Optimal control of affine connection control systems from the point of view of Lie algebroids

Abstract

The purpose of this paper is to use the framework of Lie algebroids to study optimal control problems for affine connection control systems on Lie groups. In this context, the equations for critical trajectories of the problem are geometrically characterized as a Hamiltonian vector field.

Higher-order variational calculus on Lie algebroids

“…Concretely, the family γ s (t) = γ(s, t) defines a curve in T k M by fixing s and taking k-jets [γ s ] k . 1 . Such an immersion allows to identify (k + 1)velocities with the vectors tangent to T k M which are in the image of i M k,1 .…”

“…In this paper we study optimization problems defined by a cost functional which depends on higher-order derivatives of admissible curves on a Lie algebroid. Examples of this type of problems are the optimal control of dynamical systems where the system to be controlled is a mechanical system, and hence depends on accelerations [2,1,11,8], trajectory planning problems in control theory [26], key-framed animations in computer graphics [37], and in general, problems of interpolation and approximation of curves on Riemannian manifolds [3,27]. In many of these examples the presence of symmetries is used to reduce the difficulty of the problem.…”

Lie Geometric Methods in the Study of Driftless Control Affine Systems with Holonomic Distribution and Economic Applications

“…In the same way, considering Equations ( 32) and ( 35), obtain (23). Using u 1 = ẋ1 , u 2 = ẋ2 , we obtain (24) and (25). The initial and final conditions x i (0) = 0, x i (T) = s i , i = 1, 2 lead to the following linear system: By direct computation, we obtain the solution ( 26)- (29).…”

“…Some aspects regarding the abnormality problem in control theory on Lie algebroids are presented in [23]. The link between optimal control and connection theory on Lie algebroids can be found in [24][25][26]. Lie geometric methods in control theory have been applied in many papers.…”

About simple variational splines from the Hamiltonian viewpoint

“…Moreover, a Hamiltonian version goes back to Crouch, Leite and Camarinha [32], Iyer [53,54], and more recently on Abrunheiro et al [1,2,3,4] and [44].…”