## Section: Research Program

### Optimal Control and its Geometry

Let us detail our research program concerning optimal control, evoked in section 3.1.3. Relying on Hamiltonian dynamics is now prevalent, instead of the Lagrangian formalism in classical calculus of variations. The two points of view run parallel when computing geodesics and shortest path in Riemannian Geometry for instance, in that there is a clear one-to-one correspondance between the solutions of the geodesic equation in the tangent bundle and the solution of the Pontryagin Maximum Principle in the cotangent bundle. In most optimal control problems, on the contrary, due to the differential constraints (not all direction can be the tangent of a fesable trajectory in the state space), the Lagrangian formalism becomes more involved, while the Pontryagin Maximum Principle keeps the same form, its solutions still live in the cotangent bundle, their projections are the extremals, and a minimizing curve must be the projection of such a solution.

#### Cut and conjugate loci

The cut locus —made of the points where the extremals lose optimality— is obviously crucial in optimal control, but usually out of reach
(even in low dimensions), and anyway does not have an analytic characterization because it is a non-local object. Fortunately, conjugate
points —where the extremal lose *local* optimality— can be effectively computed with high accuracy for many control systems.
Elaborating on the seminal work of the Russian and French schools (see [75], [30], [31] and
[49] among others), efficient algorithms were designed to treat the smooth case.
This was the starting point of a series of papers of members of the team culminating in the outcome of the *cotcot* software
[40], followed by the *HamPath* [51] code.
Over the years, these codes have allowed for the computation of conjugate loci in a wealth of situations including applications to space
mechanics, quantum control, and more recently swimming at low Reynolds number.

With in mind the two-dimensional analytic Riemannian framework, a heuristic approach to the global issue of determining cut points is to
search for singularities of the conjugate loci; this line is however very delicate to follow on problems stemming from applications in three
or more dimensions (see *e.g.* [52] and [37]).

Recently, computation of conjugate points was conducted in [16], [2] to determine the optimality status in swimming at low Reynolds number; because of symmetries, and of the periodicity constraint, a tailor-made notion of conjugate point had to be used, and some additional sign conditions must be checked for local minimality, see more in [63]. In all these situations, the fundamental object underlying the analysis is the curvature tensor. In Hamiltonian terms, one considers the dynamics of subspaces (spanned by Jacobi fields) in the Lagrangian Grassmannian [28]. This point of view withstands generalizations far beyond the smooth case: In ${\mathrm{L}}^{1}$-minimization, for instance, discontinuous curves in the Grassmannian have to be considered (instantaneous rotations of Lagrangian subspaces still obeying symplectic rules [56]).

The cut locus is a central object in Riemannian geometry, control and optimal transport. This is the motivation for the a series of conferences on “The cut locus: A bridge over differential geometry, optimal control, and transport”, co-organized by team members and Japanese colleagues. The first one took place during the summer, 2016, in Bangkok; the second one will take place the first week of September, 2018, in Sapporo.

#### Riemann and Finsler geometry

Studying the distance and minimising geodesics in Riemannian Geometry or Finsler Geometry is a particular case of optimal control, simpler because there are no differential constraints; it is studied in the team for the following two reasons. On the one hand, after some tranformations, like averaging (see section Section 3.4), and/or reduction, some more difficult optimal control problems lead to a Riemann or Finsler geometry problem, that have been much studied and known facts from these areas are useful. On the other hand, optimal control, mostly the Hamiltonian setting, brings a fresh viewpoint on problems in Riemann and Finsler geometry.

On Riemaniann ellipsoids of revolution, the optimal control approach allowed to decide on the convexity of the injectivity domain, which, associated with non-negativity of the Ma-Trudinger-Wang curvature tensor, ensures continuity of the optimal transport on the ambient Riemannian manifold [61], [60]. The analysis in the oblate geometry [38] was completed in [54] in the prolate one, including a preliminary analysis of non-focal domains associated with conjugate loci.

Averaging in systems coming from space mechanics control (see sections 3.4 and 4.1) with ${\mathrm{L}}^{2}$-minimization
yields a Riemannian metric, thoroughly computed in [35] together with its geodesic flow; in reduced dimension, its conjugate
and cut loci were computed in [39] with Japanese Riemaniann geometers.
Averaging the same systems for minimum time yields a Finsler Metric, as noted in [34].
In [47], the geodesic convexity properties of these two types of metrics were compared.
When perturbation (other than the control) are considered, they introduce a “drift”, *i.e.* the Finsler metric is no longer symmetric.

