# ITN SADCO "Optimal control approaches to reachability analysis: safety & avoidance" (Task 2.5)

**start of the project: **
2011
,
**end of the project: **
2014

**contract number: **
264735-SADCO (grant agreement number)

**funding institution: **
European Commission (EU)

## project members

### principal investigator

Dr. Robert Baier, Prof. Dr. Matthias Gerdts (University of the Federal Armed Forces in Munich), Dr. Oskar Ries and Christian Wegwerth (Volkswagen AG, Wolfsburg)### project members

## aims of the project

The interdisciplinary project „Optimal control approaches to reachability analysis“ aims to develop collision avoidance algorithms for cars using optimal control techniques.

The properties of reachable sets ﬁgure prominently in the study of how dynamic systems evolve under the inﬂuence of control and parameter variations. For example, changes in climate under different environmental conditions due, say, to variations in carbon dioxide concentrations or temperature, see *I. A. Chahma, Set-Valued Discrete Approximation of State-Constrained Differential Inclusions, Bayreuth. Math. Schr., 67 (2003), pp. 3–162*, can be investigated via the reachable sets of the dynamic model. While various methods for the approximation of reachable sets for linear control systems have been suggested, see e.g. *R. Baier, Ch. Büskens, A. I. Chahma, M. Gerdts: Approximation of Reachable Sets by Direct Solution Methods of Optimal Control Problems, Optim. Methods Softw. 22 (2007)*, only a few methods are known for nonlinear control systems. In this project area, we shall investigate a new method based on optimal control techniques, and representation and approximation of nonconvex sets in *R. Baier, M. Gerdts: A Computational Method for Non-Convex Reachable Sets using Optimal Control, Proceedings of the European Control Conference (ECC) 2009, Budapest (Hungary)*. We shall investigate and analyze computational methods for the approximation of reachable sets. We shall also consider applications to problems of computing optimal avoidance strategies, exempliﬁed by the problem of designing a road-vehicle driver assistance system. Here, the aim is to avoid collisions by exploiting reachable set information. More precisely, once the reachable set for the motion of the car is known, then it is easy to decide whether a collision will unavoidably occur (leading to an immediate breaking manoeuvre) or if a collision can be avoided by choosing appropriate controls. If a collision is unavoidable, then the car automatically initiates a breaking maneuver in order to reduce the speed as much as possible and to reduce the severity of the impact.

The project member will participate in the international training network SADCO with the opportunity of extended research stays at universities. The secondment (6 months) will take place in Université Paris Diderot (Paris 7) and in ENSTA ParisTech in Paris under the supervision of Olivier Bokanowski. The first part of her secondment will be from September 17, 2012 till the end of December, 2012.

## Further information is available:

- open position ESR 11 (the position is no longer available)
- homepage of the German SADCO team
- homepage of the ITN SADCO project
- International Training Network (ITN)
- Seventh Framework Programme (FP 7) of the European Commission

## publications

M. Gerdts,I. Xausa:
**Avoidance trajectories using reachable sets and parametric sensitivity analysis**

in:
*System modeling and optimization. 25th IFIP TC 7 conference on system modeling and optimization, CSMO 2011, Berlin, Germany, September 12–16, 2011. Revised Selected Papers,*
*
D. Hömberg, F. Tröltzsch
(eds.)
*

*IFIP Advances in Information and Communication Technology*

**391**, 2013, 491 - 500

DOI:
10.1007/978-3-642-36062-6_49

R. Baier, M. Gerdts, I. Xausa:
**Approximation of Reachable Sets using Optimal Control Algorithms**

*Numerical Algebra, Control and Optimization*
**3**
(3),
519 - 548,
2013,

R. Baier:
**Set-valued Euler's method with interpolated distance functions and optimal control solvers**

in:
*Proceedings on the 20th International Symposium on Mathematical Theory of Networks and Systems (MTNS 2012), July 9-13, 2012, Melbourne, Australia,*
*
*
Melbourne, Australia,
2012,
4 pages,
CD-ROM, Paper No. 85, extended abstract

Download as PDF (external link)

I. Xausa, R. Baier, M. Gerdts, M. Gonter, C. Wegwerth:
**Avoidance trajectories for driver assistance systems via solvers for optimal control problems**

in:
*Proceedings of the 20th International Symposium on Mathematical Theory of Networks and Systems (MTNS2012),*
8 pages,
Melbourne, Australia,
2012,
CD-ROM, Paper No. 294, full paper

Download as PDF (external link)

I. Kornienko, M. Gerdts, M. de Pinho:
**A new version of necessary conditions for optimal control problems with differential algebraic equations**

in:
*Proceedings on the 20th International Symposium on Mathematical Theory of Networks and Systems (MTNS 2012), July 9-13, 2012, Melbourne, Australia,*
*
*
Melbourne, Australia,
2012,
8 pages,
CD-ROM, Paper No. 43, full paper

Download as PDF (external link)

R. Baier, M. Gerdts, I. Xausa:
**Approximation of Reachable Sets using Optimal Control Algorithms**

36 pages,
University of Bayreuth, Germany,
2011,
technical report, October 2011, updated from version in April 26, 2010

Download as PDF
(File "baier_gerdts_xausa_approx_reach_sets_2011.pdf")

Download as PDF
(File "baier_gerdts_xausa_approx_reach_sets_2011_old.pdf")

R. Baier, M. Gerdts:
**A Computational Method for Non-Convex Reachable Sets using Optimal Control**

in:
*Proceedings of the European Control Conference (ECC) 2009, August 23-26, session MoA2.6,*
*
*
EUCA,
Budapest, Hungary,
2009,
97 - 102

R. Baier, C. Büskens, I. A. Chahma, M. Gerdts:
**Approximation of Reachable Sets by Direct Solution Methods of Optimal Control Problems**

*Optimization Methods and Software*
**22**
(3),
433 - 452,
2007

DOI:
10.1080/10556780600604999

Download as PDF