finding a suitable estimation of stability domain around stable equilibrium points is an important issue in the study of nonlinear dynamical systems. this paper intends to apply a set of analytical-numerical methods to estimate the region of attraction for autonomous nonlinear systems. in mechanical and structural engineering, autonomous systems could be found in large deformation problems or c...

Vladimir Ivanovich Zubov was born on April 14, 1930 in the town of Kashira, Moscow region, Russia. In 1945 he finished a secondary school. At the age of 14, Vladimir suffered in the explosion accident happened when he and his playfellows found a hand grenade. His eyes were injured and he failed eyesight soon. In 1949 he finished the Leningrad special school for blind and visually impaired child...

The main goal of this paper is to weaken the invariance assumptions in Liapunov’s direct method. Various geometric conditions resembling those provided by the collection of level surfaces of Liapunov functions are investigated. They are used to derive necessary resp. sufficient conditions for the asymptotic stability of equilibrium points of dynamical systems in Banach spaces. The main result —...

In a seminal work V.I. Zubov has described a constructive method to obtain Lyapunov functions. This method is the first to allow numerical construction of domains of attraction for general nonlinear systems. In this paper we describe recent generalizations of this method that is applicable for systems subject to perturbations or control inputs.

In a series of papers, we characterized the value function in optimal control as the unique viscosity solution of the corresponding Bellman equation that satisfies appropriate side conditions. The novelty of our results was that they applied to exit time problems with general nonnegative instantaneous costs, including cases where the instantaneous cost is not uniformly bounded below by positive...

In a series of papers, we characterized the value function in optimal control as the unique viscosity solution of the corresponding Bellman equation that satisfies appropriate side conditions. The novelty of our results was that they applied to exit time problems with general nonnegative instantaneous costs, including cases where the instantaneous cost is not uniformly bounded below by positive...

In this paper we provide generalizations of Zubov’s equation to differential games without Isaacs’ condition. We show that both generalizations of Zubov’s equation (which we call min-max and max-min Zubov equation, respectively) possess unique viscosity solutions which characterize the respective controllability domains. As a consequence, we show that under the usual Isaacs condition the respec...

This work concerns the optimal regulation of single-input-single-output nonminimum-phase nonlinear processes of relative order one. The problem of calculation of an ISE-optimal, statically equivalent, minimum-phase output for nonminimum-phase compensation is formulated using Hamilton-Jacobi theory and the normal form representation of the nonlinear system. A Newton-Kantorovich iteration is deve...

This work proposes a systematic methodology for the optimal selection of controller parameters, in the sense of minimizing a performance index which is a quadratic function of the tracking error and the control effort. The performance index is calculated explicitly as an algebraic function of the controller parameters by solving a Zubov-type partial differential equation. Standard nonlinear pro...

In recent years, the problem of determining the asymptotic stability region of autonomous nonlinear dynamic systems has been developed in several researches. Many methods, usually based on approaches using Lyapunov’s candidate functions (Davidson & Kurak, 1971) and (Tesi et al., 1996) which altogether allow for a sufficient stability region around an equilibrium point. Particularly, the method ...