We provide a formula for a stabilizing feedback law using a bounded control, under the assumption that an appropriate control-Lyapunov function is known. Such a feedback, smooth away from the origin and continuous everywhere, is known to exist via Artstein’s Theorem. As in the unbounded-control case treated in a previous note, we provide an explicit and “universal” formula given by an algebraic...

– The minimal time system design algorithm was defined as the problem of functional minimization of the control theory. By this methodology the aim of the system design process with minimal computer time is presented as a transition process of dynamic system with minimal transition time. The optimal sequence of the control vector switch points was determined as a principal characteristic of the...

This paper deals with the problem of tracking control for Takagi-Sugeno fuzzy model. An LMI (Linear Matrix Inequality) formulation is suggested to make possible the convergence of the state vector of the continuous-time system to a desired one using a new approach, called MultiQuadratic Fuzzy Lyapunov (MQFL). A fourth order unstable nonlinear system is studied to illustrate the efficiency of th...

The appropriate choice of the forms of Lyapunov functions for a positive 2D Roesser model is addressed. It is shown that for the positive 2D Roesser model: (i) a linear form of the state vector can be chosen as a Lyapunov function, (ii) there exists a strictly positive diagonal matrix P such that the matrix ATPA−P is negative definite. The theoretical deliberations will be illustrated by numeri...

We give an explicit example of a two-dimensional polynomial vector field of degree seven that has rational coefficients, is globally asymptotically stable, but does not admit an analytic Lyapunov function even locally.

This work addresses the problem of estimating the region of attraction (RA) of equilibrium points of nonlinear dynamical systems. The estimates we provide are given by positively invariant sets which are not necessarily defined by level sets of a Lyapunov function. Moreover, we present conditions for the existence of Lyapunov functions linked to the positively invariant set formulation we propo...

The notion of Lyapunov function plays a key role in design and verification of dynamical systems, as well as hybrid and cyber-physical systems. In this paper, to analyze the asymptotic stability of a dynamical system, we generalize standard Lyapunov functions to relaxed Lyapunov functions (RLFs), by considering higher order Lie derivatives of certain functions along the system’s vector field. F...

In game theory the interaction among players obligates each player to develop a belief about the possible strategies of the other players, to choose a best-reply given those beliefs, and to look for an adjustment of the best-reply and the beliefs using a learning mechanism until they reach an equilibrium point. Usually, the behavior of an individual cost-function, when such best-reply strategie...

The problem of pth mean exponential stability and stabilizability of a class of stochastic nonlinear and bilinear hybrid systems with unstable and stable subsystems is considered. Sufficient conditions for the pth mean exponential stability and stabilizability under a feedback control and stabilizing switching rules are derived. A method for the construction of stabilizing switching rules based...