An Extension of Lyapunov’s First Method to Nonlinear Systems With Non-Continuously Differentiable Vector Fields

This letter investigates the extension of Lyapunov's first method to nonlinear systems in the case where the C
¹
-regularity assumption, i.e., the underlying vector field is continuously differentiable, is not satisfied. It is shown that if this regularity assumption is not fulfilled, the Hurwitz nature of the Jacobian matrix, if it exists, does not guarantee the stability of the original nonlinear system. Under weaker assumptions than the C
¹
-regularity, namely the existence of the directional derivatives of the vector field, conditions for guaranteeing the local exponential stability of the nonlinear system are derived.