New converse Lyapunov theorems and related results on exponential stability (Q1388100)

The paper is concerned with converse Lyapunov theorems in the exponential stable case, ensuring existence of \({\mathcal C}^1\) functions satisfying \[ \omega_1| x|^m\leq V(t, x)\leq\omega_2| x|^m, \] \[ W(t, x)\leq -m\alpha V(t, x), \] where \(\alpha\), \(m\), \(\omega_1\), \(\omega_2> 0\) and \(W\) is the complete derivative function of \(V\) along the system's solutions. Most results are obtained for \(m= 2\). Applications are given as follows: linearization, cascade systems, parametrized systems for control.