We present a method for the construction of Foster-Lyapunov functions to prove ergodicity of hybrid piecewise-deterministic Markov processes. Our method searches for the Lyapunov function that maximizes a measure of the rate of convergence that appears in the theory of large deviations. We provide conditions under which such a Lyapunov function is the limit of an iterative procedure. Analytical...

We propose discrete mappings of second order that have a discrete analogue of Lyapunov function. The mappings are extensions of the integrable Quispel-RobertsThompson (QRT) mapping, and a discrete Lyapunov function of the mappings is identical to an explicit conserved quantity of the QRT mapping. Moreover we can obtain a differential and an ultradiscrete limit of the mappings preserving the exi...

Closed physical systems eventually come to rest, the reason being that due to friction of some kind they continuously lose energy. The mathematical extension of this principle is the concept of a Lyapunov function. A Lyapunov function for a dynamical system, of which the dynamics are modelled by an ordinary differential equation (ODE), is a function that is decreasing along any trajectory of th...

This paper focuses on the design of control Lyapunov function for control affine systems to guarantee the stability for the states of interest in a specified region. Without restrictive assumptions found in previous approaches, a min-max optimization problem is formulated to solve for a quadratic Lyapunov function. A derivative-free coordinate search method is employed to optimize a non-differe...

The Lyapunov stability method is the most popular and applicable stability analysis tool of nonlinear dynamic systems. However, there are some bottlenecks in the Lyapunov method, such as need for negative definiteness of the Lyapunov function derivative in the direction of the system’s solutions. In this paper, we develop a new theorem to dispense the need for negative definite-ness of Lyapunov...

In this paper, Gain-Scheduling (GS) control via parameter dependent Lyapunov function for crane is synthesized. A purpose of cranes system is to lift and carry a load to assigned positions. Cranes are required to transport the load fast, accurately and to reduce oscillations. In previous research, a method to robust control of cranes to suppress effects by variation of the rope length have been...

this paper studies the consensus problem of nonlinear leader-following multi-agent systems (mas). to do this, the error dynamics between the leader agent and follower ones are described via a takagi-sugeno (ts) fuzzy model. if the obtained ts fuzzy model is stable, then all of the nonlinear agents reach consensus. the consensus problem is investigated based on the parameterized or fuzzy lyapuno...

In this paper, we use a state-dependent Lyapunov function to reduce the conservatism in stability analysis of Takagi-Sugeno (T-S) fuzzy systems via Linear Matrix Inequalities (LMIs). Our approach uses T-S fuzzy rules to deene Lyapunov functions, and therefore substantially enlarges the class of Lyapunov functions which can be used to prove stability of T-S fuzzy systems. This approach greatly r...

We develop tools for investigation of input-to-state stability (ISS) of infinite-dimensional control systems. We show that for certain classes of admissible inputs the existence of an ISS-Lyapunov function implies the inputto-state stability of a system. Then for the case of the systems described by abstract equations in Banach spaces we develop two methods of construction of local and global I...