The set SF(x0;T) of states y reachable from a given state x0 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 x0 at T in reverse time of the target {x0} 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.
References
[1]
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.
[2]
R. Baier, “Set-Valued Integration and Discrete Approximation. Reachable Sets,” Bayreuth Mathematical Reports 50, University of Bayreuth, Bayreuth, 1995.
[3]
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
[4]
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
[5]
J.-P. Aubin, “A concise introduction to Viability Theory, Optimal Control and Robotics,” école Normale Supérieure de Cachan, Cachan, 2003.
[6]
P. Saint-Pierre, “Approximation of the Viability Kernel,” Applied Mathematics and Optimization, Vol. 29, No. 2, 1994, pp. 187-209. doi:10.1007/BF01204182
[7]
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
