Groupe d’études et de recherche en analyse des décisions

Approximating viability kernels and capture basins with simplex star surfaces

Guillaume Deffuant Irstea − Lisc, France

The algorithms approximating viability kernels or capture basins are based on mathematical objects that approximate a hypersurface in a multi-dimensional space. The initial algorithms use points on a grid that approximate the interior of the hypersurface. More recent approaches use convex polytopes, ellipsoids or support vector machines (some of these approaches are based on the link between viability and reachability which opened the possibility to use well known reachability techniques for computing viability kernels). In this talk, we present an alternative mathematical object, the simplex star surfaces, and new algorithms of viability kernel and capture basin approximation. The approach is also based on the link between reachability and viability. We argue that simplex star surfaces offer more local flexibility than support vector machines and other approaches. Moreover, they provide a good accuracy of approximation for a relatively low number of parameters to tune. Finally we can define intersections of simplex star surfaces that provide good approximations of singularities on the surfaces. This opens new possibilities to get local information on the viability kernel or the capture basin that could be useful in defining feedback laws.