Topological obstructions to distributed feedback stabilization to a submanifold

Abstract

We consider the problem of local asymptotic feedback stabilization – via a continuously differentiable feedback law – of a control system ẋ = f(x,u) defined in Euclidean space R n (with f being continuously differentiable) to a compact, connected, oriented p−dimensional submanifold P of R, subject to the constraint that the scalar entries of the system function f and of the feedback law u depend only on selected subsets of the state variables. Such constraints arise naturally in the context of distributed control systems, typically consisting of multiple agents with only local communication between the various agents. We obtain topological necessary conditions for the existence of such a stabilizing feedback control law when the submanifold to be stabilized to is even-dimensional; these topological conditions are expressed in terms of the generators of the homology groups of certain topological spaces naturally associated with the control problem, as well as the topology of the submanifold to which stabilization is to be performed.