The dynamic interpolation problem: on Riemannian manifolds, Lie groups, and symmetric spaces. (English) Zbl 0946.58018

Crouch, P.; Silva Leite, F.

J. Dyn. Control Syst. 1, No. 2, 177-202 (1995).

Summary: We consider the dynamic interpolation problem for nonlinear control systems modeled by second-order differential equations whose configuration space is a Riemannian manifold \(M\). In this problem we are given an ordered set of points in \(M\) and would like to generate a trajectory of the system through the application of suitable control functions, so that the resulting trajectory in configuration space interpolates the given set of points. We also impose smoothness constraints on the trajectory and typically ask that the trajectory be also optimal with respect to some physically interesting cost function. Here, we are interested in the situation where the trajectory is twice continuously differentiable and the Lagrangian in the optimization problem is given by the norm squared acceleration along the trajectory. The special cases where \(M\) is a connected and compact Lie group or a homogeneous symmetric space are studied in more detail.

MSC:

58E25	Applications of variational problems to control theory
93B27	Geometric methods
49K27	Optimality conditions for problems in abstract spaces

Keywords:

optimal control; dynamic interpolation problem; nonlinear control systems; second-order differential equations; Riemannian manifold; Lie group; symmetric space

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