Convergence Properties of Monotone Measure Differential Inclusions
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In this chapter, we present theorems which give sufficient conditions for the convergence of measure differential inclusions with certain maximal monotonicity properties. The framework of measure differential inclusions allows us to describe systems with state discontinuities, as has been shown in the previous chapters. The material presented in this chapter is based on the paper [104]. The chapter is organised as follows. First, we define the convergence property of dynamical systems in Section 8.1 and state the associated properties of convergent systems. Theorems are presented in Section 8.2 which give sufficient conditions for the convergence of measure differential inclusions with certain maximal monotonicity properties. Furthermore, we illustrate in Section 8.3 how these convergence results for measure differential inclusions can be exploited to solve tracking problems for certain classes of non-smooth mechanical systems with friction and one-way clutches. Illustrative examples of convergent mechanical systems are discussed in detail in Section 8.4. Finally, Section 8.5 presents concluding remarks.
8.1 Convergent Systems In this section, we will briefly discuss the definition of convergence and certain properties of convergent systems. In the definition of convergence, the Lyapunov stability of solutions of (8.1) plays a central role. Definitions of (uniform) stability and attractivity of measure differential inclusions are given in the Section 6.4. The definitions of convergence properties presented here are based on and extend the definition given in [44] (see also [132]). Consider a system described by the measure differential inclusion dx ∈ dΓ (t, x), where x ∈ Rn , t ∈ R.
(8.1)
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8 Convergence Properties of Monotone Measure Differential Inclusions
Let us formally define the property of convergence. Definition 8.1. System (8.1) is said to be •
¯ satisfying the following conditions: convergent if there exists a solution x(t) ¯ (i) x(t) is defined for almost all t ∈ R, ¯ ¯ (ii) x(t) is bounded for all t ∈ R for which x(t) exists, ¯ (iii) x(t) is globally attractively stable. ¯ • uniformly convergent if it is convergent and x(t) is globally uniformly attractively stable. ¯ • exponentially convergent if it is convergent and x(t) is globally exponentially stable.
The wording ‘attractively stable’ has been used instead of the usual term ‘asymptotically stable’, because attractivity of solutions in (measure) differential inclusions can be asymptotic or symptotic (finite-time attractivity), see page 121. ¯ The solution x(t) is called a steady-state solution. As follows from the definition of convergence, any solution of a convergent system “forgets” its initial condition and converges to some steady-state solution. In general, the ¯ steady-state solution x(t) may be non-unique. But for any two steady-state ¯ 2 (t) it holds that x ¯ 1 (t) − x ¯ 2 (t) → 0 as t → +∞. At ¯ 1 (t) and x solutions x the same time, for uniformly convergent systems the steady-state solution is unique, as formulated below. Property 8.2 ( [131, 132]). I
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