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...

Some new Lyapunov-type theorems for stochastic differential equations of neutral type are proved. It is shown that these theorems simplify an application of Kolmanovskii and Shaikhet's general method of Lyapunov functionals construction for stability investigation of different mathematical models.

Iterative learning control methods are represented as powerful tools to control dynamics nowadays. Our new controller based on particular case of iterative learning control is radically different from the presented conventional method, which attempts to stabilize a class of nonlinear systems by satisfying the conditions of Lyapunov Stability Theorem. Since our algorithm is model based, its robu...

Formations that contain a small number of robots are modeled as controlled Lagrangian systems on Jacobi shape space. This allows a block-structured control of position, orientation and shape of the formation. Feedback control laws are derived using control Lyapunov functions. The controlled dynamics converges to the invariant set where desired shape is achieved. Controllers are implemented in a...

Lyapunov exponents of chaotic attractors are hard to estimate, especially for non-smooth systems. One method to estimate the maximal Lyapunov exponent is by using its relationship with the synchronization properties of coupled systems. The maximal Lyapunov exponent is equal to the minimal proportional feedback gain necessary to achieve full state synchronization with a replica system. In this p...

This paper is concerned with the problem of improving software products and investigates how to base that process on solid empirical foundations. Our key contribution is a user-centered, contextual method which provides a means of identifying new features, to support the discovered and currently unsupported ways of working, and a means of evaluating the usefulness of proposed features. Standard...

Rather than designing engineering systems from the ground up, engineers often redesign strategic portions of existing systems to accommodate emerging needs. In the redesign of mechatronic systems, engineers typically seek to meet the requirements of a new application via control redesign only, but this is often insufficient and physical system (plant) design changes must be explored. Here, an i...

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 chapter is about numerical methods for a particular type of equation expressed as a matrix equality. The Lyapunov equation is the most common problem in the class of problems called matrix equations. Other examples of matrix equations: Sylvester equation, Stein equation, Riccati equation. Definition 5.0.1 Consider two square matrices A, W ∈ Rn×n. The problem to find a square matrix X ∈ Rn×...

The basin of attraction of an asymptotically stable fixed point of the discrete dynamical system given by the iteration xn+1 = g(xn) can be determined through sublevel sets of a Lyapunov function. In [3] a Lyapunov function is constructed by approximating the solution of a difference equation using radial basis functions. However, the resulting Lyapunov function is non-local, i.e. it has no neg...