Title Page -- Contents -- Introduction -- Chapter 1. Some Simple Examples -- 1.1. Introduction -- 1.2. Frictions -- 1.2.1. Coulomb's law -- 1.2.2. Differential equation with univalued operator and usual sign -- 1.2.3. Differential equation with multivalued term: differential inclusion -- 1.2.4. Other friction laws -- 1.3. Impact -- 1.3.1. Difficulties with writing the differential equation -- 1.3.2. Ill-posed problems -- 1.4. Probabilistic context -- Chapter 2. Theoretical Deterministic Context -- 2.1. Introduction -- 2.2. Maximal monotone operators and first result on differential inclusions (in R) -- 2.2.1. Graphs (operators) definitions -- 2.2.2. Maximal monotone operators -- 2.2.3. Convex function, sub-differentials and operators -- 2.2.4. Resolvent and regularization -- 2.2.5. Taking the limit -- 2.2.6. First result of existence and uniqueness for a differential inclusion -- 2.3. Extension to any Hilbert space -- 2.4. Existence and uniqueness results in Hilbert space -- 2.5. Numerical scheme in a Hilbert space -- 2.5.1. The numerical scheme -- 2.5.2. State of the art summary and results shown in this publication -- 2.5.3. Convergence (general results and order 1/2) -- 2.5.4. Convergence (order one) -- 2.5.5. Change of scalar product -- 2.5.6. Resolvent calculation -- 2.5.7. More regular schemes -- Chapter 3. Stochastic Theoretical Context -- 3.1. Introduction -- 3.2. Stochastic integral -- 3.2.1. The stochastic processes background -- 3.2.2. Stochastic integral -- 3.3. Stochastic differential equations -- 3.3.1. Existence and uniqueness of strong solution -- 3.3.2. Existence and uniqueness of weak solution -- 3.3.3. Kolmogorov and Fokker-Planck equations -- 3.4. Multivalued stochastic differential equations -- 3.4.1. Problem statement -- 3.4.2. Uniqueness and existence results -- 3.5. Numerical scheme.

3.5.1. Which convergence: weak or strong? -- 3.5.2. Strong convergence results -- 3.5.3. Weak convergence results -- Chapter 4. Riemannian Theoretical Context -- 4.1. Introduction -- 4.2. First or second order -- 4.3. Differential geometry -- 4.3.1. Sphere case -- 4.3.2. General case -- 4.4. Dynamics of the mechanical systems -- 4.4.1. Definition of mechanical system -- 4.4.2. Equation of the dynamics -- 4.5. Connection, covariant derivative, geodesics and parallel transport -- 4.6. Maximal monotone term -- 4.7. Stochastic term -- 4.8. Results on the existence and uniqueness of a solution -- Chapter 5. Systems with Friction -- 5.1. Introduction -- 5.2. Examples of frictional systems with a finite number of degrees of freedom -- 5.2.1. General framework -- 5.2.2. Two elementary models -- 5.2.3. Assembly and results in finite dimensions -- 5.2.4. Conclusion -- 5.2.5. Examples of numerical simulation -- 5.2.6. Identification of the generalized Prandtl model (principles and simulation) -- 5.3. Another example: the case of a pendulum with friction -- 5.3.1. Formulation of the problem, existence and uniqueness -- 5.3.2. Numerical scheme -- 5.3.3. Numerical estimation of the order -- 5.3.4. Example of numerical simulations -- 5.3.5. Free oscillations -- 5.3.6. Forced oscillations -- 5.3.7. Transition matrix and calculation of the Lyapunov exponents -- 5.3.8. Melnikov's method, transitory chaos and Lyapunov exponents -- 5.4. Elastoplastic oscillator under a stochastic forcing -- 5.4.1. Introduction -- 5.4.2. Modeling -- 5.4.3. Numerical scheme -- 5.4.4. Numerical results -- 5.5. Spherical pendulum under a stochastic external force -- 5.5.1. Establishment of the model -- 5.5.2. Numerical aspects -- 5.6. Gephyroidal model -- 5.6.1. Introduction -- 5.6.2. Description and transformation of the model -- 5.6.3. Quasi-static problems -- 5.6.4. Numerical simulations.

