic stability of bimodal higher-order piecewise linear systems

We introduce bimodal higher-order piecewise linear systems, i.e. the sets of solutions of two n-th order linear differential equations with n ≥ 1, coupled with an inequality constraint defined by a polynomial differential operator acting on the system trajectories. Under suitable assumptions on the characteristic polynomials of the differential equations and the polynomial associated with the inequality constraint, we prove that a solution always exists and is unique given the initial conditions, that no forward Zeno-behavior occurs, and that the trajectories are continuous together with their first n - 1 derivatives. Moreover, we prove that such systems are quadratically stable and we provide an algorithm based on polynomial algebra to compute a Lyapunov function.