Scientific objectives
More details
List of hotels and registration form


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Main objectives

 The topic of this school is  nonsmooth systems and their applications. Roughly speaking, nonsmooth systems are those systems whose solutions are not everywhere differentiable, and may even possess discontinuities. Their study requires quite specific tools that people working with smooth systems are usually not familiar with. Mechanical Engineers and Applied Mathematicians have long studied this class of dynamical systems (Lagrangian systems with impacts, friction, variational inequalities, differential inclusions). Other scientific communities (Systems and Control, Numerical Analysis) as well as software development, have recently shown a strong interest in nonsmooth dynamical systems. Fields of application are numerous: aeronautics, robotics, electrical circuits, virtual reality, haptic interfaces, automotive systems, nuclear industry, etc, as well as open theoretical issues: modelling, analysis for control -- controllability, observability, identification --, stability, bifurcation analysis, etc. This school will focus on the  mathematical aspects (formalisms and their properties, existence/uniqueness of solutions, convex and nonsmooth analysis tools, complementarity theory), stability/control aspects (many problems being still open in this field), numerical analysis and software development (mainly mechanical and electrical systems), and specific applications (mechanics, electricity, bio-processes, genetic networks).

This school is therefore intended to all researchers in the fields of Control, Applied Mathematics, Virtual Reality, Mechanics, Robotics, Bifurcations and Chaos, and even Mathematical Biology, who want to get acquainted with nonsmooth dynamical systems.

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Preliminary Programme

Mathematical tools  M. Monteiro-Marques (CMAF Lisbon), H. Schumacher(Tilburg university)

Convex and nonsmooth analysis, basic useful tools   (2 h. MDPMM)
Evolution variational inequalities, Kato's theorem and extensions    (2 h. MDPMM)
Differential inclusions: standard and unbounded inclusions, Moreau's sweeping process    (2 h. MDPMM)
Complementarity systems and complementarity theory, Relationships between the various formalisms  (2 h. HS)

Stability, Control, Bifurcation theory  R. Leine (ETH Zurich), H. Schumacher (Tilburg university), S. Adly (LACO)  B. Brogliato (INRIA)    

Lyapunov stability of evolution variational inequalities (1 h. SA)
Lyapunov stability of nonsmooth mechanical systems (1 h. SA)
Controllability  issues in complementarity systems  (1 h. HS)
Stability issues in complementarity systems (1h. HS)
Controllability  of juggling systems (1 h. BB)
Bifurcation and chaos in nonsmooth mechanical systems  (3 h. RL)

Numerical analysis and simulation  V. Acary (INRIA), M. Jean (CNRS, Marseille)

Formulations of nonsmooth dynamical systems
Numerical methods for time integration of NSDS (NonSmooth Dynamical Systems)
Numerical methods for the one-step discretized problem
Software codes

(6 h.)

Applications  H. de Jong (INRIA), J.L. Gouzé (INRIA), H. Schumacher (Tilburg university)

Genetic networks   (3 h. HdJ and JLG)
Economics and transportation sciences (1 h. HS)

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                                  Monday             Tuesday           Wednesday            Thursday            Friday

10:00-11:00            intro (BB)                     MDPMM                    VA, MJ                             
VA, MJ                      RL

11:00-12:00            MDPMM                      MDPMM                  
VA, MJ                              VA, MJ                       RL

12:00-12:15            break                               break                            break                                 break                      break

12:15-13:15           MDPMM                         JMS                          
VA, MJ                             VA, MJ                      JMS

13:15-14:30            lunch                                lunch                          lunch                                  lunch                    lunch

14:30-15:30            JMS                                 JLG, HdJ                                                                   SA                          SA

15:30-16:30            MDPMM                        JLG, HdJ                                                                   SA                      conclusions

16:30-17:00            break                                break                                                                     break
17:00-18:00            JMS                                
JLG, HdJ                                                                    RL

More details on the talks

Hans Schumacher, Dept. of Econometrics and Operations Research, Tilburg University

Introduction to complementarity systems

Complementarity systems: what are they and what are they good for.
Formulation of the dynamics; comparison to other frameworks. Special
classes of complementarity systems (linear, passive,...). Categorization
by index.

 Dynamics of complementarity systems

Notions of solution. Typical nonsmooth phenomena, such as
impulsive behavior, convergence to zero in finite time, merging of
solutions, irreversibility of time, Zeno solutions. Conditions for
existence and uniqueness of solutions; the rational complementarity
problem. Numerical approximation of solutions.

