Search

Sufficient conditions are proved to guarantee existence and uniqueness of the directional anti-derivative of a smooth (possibly, analytic) scalar function along a smooth (possibly, analytic) vector field. In particular, it is shown that smooth directional anti-derivatives always exist in the neighbourhood of regular points, whereas their existence in the neighbourhood of singular points is guaranteed just in the case that the eigenvalues of the corresponding linearised system are linearly independent over the set of non-negative integer numbers. If, additionally, the scalar function and the vector field are both analytic and the eigenvalues of the linearised system belong to the Poincaré domain, then the directional anti-derivative is analytic. Based on the theoretical results, techniques are proposed to either compute or approximate such a function. Several examples are proposed to illustrate how the directional anti-derivative can be used to solve some relevant control problems, such as determining a change of coordinates that recasts scalar and triangular nonlinear systems into linear form, and evaluating a performance index on the transient behaviour of an asymptotically stable closed loop system. [ABSTRACT FROM AUTHOR]

In this paper, a feedback control law is proposed to solve the asymptotic tracking problem for a ball bouncing in a triangular billiard. This controller is designed by firstly transforming the billiard table into a surface on which the ball moves without experiencing any impact, i.e. by reflecting the billiard table rather than the ball trajectory and designing a position feedback controller based on such an unfolded billiard table. Furthermore, it is shown how such an infinite-billiard table can be mapped to the surface of a torus, thus leading to bounded trajectories of the ball. [ABSTRACT FROM AUTHOR]

The main objective of this paper is the characterisation of the notion of uniform algebraic observability for multi-switching linear systems. In particular, it is shown that a multi-switching linear system is 'generically' uniformly algebraically observable if and only if the number of available measurements is greater than the number of switching signals acting on the systems. This result is established without requiring any knowledge about the (possibly, stochastic) process generating the switching pattern. Moreover, since such systems can be used to model sensor networks, the results given in this paper can 'generically' be used to design simple observers for sensor networks able to cope with sensor failures and data losses. [ABSTRACT FROM AUTHOR]

In this paper, an algorithm based on linear algebra tools is proposed to compute a weighted sum of squares decomposition of a given form whose length is lower than the number of variables. Such an objective is pursued by using linear algebra techniques to perform tasks that are usually carried out through computational algebraic geometry tools. Several examples are reported to show that the use of linear algebra rather than algebraic geometry leads to a reduction of the execution times, without affecting the effectiveness of the algorithm. Applications of the given procedure to system analysis and to control design problems are reported as well as a detailed complexity analysis. [ABSTRACT FROM AUTHOR]

In this paper, the single-gene regulatory system for the lac operon is considered and a method to build a Boolean network able to represent its dynamics is given. A procedure, based on techniques borrowed from algebraic geometry, to compute an optimal control policy minimising a cost function is given for such a Boolean network. A tool to apply such an optimal control policy to the continuous-time system is reported. The proposed control method is tested in two simulative examples. [ABSTRACT FROM PUBLISHER]

The primary goal of this paper is to describe the classes of all polynomial vector fields such that the associated dynamical system admits some polynomial Lyapunov functionv(x). By fixing or constraining the time derivative ofv(x) in different ways, several different cases are studied. Hints are given about how to use the results for stability analysis and for non-polynomial systems. [ABSTRACT FROM PUBLISHER]

Semi-invariants and relative characteristic functions extend to nonlinear systems the concept of eigenvector-eigenvalue pair for linear systems, and are, therefore, very useful to depict the behaviour of the system. In this article, semi-invariants are used to construct explicitly Lyapunov functions useful for studying the stability of the origin, for continuous-time systems. Moreover, using well-known tools from differential geometry such as (orbital) symmetries, it is shown how semi-invariants can be found for several classes of systems. Important connections with centre manifold theory are pointed out. By using the proposed general techniques, a new proof of a result claimed to Bendixson (Bendixson, I. (1901), 'Sur les courbes definies par des equations differentielles', Acta Mathematics, 24, 1-88) is given. [ABSTRACT FROM AUTHOR]