Lyapunov Stability of Measure Driven Impulsive Systems

Abstract

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 ẋ = f(x, u), x(0) = x0, or, in terms of differential inclusions, ẋ ∈ 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. Despite numerous publications on the stability of ordinary control systems and active research in the field of measure driven dynamical systems, little is known about the stability of impulsive dynamical systems. In [6], the stability problem for impulsive systems was considered in the context of the robustness of the solution under perturbations of the control measure. Necessary and sufficient stability conditions were obtained after the introduction of an appropriate notion of the closure of a set of trajectories. In the next section, we represent the result [7] on the robustness of a solution of a differential inclusion with a measure in the framework of the approach accepted in the present paper. In the context of a model substantially different from the one used in the present paper, the paper [8] gives stability conditions for dynamical systems with discontinuous trajectories via inequalities containing some function, referred to as a Lyapunov function, which is a contingent solution of Hamilton–Jacobi variational inequalities.