Feedback Classification and Optimal Control with Applications to the Controlled Lotka-Volterra Model

Bernard Bonnard (1, 2) , Jérémy Rouot (3)

1 McTAO - Mathematics for Control, Transport and Applications
2 IMB - Institut de Mathématiques de Bourgogne [Dijon]
3 LMBA - Laboratoire de Mathématiques de Bretagne Atlantique

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

Keywords: Feedback classification Nonlinear systems Lotka-Volterra model Optimal control Direct numerical methods; Feedback classification; Nonlinear systems; Lotka-Volterra model; Optimal control; Direct numerical methods

Domains: Mathematics [math]; Optimization and Control [math.OC]

Bernard Bonnard, Jérémy Rouot. Feedback Classification and Optimal Control with Applications to the Controlled Lotka-Volterra Model. 2023. ⟨hal-03917363⟩