, The case when x + ?D M 705 or x ? ?D M is located outside K requires a specic management. The estimation 706 of the normal vector to ? K? q at point x, is based on the estimation (by successive 707 dichotomies) of d ? 1 anely independent points of ? K? q at a given distance from 708 x, from which the normal vector can be derived. The method includes a specic 709 tratment for the cases, ?V iab(K) with the segment [x, x + ?D M ] or with the segment [x, x ? ?D M ] (D M 704 being a parameter) by performing successive dichotomies

, It uses indeed 713 the resistar surface denoted H , dened from the boundary points b = (v+v )/2 where 714 v and v dene a grid edge [v, v ] such that v ?? q and v / ?? q . It estimates the 715 distance from the centres of the cubes containing boundary points of H to ?V iab(K) 716 using the normal to H, Nearest vertex approximation. The set? q is now the nearest vertex 711 approximation

, Figure 1 shows the sets? i for all the iterations of algorithm 722 3.1 applied on the 2D problems. The nal result can be visually compared with 723 the theoretical viability kernel (quantied evaluations of the Hausdor distance are 724 shown on gure 3). Note that the algorithm stops after 4 iterations for the population 725 and consumption problems and after 6 iterations for the spirals problem, p.726

, The number of iterations is similar in higher dimensionality. Figure 2 727 shows examples of nal results on the 3D problems. The smooth non-linearity along 728 the x 2 axis ruled by parameters ? and ? appears in the viability kernel approximations 729 of population and consumption problems. Panels (a), (b) and (c) of Figure 4 show the 730 intersection with three chosen hyperplanes of viability kernel resistar approximations 731 of algorithm 3

, 25 for the 738 5D problems. For nearest vertex approximations, the values 7 and 9 are not tested 739 because they are too small for dening properly the sets B i in, panels (a), (b) and (c) show the estimated Hausdor distance be-733 tween the viability kernel and its approximation, vol.3

, These results are in good agreement with the theoretical prediction of an Hausdor 744 distance decreasing like n ?1 for the approximation with the nearest vertex and like, vol.745, p.19

. Overall,

, Even when using the nearest vertex 755 approximation, we expect our algorithm to then outperform the current techniques 756 approximating viability kernels for three reasons. Firstly, the convergence to the 757 viability kernel is ensured without decreasing the time step to 0, which is a major 758 dierence. Secondly, for a given time step, our algorithm requires a lower number 759 of iterations, especially when the supremum of time steps in K for the non

, When using resistars as set approximation technique, if the best conditions are 764 satised, the convergence rate of our algorithm is like O(n ?2 ) which signicantly 765 increases the advantage over the standard methods

, the standard methods need a grid of at least n 2d points. For instance, we have 768 shown that it is possible to run resistars approximations in 5 dimensions using a grid 769 of 25 5 (about 8 10 6 ) points, order to be as accurate as a resistar approximation using a grid of n d 767 points

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