the present article deals with the inter specific competition and intra-specific competition among predator populations of a prey-dependent three component food chain model consisting of two competitive predator sharing one prey species as their food. the behaviour of the system near the biologically feasible equilibria is thoroughly analyzed. boundedness and dissipativeness of the system are e...

Lyapunov functions are a fundamental tool to investigate stability properties of equilibrium points of linear and nonlinear systems. The existence of Lyapunov functions for asymptotically stable equilibrium points is guaranteed by converse Lyapunov theorems. Nevertheless the actual computation (of the analytic expression) of the function may be difficult. Herein we propose an approach to avoid ...

We introduce the concept of sos-convex Lyapunov functions for stability analysis of discrete time switched systems. These are polynomial Lyapunov functions that have an algebraic certificate of convexity, and can be efficiently found by semidefinite programming. We show that sos-convex Lyapunov functions are universal (i.e., necessary and sufficient) for stability analysis of switched linear sy...

An artificial neural network is proposed as a function approximator for empirical modeling of a Lyapunov function for a nonlinear dynamic system that projects stable behavior as potentially observable in its state space. Theoretical framework for the methodology of designing the so-called Lyapunov neural network, which empirically models a Lyapunov function, is described. Algorithms for trainin...

This paper consists of two main parts. In the first part, we provide a framework for stabilization of arbitrary (not necessarily compact) closed sets for sampled-data nonlinear differential inclusions via their approximate discrete-time models. We generalize [19, Theorem 1] in several different directions: we consider stabilization of arbitrary closed sets, plants described as sampled-data diff...

This paper studies the exponential stabilization problem for discrete-time switched linear systems based on a control-Lyapunov function approach. A number of versions of converse control-Lyapunov function theorems are proved and their connections to the switched LQR problem are derived. It is shown that the system is exponentially stabilizable if and only if there exists a finite integer N such...

This paper studies the exponential stabilization problem for discrete-time switched linear systems based on a control-Lyapunov function approach. A number of versions of converse control-Lyapunov function theorems are proved and their connections to the switched LQR problem are derived. It is shown that the system is exponentially stabilizable if and only if there exists a finite integer N such...

We develop a direct Lyapunov method for the almost sure open-loop stabilizability and asymptotic stabilizability of controlled degenerate diffusion processes. The infinitesimal decrease condition for a Lyapunov function is a new form of Hamilton-Jacobi-Bellman partial differential inequality of 2nd order. We give local and global versions of the First and Second Lyapunov Theorems assuming the e...

Recently the authors proved the existence of piecewise affine Lyapunov functions for dynamical systems with an exponentially stable equilibrium in two dimensions [7]. Here, we extend these results by designing an algorithm to explicitly construct such a Lyapunov function. We do this by modifying and extending an algorithm to construct Lyapunov functions first presented in [17] and further impro...