Controllability and stability of complementarity systems

Conditions for controllability and stability of some classes of linear
complementarity systems and related classes of nonsmooth dynamical systems.

Applications of complementarity systems in economics

Examples of optimization and equilibrium problems that give rise to
complementarity systems: dynamic optimization under inequality constraints,
stochastic optimal control (optimal stopping, optimal depletion), traffic
networks, oligopolies.

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Manuel Monteiro Marques, CMAF Lisbon


Remco Leine, Mechanical Engineering, ETH Zurich        SLIDES_LEINE

Introduction to Nonsmooth dynamical systems

Nonsmooth continuous systems, Filippov systems, systems with state discontinuities, iterated maps.
Definitions from Nonlinear Dynamics: equilibria, fixed points of iterated maps, periodic solutions, limit cycles, quasi-periodic solutions and chaos.
The Poincaré map
The fundamental solution matrix and Floquet theory.


Definitions of bifurcation: why are some definitions better than others?
Bifurcations of equilibria in smooth and nonsmooth continuous dynamical systems.
Classical bifurcations of periodic solutions.
Bifurcations of periodic solutions in Filippov systems.
Bifurcations op periodic solutions in mechanical systems with impact and friction.

Numerical methods

Shooting method to find periodic solutions.
Path-continuation of branches of periodic solutions.

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Jean Luc Gouzé (Comore project, INRIA)  and Hidde de Jong  (Helix project, INRIA)

Genetic networks I: Biology and modeling  (HdJ)

                   Introduction to modeling of genetic networks:

Genes and proteins, gene regulation, genetic networks.
Challenges and constraints in modeling of genetic networks, major approaches towards modeling of genetic networks.

                   Qualitative analysis of genetic networks:

Motivations for qualitative analysis, piecewise-linear differential equation models of genetic networks.
Discrete abstraction of continuous dynamics of piecewise-linear differential equations, transition graph, invariance of state transition graph in parameter space, computation of state transition graph, Genetic Network Analyzer (GNA).


Nutritional stress response in Escherichia coli, description of piecewise-linear differential equation model of nutritional stress response, qualitative analysis of model using GNA, experimental verification of model predictions.

Genetic networks II: Mathematical analysis  (JLG)

Mathematical analysis of piecewise-linear differential equations: Solutions of piecewise-linear differential equations, Filippov solutions, regular and singular equilibria, stability, transition graph, links between transition graph and stability.
Application: Analysis of piecewise-linear differential equation model of nutritional stress response in E. coli, comparison with results for ordinary differential equation model, numerical simulation


Vincent Acary (Bipop project INRIA) and Michel Jean  (CNRS LMA)

Lecture 1 V. Acary
Lecture 2 V. Acary
Lecture 3 V. Acary

      Formulations of nonsmooth dynamical systems

Dynamical systems with unilateral consraints; the linear and nonlinear complementarity problems; Lagrangian dynamical systems with unilateral constraints; formulation as measure differential inclusions for general constrained dynamics; enhanced nonsmooth laws; some mathematical properties in practical conditions;

     Numerical methods for time integration of NSDS (NonSmooth Dynamical Systems)

Event driven schemes; Time stepping schemes; comparisons of the two appraoches;

     Numerical methods for the one-step discretized problem

The linear complementarity problem; the quadratic programming approach; the nonlinear complementarity problem; the special case of Coulomb's friction; Relay and piecewise linear mapping; some comparisons and advices;


indeterminacy; stability;

     Software codes

LMGC90; SICONOS; HUMANS; hybrid approaches;


Samir Adly, university of Limoges, LACO

Introduction to Evolution Variational Inequalities

Definition of  Evolution Variational Inequalities, existence results, Kato's Theorem and extensions

Lyapunov's stability of first order evolution variational inequalities

Definition of stability and asymptotic stability, attractivity of  the set of stationary solutions, LaSalle's invariance theory for evolution variational inequalities, necessary conditions for asymptotic stability.

Applications to second order dynamical systems with friction

Examples in nonsmooth mechanics


Bernard Brogliato, Bipop project, INRIA

An introduction to nonsmooth models in Mechanics and electrical circuits (differential inclusions, variational inequalities, complementarity systems, projected systems).


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Short Bibliography

S. Adly,
D. Goeleven, "A stability theory for second-order nonsmooth dynamical systems with application to friction problems", Journal de Mathématiques Pures et Appliquées, vol.83, pp.17-51, 2004.

S. Adly, "Attractivity theory for second order non-smooth dynamical systems with application to dry friction",  Journal of Mathematical Analysis and Applications, in press, 2006.

