About the minimal time control of passive tracers in presence of a single point vortex

Olivier Cots (1, 2) , Joseph Gergaud , B Wembe

1 IRIT-APO - Algorithmes Parallèles et Optimisation
2 Toulouse INP - Institut National Polytechnique (Toulouse)

In this work, we are interested in time-optimal displacements of passive tracers (that is point vortices with zero circulation) in presence of a single point vortex (solution of the two-dimensional incompressible Euler equation). This problem from fluid mechanics, is formulated as an optimal control problem in Mayer form that can be seen as a Zermelo-like deformation (presence of a drift) of a Riemannian situation. We first prove the controllability of the control system and the existence of optimal solutions, and then we apply the Pontryagin Maximum Principle (PMP) in order to reduce the set of candidates as minimizers. The transcription of the PMP gives us a set of nonlinear equations to solve, the so-called shooting equations. We numerically check the absence of conjugate points along the computed extremals in relation with second-order conditions of local optimality. This is a first step before computing the optimal synthesis which essentially depends on the existence of abnormal extremals and which consists, for a part, in computing the cut locus, where the extremals ceases to be optimal.

Domaines: Optimisation et contrôle [math.OC]

Olivier Cots, Joseph Gergaud, B Wembe. About the minimal time control of passive tracers in presence of a single point vortex. 2019. ⟨hal-02070786⟩