## Section: New Results

### Non-Linear Computational Geometry

Participants : Sény Diatta, Laurent Dupont, George Krait, Sylvain Lazard, Guillaume Moroz, Marc Pouget.

#### Reliable location with respect to the projection of a smooth space curve

Consider a plane curve $\mathcal{B}$ defined as the projection of the intersection of two analytic surfaces in ${\mathbb{R}}^{3}$ or as the apparent contour of a surface. In general, $\mathcal{B}$ has node or cusp singular points and thus is a singular curve. Our main contribution [9] is the computation of a data structure for answering point location queries with respect to the subdivision of the plane induced by $\mathcal{B}$. This data structure is composed of an approximation of the space curve together with a topological representation of its projection $\mathcal{B}$. Since $\mathcal{B}$ is a singular curve, it is challenging to design a method only based on reliable numerical algorithms.

In a previous work [39], we have shown how to describe the set of singularities of $\mathcal{B}$ as regular solutions of a so-called ball system suitable for a numerical subdivision solver. Here, the space curve is first enclosed in a set of boxes with a certified path-tracker to restrict the domain where the ball system is solved. Boxes around singular points are then computed such that the correct topology of the curve inside these boxes can be deduced from the intersections of the curve with their boundaries. The tracking of the space curve is then used to connect the smooth branches to the singular points. The subdivision of the plane induced by $\mathcal{B}$ is encoded as an extended planar combinatorial map allowing point location. We experimented our method and show that our reliable numerical approach can handle classes of examples that are not reachable by symbolic methods.

#### Computing effectively stabilizing controllers for a class of $n$D systems

In this paper [1], we study the internal stabilizability and internal stabilization problems for multidimensional ($n$D) systems. Within the fractional representation approach, a multidimensional system can be studied by means of matrices with entries in the integral domain of structurally stable rational fractions, namely the ring of rational functions which have no poles in the closed unit polydisc ${\overline{\mathbb{U}}}^{n}=\left\{z=({z}_{1},...,{z}_{n})\in {\u2102}^{n}\phantom{\rule{4pt}{0ex}}\left|\phantom{\rule{4pt}{0ex}}\right|{z}_{1}|\le 1,...,|{z}_{n}|\le 1\right\}$.

It is known that the internal stabilizability of a multidimensional system can be investigated by studying a certain polynomial ideal $I=\langle {p}_{1},...,{p}_{r}\rangle $ that can be explicitly described in terms of the transfer matrix of the plant. More precisely the system is stabilizable if and only if $V\left(I\right)=\{z\in {\u2102}^{n}\phantom{\rule{0.277778em}{0ex}}|\phantom{\rule{0.277778em}{0ex}}{p}_{1}\left(z\right)=\cdots ={p}_{r}\left(z\right)=0\}\cap {\overline{\mathbb{U}}}^{n}=\varnothing $. In the present article, we consider the specific class of linear $n$D systems (which includes the class of 2D systems) for which the ideal $I$ is zero-dimensional, i.e., the ${p}_{i}$'s have only a finite number of common complex zeros. We propose effective symbolic-numeric algorithms for testing if $V\left(I\right)\cap {\overline{\mathbb{U}}}^{n}=\varnothing $, as well as for computing, if it exists, a stable polynomial $p\in I$ which allows the effective computation of a stabilizing controller. We illustrate our algorithms through an example and finally provide running times of prototype implementations for 2D and 3D systems.