Lyapunov functions: necessary and sufficient conditions for practical stability

We prove the necessary and sufficient conditions for practical stability of nonlinear dynamical system at general phase restrictions. In such a case the Lyapunov function is nondifferentiable. But if the set of initial data is starry compact, then it is possible building Lyapunov function which belongs to differentiable functions class. We also estimate the optimal sets of initial conditions in structural forms for linear system and concrete phase restrictions.