Abstract
We consider second-order differential inclusions on a Riemannian manifold with lower semicontinuous right-hand sides. Several existence theorems for solutions of two-point boundary value problem are proved to be interpreted as controllability of special mechanical systems with control on nonlinear configuration spaces. As an application, a statement of controllability under extreme values of controlling force is obtained.
Highlights
Introduction and motivationThe main object of research in this paper is a mechanical system with set-valued force given in geometrically invariant terms
This language allows us to consider, from unique mathematical point of view, a broad class of real mechanical systems including those on curved nonlinear configuration spaces, forces with control, and so forth
First we introduce some basic notions in order to set up the problem
Summary
The main object of research in this paper is a mechanical system with set-valued force given in geometrically invariant terms. We investigate the two-point boundary value problem for (1.2), that is, the existence of a solution m(t) such that for given points m0, m1 ∈ M and time instants t0, t1 the relations m(t0) = m0 and m1 = m1 hold If such a trajectory exists, there exists a curve in the domain of controlling parameter such that using this (time-dependent) control, we can derive the trajectory to m1 at t1 from m0 at t0. The solvability of the two-point boundary value problem with the force ExtA(t,m,X) (see Theorem 3.6) means controllability of the system with force A(t,m,X) for given points under extreme values of controlling force This fact cannot be covered by previous existence theorems for upper semicontinuous forces with convex images (see, e.g., [5]).
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