Computing Reachable Sets as Capture-Viability Kernels in Reverse Time

ABSTRACT

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.

The set

Cite this paper

N. Bonneuil, "Computing Reachable Sets as Capture-Viability Kernels in Reverse Time,"*Applied Mathematics*, Vol. 3 No. 11, 2012, pp. 1593-1597. doi: 10.4236/am.2012.311219.

N. Bonneuil, "Computing Reachable Sets as Capture-Viability Kernels in Reverse Time,"

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

[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