Feedback classification and optimal control with applications to the controlled Lotka–Volterra model

Let M be a σ-compact C∞ manifold of dimension n≥2 and consider a single-input control system: x˙(t)=X(x(t))+u(t)Y(x(t)), where X, Y are C∞ vector fields on M. We prove that there exists an open set of pairs (X,Y) for the C∞-Whitney topology is such that they admit singular abnormal rays so that the spectrum of the projective singular Hamiltonian dynamics is feedback invariant. This result is applied to the controlled Lotka–Volterra dynamics, where such rays are related to shifted equilibria of the free dynamics.