Lyapunov Stability of Measure Driven Differential Inclusions

• Type: Article in International Scientific Journal
• Date: 2004

Authors

Fernando Lobo Pereira (Author) FEUP; G. N. Silva (Author) FEUP

Journal

Title: Journal of Differential Equations

Pages: 1122-1130

ISSN: 0022-0396

Publisher: Elsevier

Scientific classification

FOS: Engineering and technology > Electrical engineering, Electronic engineering, Information engineering

Other information

Abstract (EN): In the present paper, we analyze the stability of equilibria of impulsive control systems whose dynamics is determined by a differential inclusion driven by a vector-valued measure. The notion of solution given in [1] provides the meaningfulness of the stabilization problem under very general assumptions on the conditions of the problem. This notion of solution has the important property that it covers systems whose singular dynamics does not satisfy the so-called Frobenius condition. It turns out that for each admissible solution the trajectory joining boundary points of discontinuity is determined by the singular dynamics. Note that the above-mentioned notion of solution is caused by practical engineering considerations. For important classes of applied problems, it is of interest to control a dynamical system that can operate in several viable configurations. Although the transitions between configurations, modelled by jumps (discontinuities) of the trajectories, are unproductive and their duration is negligible, their nature can affect the general properties of the system. Therefore, it is advisable to consider the jump dynamics as integral part of the dynamical optimization problem. This class of problems arises in various applications such as finance, mechanics of vibroshock systems, renewable resource management, or aerospace navigation, where the solution is contained in the set of control processes with trajectories of bounded variation. This has naturally given an impetus to the recent rapid development of the theory of such systems and numerical schemes implementing the control strategies. There is a wide literature on the stability of ordinary control systems ˙x = f(x, u), x(0) = x0, or, in terms of differential inclusions, ˙x ∈ F(x) (for a detailed list, see, e.g., [2–4], and a brief survey can be found in [5]). The stability conditions were stated in [5] in terms of a controllable Lyapunov pair of functions satisfying the uniform decay condition. This is due to the fact that these conditions were found by applying ordinary stability theory to a standard problem obtained by a reparametrization of the original control system. However, these conditions are useless in numerous cases. Therefore, we weaken this result and extend the notion of a controllable Lyapunov pair of functions in such a way that V increases at each jump. The price we pay for this approach is that we have to consider only control problems with a control measure such that either the total variation of its singular component is finite or its total variation on any finite interval tends to zero as its lower bound tends to infinity. This is a rather general scheme from the viewpoint of applications, although it might seem restrictive.

Language: English

Type (Professor's evaluation): Scientific

No. of pages: 9