Asymptotic stabilization of submanifolds embedded in Riemannian manifolds

We study control problems in which systems whose states spaces are Riemannian manifolds must be steered towards embedded submanifolds of their state spaces in a stable fashion, i.e., feedback laws rendering the given submanifold asymptotically stable are sought. Bringing the closed loop to the form of a gradient system with drift, we find that the gradient part can be scaled in a fashion so as to guarantee asymptotic stability of the desired submanifold. Under this circumstance, we show that solutions of the system can be brought to an arbitrarily small neighborhood of the submanifold in arbitrary time. Thereafter, we derive an algebraic condition under which the control vector fields can be brought to the desired form. Lastly, we recast our algebraic condition in terms of vertical and horizontal spaces for the case that the submanifold is an equivalence class.