Navigation Functions on 3-Manifold With Boundary as a Disjoint Union of Hopf Tori

This article concerns the construction of a scalar valued analytic map, which we term as the navigation function, on three-manifold with boundary as a disjoint union of Hopf tori. Although it is well known that the existence of smooth navigation functions is guaranteed on any smooth manifold, the exact form of the navigation function on three-manifold with a disjoint union of Hopf tori as its boundary is yet to be explored. To bridge this gap, we propose a novel scalar map on the specified manifold with boundary, which is proved to be a valid navigation function under mild condition. Moreover, it is shown that the gradient of the proposed navigation function can be utilized to solve the constrained attitude control problem of a rigid body. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed control approach.