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

Geometric optimal control of the generalized Lotka–Volterra model of the intestinal microbiome

“…For higher codimensional cases the local time optimal policy can also be determined using the dictionary of syntheses presented in Section 3.4 and the method described in Reference 12. In our example, the case $$$ x(0)={x}_{es} $$$ highlights such higher codimensional case and was analyzed in Reference 22.…”

Zermelo navigation problems on surfaces of revolution and geometric optimal control

“…The aim of this section is to provide algorithms to compute conjugate points. The regular case is covered in [14] whereas the recent article [15] analyzes in a general framework conjugate points along non-immersed abnormal geodesics in Zermelo navigation problems. Note that in both cases, the integrability property of the geodesic flow is not required, but in fine the semi-normal form is integrable cf.…”

“…Next, we use [15] to describe the cusp singularity of the abnormal geodesics when meeting the transition between the strong and weak current domains. It is based on [32] in the algebraic case and [3] in the C ∞ -case.…”

“…From [32], p. 56, an algebraic model in R[x, y] at an ordinary cusp γ(t cusp ) is given by the equation x 3 − y 2 = 0. Moreover it is the transition between a R-node solution of the equation x 3 − x 2 + y 2 = 0, where the origin is a double point with two distinct tangents at 0: x ± y = 0 and a C−node solution of x 3 + x 2 + y 2 = 0 with two complex tangents at 0 given by x ± iy = 0 and with two distinct components x = y = 0 and a smooth real branch (see also [15] for more details and Fig. 2…”

“…But we extend the analysis to the case of a strong current, and we show that the cut locus splits into two branches. This phenomenon is related to the optimality status of the abnormal geodesics [15] and to the shape of the small-time balls [12]. The third case study concerns the extension of the evolution of a passive tracer near a vortex and it was analyzed in details in [13].…”