The set S_{F}(x_{0};T) of states y reachable from a given state x_{0} at time T under a set-valued dynamic x’(t)∈F(x (t)) and under constraints x(t)∈K where K is a closed set, is also the capture-viability kernel of x_{0} at T in reverse time of the target {x_{0}} while remaining in K. In dimension up to three, Saint-Pierre’s viability algorithm is well-adapted; for higher dimensions, Bonneuil’s viability algorithm is better suited. It is used on a large-dimensional example.

R. Baier and M. Gerdts, “A Computational Method for Non-Convex Reachable Sets Using Optimal Controls,” Proceedings of the European Control Conference 2009, Budapest, 23-26 August 2009, pp. 97-101.

R. Baier, “Set-Valued Integration and Discrete Approximation. Reachable Sets,” Bayreuth Mathematical Reports 50, University of Bayreuth, Bayreuth, 1995.

R. Baier, M. Gerdts and I. Xausa, “Approximation of Reachable Sets Using Optimal Control Algorithms,” 2011.
http://num.math.uni-bayreuth.de/en/publications/2012/baier_gerdts_xausa_approx_reach_sets_2011/index.html

N. Bonneuil, “Computing the Viability Kernel in Large State Dimension,” Journal of Mathematical Analysis and Applications, Vol. 323, No. 2, 2006, pp. 1444-1454.
doi:10.1016/j.jmaa.2005.11.076

P. Saint-Pierre, “Approximation of the Viability Kernel,” Applied Mathematics and Optimization, Vol. 29, No. 2, 1994, pp. 187-209. doi:10.1007/BF01204182

N. Bonneuil, “Maximum under Continuous-Discrete-Time Dynamic with Target and Viability Constraints,” Optimization, Vol. 61, No. 8, 2012, pp. 901-913.
doi:10.1080/02331934.2011.605127