5.6.5. Conclusion -- 5.7. Chain -- 5.7.1. Introduction -- 5.7.2. Description of the model -- 5.7.3. Transformation of the equations -- 5.7.4. Conclusion -- 5.8. An infinity of internal variables: continuous generalized Prandtl model -- 5.8.1. Introduction -- 5.8.2. Description of the continuous model -- 5.8.3. Existence, uniqueness and regularity results -- 5.8.4. Application to the discrete case, and convergence of the discrete model to the continuous model -- 5.8.5. Numerical scheme -- 5.8.6. Study of hysteresis loops -- 5.8.7. Numerical simulations -- 5.9. Locally Lipschitz continuous spring -- 5.9.1. Introduction -- 5.9.2. The studied model -- 5.9.3. Results for the existence and uniqueness of the solutions -- 5.9.4. Convergence results for the numerical schemes -- 5.9.5. The locally Lipschitz continuous case -- 5.9.6. Identification of the parameters from the hysteresis loops -- 5.9.7. Numerical simulations -- Chapter 6. Impact Systems -- 6.1. Existence and uniqueness for simple problems (one degree of freedom) -- 6.1.1. The work of Schatzman-Paoli -- 6.1.2. Simple case with one degree of freedom, forcing and impact: piecewise analytical solutions -- 6.1.3. Adaptation of some classical methods -- 6.1.4. Movement with the accumulation of impacts and a sticking phase -- 6.1.5. Behavior of the numerical methods -- 6.1.6. Convergence and order of one-step numerical methods applied to non-smooth differential systems -- 6.1.7. Results of numerical experiments -- 6.2. A particular behavior: grazing bifurcation -- 6.2.1. Approximation of the map in the general case -- 6.2.2. Particular case -- 6.2.3. Stability of the non-differentiable fixed point -- 6.2.4. Numerical example -- Chapter 7. Applications-Extensions -- 7.1. Oscillators with piecewise linear coupling and passive control -- 7.1.1. Description of the model.

7.1.2. Free oscillations of the system -- 7.1.3. Order є1 -- 7.1.4. Case of periodic forcing -- 7.1.5. Conclusion -- 7.2. Friction and passive control -- 7.2.1. Introduction -- 7.2.2. Introduction to the models: smooth and non-smooth systems -- 7.3. The billiard ball -- 7.3.1. Maximal monotone framework -- 7.3.2. More realistic but non-maximal monotone framework -- 7.4. An industrial application: the case of a belt tensioner -- 7.4.1. The theory -- 7.4.2. The tensioner used -- 7.4.3. Identification of the parameters -- 7.4.4. Validation -- 7.5. Problems with delay and memory -- 7.5.1. Theory -- 7.5.2. Applications -- 7.6. Other friction forces -- 7.6.1. More general forms (variable dynamical coefficient) -- 7.6.2. With a variable static coefficient -- 7.6.3. With variable static and dynamical coefficients -- 7.7. With the viscous dissipation term -- 7.8. Ill-posed problems -- 7.8.1. First model: limit of a well-posed friction law -- 7.8.2. Second model: a differential inclusion without uniqueness -- 7.8.3. Conclusion -- Appendix 1. Mathematical Reminders -- A1.1. Two Gronwall's lemmas -- A1.2. Norms, scalar products, normed vector space, Banach and Hilbert space -- A1.2.1. Scalar products, norms -- A1.2.2. Banach and Hilbert space, separable space -- A1.3. Symmetric positive definite matrices -- A1.4. Differentiable function -- A1.5. Weak limit -- A1.6. Continuous function spaces -- A1.7. Lp space of integrable functions -- A1.7.1. Lp(Ω) space -- A1.7.2. Lp(Ω, Rq ) space -- A1.7.3. Lp(Ω -- H) spaces -- A1.8. Distributions -- A1.8.1. Real values distributions -- A1.8.2. Distributions with values in Rq -- A1.8.3. Distributions with values in Hilbert space -- A1.9. Sobolev space definition -- A1.9.1. Functions with real values -- A1.9.2. Functions with values in Hilbert space -- Appendix 2. Convex Functions -- A2.1. Functions defined on.

A2.2. Functions defined on Hilbert space -- A2.2.1. Any Hilbert space -- A2.2.2. Particular case of the finite dimension -- Appendix 3. Proof of Theorem 2.20 -- Appendix 4. Proof of Theorem 3.18 -- Appendix 5. Research of Convex Potential -- A5.1. Method used -- A5.2. Lemma 5.1 -- A5.3. Lemma 5.4 -- A5.4. Lemma 7.1 -- Bibliography -- Index.