From 1836 to 1844, Jacobi developed a theory of differential equations that was meant to take place in a book entitled “Phoronomia”. After renouncing to this project, some manuscripts remained unpublished in his lifetime. Among them, a bound, still conjectural in the general case, on the order of a differential system, expressed by a “tropical determinant”, i.e.  replacing in the determinant formula, products by sums and sums by max.

Jacobi gave a polynomial time algorithm to compute it, instead of trying the n! number of permutations. This algorithm relies on the notion of “minimal canon”, which is computed by building a “path relation”. In modern terms, Jacobi builds a weighted oriented graph and computes shortest paths. He gave indeed an algorithm to compute the minimal canon, knowing a canon that is equivalent to Dijkstra’s algorithm, and a second algorithm to compute a canon, knowing a permutation that gives the maximum, which is equivalent to Bellman–Ford algorithm.

An equivalent algorithm, the “Hungarian method”, that uses Egerváry’s “minimal covers”, was obtained by Kuhn in 1955 to solve the “assignment problem”, which is to assign in the best way n workers to n jobs, with many applications, for example to optimize transportation problems.

When there are more functions than equations, under some genericity hypotheses, we can still bound the order using the maximum of all tropical determinants obtained by considering square submatrices of the order matrix, and this may be done in polynomial time. But what about the “saddle Jacobi number”, which is the minimum of those tropical determinants? No polynomial time algorithm seems to be known to compute it. Yet, in a work with Y. Kaminski, we could provide a polynomial time algorithm to test is the saddle Jacobi number is 0. In such a case, if the associated Jacobian determinant does not vanish, the system is “flat”. This important property of control systems, introduced by Fliess, Lévine, Martin and Rouchon, means that one may parametrize the solutions of the system, using functions of some “flat outputs” and their derivatives. Here, the flat outputs are those functions in the system that were set aside to restrict to a square system with a square order matrix.

Flat systems with such a property are “oudephippical systems”, that generalize many notions of chained systems. We can easily define for them sets of flat outputs and their regularity conditions. This allows to choose flat outputs adapted for the desired trajectory. With some simplifications, aircraft models are flat as shown by Martin in 1991, but the flat outputs he introduced do not work for 0-gravity flight. We were able to provide a new set of flat outputs that work in that case and to show that the only possible flat singularities of the simplified aircraft model correspond to stalling conditions.

Joint work with Yirmeyahu J. Kaminski, Holon Institute of Technology, Israel

For more details : http://www.lix.polytechnique.fr/%7Eollivier/PRODUCTION_SCIENT/publications/jacobi3V17.2.pdf http://www.lix.polytechnique.fr/%7Eollivier/PRODUCTION_SCIENT/publications/aircraft-hal-V2.pdf