# INNER AND OUTER CAPTURE BASIN APPROXIMATION WITH SUPPORT VECTOR MACHINES

### Laetitia Chapel, Guillaume Deffuant

#### Abstract

We propose a new approach to solve target hitting problems, that iteratively approximates capture basins at successive times, using a machine learning algorithm trained on points of a grid with boolean labels. We consider two variants of the approximation (from inside and from outside), and we state the conditions on the machine learning procedure that guarantee the convergence of the approximations towards the actual capture basin when the resolution of the grid decreases to 0. Moreover, we define a control procedure which uses the set of capture basin approximations to drive a point into the target. When using the inner approximation, the procedure guarantees to hit the target, and when the resolution of the grid tends to 0, the controller tends to the optimal one (minimizing time to hit the target). We use Support Vector Machines as a particular learning method, because they provide parsimonious approximations, from which one can derive fast and efficient controllers. We illustrate the method on two simple examples, Zermelo and car on the hill problems.

#### References

- Aubin, J.-P. (1991). Viability theory. Birkhäuser.
- Bayen, A., E., C., and & Tomlin, C. (2002). Guaranteed overapproximations of unsafe sets for continuous and hybrid systems: Solving the hamilton-jacobi equation using viability techniques. Lecture Notes In Computer Science, 2289:90-104.
- Cardaliaguet, P., Quincampoix, M., and Saint-Pierre, P. (1998). Set-Valued Numerical Analysis for Optimal control and Differential Games. Annals of the International Society of Dynamic Games.
- Chang, C.-C. and Lin, C.-J. (2001). Libsvm: a library for support vector machines.
- Deffuant, G., Chapel, L., and Martin, S. (2007). Approximating viability kernels with support vector machines. IEEE transactions on automatic control, 52(5):933- 937.
- Frankowska, H. (1989). Optimal trajectories associated with a solution of the contingent hamilton-jacobi equation. Applied Mathematics and Optimization, 19(1):291-311.
- Lhommeau, M., Jaulin, L., and Hardouin, L. (2007). Inner and outer approximation of capture basin using interval analysis. In 4th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2007).
- Lygeros, J. (2004). On reachability and minimum cost optimal control. Automatica, 40:917-927.
- Martin, S. (2004). The cost of restoration as a way of defining resilience: a viability approach applied to a model of lake eutrophication. Ecology and Society, 9(2).
- Mitchell, I., Bayen, A., and Tomlin, C. (2005). A timedependent Hamilton-Jacobi formulation for reachable sets for continuous dynamic games. IEEE Transactions on Automatic Control, 50(7):947-957.
- Moore, A. and Atkeson, C. (1995). The parti-game algorithm for variable resolution reinforcement learning in multidimensional state-spaces. Machine Learning, 21:199-233.
- Saint-Pierre, P. (1994). Approximation of the viability kernel. Applied Mathematics & Optimization, 29(2):187- 209.
- Saint-Pierre, P. (2001). Approche ensembliste des systèmes dynamiques, regards qualitatifs et quantitatifs. Matapli, Société de Mathématiques appliquées et Industrielles, page 66.
- Scholkopf, B. and Smola, A. (2002). Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond. MIT Press, Cambridge, MA, USA.
- Vapnik, V. (1995). The nature of statistical learning theory. Springer Verlag.

#### Paper Citation

#### in Harvard Style

Chapel L. and Deffuant G. (2011). **INNER AND OUTER CAPTURE BASIN APPROXIMATION WITH SUPPORT VECTOR MACHINES** . In *Proceedings of the 8th International Conference on Informatics in Control, Automation and Robotics - Volume 1: ICINCO,* ISBN 978-989-8425-74-4, pages 34-40. DOI: 10.5220/0003536000340040

#### in Bibtex Style

@conference{icinco11,

author={Laetitia Chapel and Guillaume Deffuant},

title={INNER AND OUTER CAPTURE BASIN APPROXIMATION WITH SUPPORT VECTOR MACHINES},

booktitle={Proceedings of the 8th International Conference on Informatics in Control, Automation and Robotics - Volume 1: ICINCO,},

year={2011},

pages={34-40},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0003536000340040},

isbn={978-989-8425-74-4},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the 8th International Conference on Informatics in Control, Automation and Robotics - Volume 1: ICINCO,

TI - INNER AND OUTER CAPTURE BASIN APPROXIMATION WITH SUPPORT VECTOR MACHINES

SN - 978-989-8425-74-4

AU - Chapel L.

AU - Deffuant G.

PY - 2011

SP - 34

EP - 40

DO - 10.5220/0003536000340040