M.D.P. Monteiro Marques, Differential Inclusions in Nonsmooth Mechanical Problems: Shocks and Dry Friction, Birkhauser, Boston, PNLDE 9, 1993.

J.M. Schumacher, "Complementarity systems in optimization", Mathematical Programming Series B, 101, pp.263-295, 2004.

M. Jean, "The nonsmooth contact dynamics approach", Computer Methods in Applied Mechanics and Engineering, vol.177, pp.235-277, 1999.

J.J. Moreau, "Unilateral contact and dry friction in finite freedom dynamics", in Nonsmooth Mechanics and Applications (J.J. Moreau, P.D. Panagiotopoulos, Eds.), CISM Courses and Lectures no 302, pp.1-82, 1988.

J.J. Moreau, "Numerical aspects of the sweeping process. Computational modeling of contact and friction",
Computer Methods in Applied Mechanics and Engineering, vol.177, pp. 329-349, 1999.

H. de Jong, "Modeling and simulation of genetic regulatory systems: A literature survey", J. Comput. Biol., vol.9, no 1, pp.67-103, 2002.

D. Goeleven, B. Brogliato, "Stability and instability matrices for linear evolution variational inequalities", IEEE transactions on Automatic Control, vol.49, no 4, pp.521-534, April 2004.

B. Brogliato, "Some perspectives on the analysis and control of complementarity systems", IEEE transactions on Automatic Control, vol. 48, no 6, pp.918-935, June 2003.

B. Brogliato, A. Daniilidis, C. Lemaréchal, V. Acary, "On the equivalence between complementarity systems, projected systems and differential inclusions", Systems and Control Letters, vol.55, no 1, pp.45-51, January 2006.

B. Brogliato, Nonsmooth Mechanics, Springer London, CCE Series, second edition, 1999.

R. Leine,  H. Nijmeijer, Dynamics and Bifurcations of Non-smooth Mechanical Systems, Lecture Notes in Applied and Computational Mechanics, vol.18, Springer-Verlag, Heidelberg, 2004.


There are shuttles which go from the Versailles train (RER) stations to the INRIA.  There is also a bus line which goes from Versailles to the INRIA (line B).

Hôtel Résidence Montparnasse
14, rue Stanislas
75006 Paris
Téléphone : 01 45 44 55 09
Hôtel Jardin de Paris Saint-Lazare
53, rue Amsterdam
75008 Paris
Téléphone : 01 48 74 79 74 
Comfort Hotel Gare de l’Est
31, boulevard de Strasbourg
75010 Paris
Téléphone : 01 47 70 25 00
Picardy Hotel
9 rue de Dunkerque
75010 Paris
Téléphone : 01 42 81 00 66  
Hôtel des 3 nations
13, rue Château d’Eau
75010 Paris
Téléphone : 01 42 38 18 18
Hotel Little Regina**Gare de l'Est89 bd de Strasbourg75010 ParisTéléphone : 01 40 37 72 30
Hôtel Bastille de Launay
42, rue Amelot
75011 Paris
Téléphone : 01 47 00 88 11
Hôtel Hibiscus
66, rue de Malte
75011 Paris
Téléphone : 01 47 00 34 34
Relais de Paris Lyon Bastille
35, rue de Citeaux
75012 Paris
Téléphone : 01 43 07 77 28
Timhotel Italie Bercy
22, rue Barrault
75013 Paris
Téléphone : 01 45 80 67 67
Hôtel La Manufacture
8, rue Philippe de Champagne
75013 Paris
Téléphone : 01 45 35 45 25
Hôtel Beaunier
31, rue Beaunier
75014 Paris
Téléphone : 01 45 39 36 45
Hôtel du Parc
6, rue Jolivet
75014 Paris
Téléphone : 01 43 20 95 54
Hôtel Crimée
188, rue Crimée
75019 Paris
Téléphone : 01 40 36 75 29

Hôtel Le Paris
14, avenue de Paris
78000 Versailles
Téléphone : 01 39 50 56 00
Hôtel Richaud
16, rue Richaud
78000 Versailles
Téléphone : 01 39 50 10 42
Hôtel du Cheval Rouge
18, rue André Chénier
78000 Versailles
Téléphone : 01 39 50 03 03
Hôtel Home Saint-Louis
28, rue St Louis
78000 Versailles
Téléphone : 01 39 50 23 55
Hotel d'Angleterre **
2 bis rue de Fontenay
78000 Versailles
Téléphone : 01 39 51 43 50

  Registration Form

(this registration form is to be sent to the organizers at Rocquencourt, see the web site at the top of this page)