Completeness of Riemannian metrics: an application to the control of constrained mechanical systems

We introduce a mathematical technique based on modifying a given Riemannian metric and we investigate its applicability to controlling and stabilizing constrained mechanical systems, regardless of actuation. Our result relies on constructing a complete Riemannian metric on the modified space incorporating the constraint. Essentially, this technique modifies incomplete Riemannian metrics with complete ones, altering evolution near a boundary, thereby leveraging the design of a controller to meet a specified criterion. A byproduct is its applicability to the controlled Lagrangians method, by modifying its metric to control mechanical systems with configuration constraints.