Accessibility properties of abnormal geodesics in optimal control illustrated by two case studies

Abstract

In this article, we use two case studies from geometry and optimal control of chemical network to analyze the relation between abnormal geodesics in time optimal control, accessibility properties and regularity of the time minimal value function. Introduction. In this article, one considers the time minimal control problem for a smooth system of the form dq dt = f(q, u), where q ∈ M is an open subset of R n and the set of admissible control is the set U of bounded measurable mapping u(·) valued in a control domain U , where U is a two-dimensional manifold of R with boundary. According to the Maximum Principle [14], time minimal solutions are extremal curves satisfying the constrained Hamiltonian equation q̇ = ∂H ∂p , ṗ = − ∂q , H(q, p, u) = M(q, p), (1) where H(q, p, u) = p ·F (q, u) is the pseudo (or non maximized) Hamiltonian, while M(q, p) = maxv∈U H(q, p, u) is the true (maximized) Hamiltonian. A projection of an extremal curve z = (q, p) on the q-space is called a geodesic. Moreover since M is constant along an extremal curve and linear with respect to p, the extremal can be either exceptional (abnormal) if M = 0 or non exceptional if M 6= 0. To refine this classification, an extremal subarc can be either regular if the control belongs to the boundary of U or singular if it belongs to the interior and satisfies the condition ∂H ∂u = 0. Taking q(0) = q0 the accessibility set A(q0, tf ) in time tf is the set ∪u(·)∈U q(tf , x0, u), where t 7→ q(·, q0, u) denotes the solution of the system, with q(0) = q0 and clearly since the time minimal trajectories belongs to the boundary of the accessibility set, the Maximum Principle is a parameterization of this boundary. 2020 Mathematics Subject Classification. Primary: 49K15, 49L99, 53C60, 58K50.