Accessible Points and Sub-Manifolds of Mechanical Systems. Controllability

Abstract

In this chapter we study the question of whether or not two points m ₀ and m ₁ in the configuration space of a mechanical system can be connected by a trajectory. It is known (see, e.g., Hartman (1964)) that for a second order differential equation (i.e., in particular, for Newton’s law) on Euclidean space such a trajectory exists provided that the right-hand side of the differential equation is bounded and continuous. More precisely, for any two points m ₀ and m ₁ and any interval [a,b], there exists a solution m(t) such that m(a)=m ₀ and m(b)=m ₁. When the right-hand side is linearly bounded, some similar results are known for small time intervals.