Here, a predator-prey model with Hassell-Varley type functional responses is studied. Some sufficient conditions are obtained for the permanence and global asymptotic stability of the system by using comparison theorem and constructing a suitable Lyapunov functional. Moreover, an example is illustrated to verify the results by simulation.

In this paper a new mathematical model is developed for the dynamics between tumor cells, normal cells, immune cells, chemotherapy drug concentration and drug toxicity. Then, the theorem of Lyapunov stability is applied to design treatment strategies for drug protocols that ensure a desired rate of tumor cell kill and push the system to the area with smaller tumor cells. Using of this theorem a...

In this paper we make use of the alternative converse Lyapunov theorem presented in [1] for specific classes of systems. We show that the developed converse Lyapunov theorem can be used to establish non–conservatism of a particular type of Lyapunov functions. Most notably, a proof that the existence of conewise linear Lyapunov functions are non– conservative for globally exponentially stable (G...

One of the most fundamental results in analysing the stability properties of periodic orbits and limit cycles of dynamical systems is Poincaré’s theorem. The proof of this result involves system analytic arguments along with the Hartman–Grobman theorem. Using the notions of stability of sets, lower semicontinuous Lyapunov functions are constructed to provide a Lyapunov function proof of Poincar...

In this article we use Lyapunov Second Theorem as well as LaSalle Theorem in order to determine the attraction region of an optical trap and to design an steering strategy to move a microscopic particle by means of optical tweezers. A variational approach based on Lagrangian formulation is used to obtain the equations of motion of a particle in the optical trap[1]. System is proved to be locall...

Theorem of the alternatives for linear matrix inequality based on the separation theorems are proposed to provide systematic and unified proofs of necessary and sufficient conditions for solvability of Lyapunov inequality on P . As an application of the Theorem of the alternatives, we investigate the conditions for the existence of feasible solutions to Lyapunov inequality on P and obtain a res...

A qualitative method is explored for analyzing the stability of systems. The approach is a generalization of the celebrated Lyapunov method. Whereas classically, the Lyapunov method is based on the simple comparison theorem, deriving suitable candidate Lyapunov functions remains mostly an art. As a result, in the realm of delay equations, such Lyapunov methods can be quite conservative. The gen...

We prove a converse Lyapunov theorem for almost sure stabilizability of controlled diffusions: given a stochastic system a.s. open loop stabilizable at the origin, we construct a lower semicontinuous positive definite function whose level sets form a local basis of viable neighborhoods of the equilibrium. This result provides, with the direct Lyapunov theorem proved in a companion paper, a comp...