#### Sub-Riemannian Geometry

Optimal control problems that pertain to sub-Riemannian Geometry bear all the difficulties of optimal control, like the role of singular/abnormal trajectories, while having some useful structure. They lead to many open problems, like smoothness of minimisers, see the recent monograph [66] for an introduction. Let us detail one open question related to these singular trajectories: the Sard conjecture in sub-Riemannian geometry.

Given a totally non-holonomic distribution on a smooth manifold, the Sard Conjecture is concerned with the size of the set of points that can be reached by singular horizontal paths starting from a given point. In the setting of rank-two distributions in dimension three, the Sard conjecture is that this set should be a subset of the so-called Martinet surface, indeed small both in measure and in dimension. In [33], it has been proved that the conjecture holds in the case where the Martinet surface is smooth. Moreover, the case of singular real-analytic Martinet surfaces was also addressed. In this case, it was shown that the Sard Conjecture holds true under an assumption of non-transversality of the distribution on the singular set of the Martinet surface. It is, of course, very intersting to get rid of the remaining technical assumption, or to go to higher dimension. Note that any that Sard-type result has strong consequences on the regularity of sub-Riemannian distance functions and in turn on optimal transport problems in the sub-Riemannian setting.

#### Singularities

The analysis of singularities in optimal control yields some more interplay with Hamiltonian dynamics. The Hamiltonian setting, much more than the Lagrangian one used in Riemannian geometry, is instrumental to treat such degeneracies. In fact, the latter do not really create singularities in the Pontryagin Maximum Principle equations. Almost-Riemannian metrics on the two-sphere appear after averaging the Pontryagin Maximum Principle for a quadratic cost (these metrics on the two-sphere are thoroughly described, with their degenracies, in [36]), or in the control of a quantum system with Ising coupling of three spins [45].

Another example comes from the analysis of singularities arising in minimum time systems. Consider a control affine system in dimension four with control on the disc such that the controlled fields together with their first order Lie brackets with the drift have full rank. There is a natural stratification of the codimension two singular set in the cotangent bundle leading to a local classification of extremals in terms of singular and bang arcs. This analysis was done in [52] using the nilpotent model, and extended in [27] by interpreting the singularities of the extremal flow as equilibrium points of a regularized dynamics to prove the continuity of the flow. One can actually treat these singularities as connections of pairs of normally hyperbolic invariant manifolds in order to find a suitable stratification of the flow and prove finer regularity properties. Another issue is to be able to give global bounds on the number of these heteroclinic connections. This work in progress is part of M. Orieux PhD thesis in collaboration with J. Féjoz at U. Paris-Dauphine.

#### Optimality of periodic solutions/periodic controls.

When seeking to minimize a cost with the constraint that the controls and/or part of the states are periodic (and with other initial and final conditions), the notion of conjugate points is more difficult than with straightforward fixed initial point. In [43], for the problem of optimizing the efficiency of the displacement of some micro-swimmers (see section 4.3) with periodic deformations, we used the sufficient optimality conditions established by Vinter's group [79], [63] for systems with non unique minimizers due to the existence of a group of symmetry (always present with a periodic minimizer-candidate control). This takes place in a long term collaboration with P. Bettiol (Univ. Bretagne Ouest) on second order sufficient optimality conditions for periodic solutions, or in the presence of higher dimensional symmetry groups, following [79], [63].

Another question relevant to locomotion is: how minimizing is it to use periodic deformations ? Observing animals (or humans), or numerically solving the optimal control problem associated with driftless micro-swimmers for various initial and final conditions, we remark that the optimal strategies of deformation seem to be periodic, at least asymptotically for large distances. This observation is the starting point for characterizing dynamics for which some optimal solutions are periodic, and are asymptotically attract other solutions as the final time grows large; this is reminiscent of the “turnpike theorem” (classical, recently applied to nonlinear situations in [77]).

#### Software

These applications (but also the development of theory where numerical experiments can be very enlightening) require many algorithmic and numerical developments that are an important side of the team activity.
The software *HamPath* (see section 5.1) is maintained by former members of the team in close collaboration with McTAO.
We also use direct discretization approaches (such as the Bocop solver developed by COMMANDS) in parallel.
Apart from this, we develop on-demand algorithms and pieces of software, for instance we have to interact with a production software developed by Thales Alenia Space.

A strong asset of the team is the interplay of its expertise in geometric control theory with applications and algorithms (see sections 4.1 to 4.3) on one hand, and with optimal transport, and more recently Hamiltonian dynamics, on the other.