Finite-time partial stability theory and fractional Lyapunov differential inequalities

In many practical applications, stability with respect to part of the system´s states is often necessary with finite-time convergence to the equilibrium state of interest. Finite-time partial stability involves dynamical systems whose part of the trajectory converges to an equilibrium state in finite time. Since finite-time convergence implies non-uniqueness of system solutions in backward time, such systems possess non- Lipschitzian dynamics. In this paper, we address finite-time partial stability and uniform finite-time partial stability for nonlinear dynamical systems. Specifically, we provide Lyapunov conditions involving a Lyapunov function that is positive definite and decrescent with respect to part of the system state, and satisfies a differential inequality involving fractional powers for guaranteeing finite-time partial stability. In addition, we show that finite-time partial stability leads to uniqueness of solutions in forward time and we establish necessary and sufficient conditions for continuity of the settling-time function of the nonlinear dynamical system.