R. Bruck, Asymptotic convergence of nonlinear contraction semigroups in Hilbert spaces

, J. Funct. Anal, vol.18, p.1526, 1975.

R. Cominetti, J. Peypouquet, and S. Sorin, Strong asymptotic convergence of evolution equations governed by maximal monotone operators with Tikhonov regularization, J. Dierential Equations, vol.245, p.37533763, 2008.

S. Reich, Nonlinear evolution equations and nonlinear ergodic theorems, Nonlinear Anal, vol.1, p.319330, 1976.

J. Peypouquet and S. Sorin, Evolution equations for maximal monotone operators : asymptotic analysis in continuous and discrete time, J. Convex Anal, vol.17, p.11131163, 2010.

J. Peypouquet and S. Sorin, Evolution equations for maximal monotone operators : asymptotic analysis in continuous and discrete time, J. of Convex Analysis, vol.17, pp.1113-1163, 2010.

H. Attouch and B. F. Svaiter, A continuous dynamical Newton-like approach to solving monotone inclusions, SIAM J. Control Optim, vol.49, p.574598, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00803137

H. Attouch, P. Redont, and B. F. Svaiter, Global convergence of a closed-loop regularized Newton method for solving monotone inclusions in Hilbert spaces, J. Optim. Theory Appl, vol.157, p.624650, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00803194

A. Beck and M. Teboulle, Gradient-based algorithms with applications in signal recovery problems, Convex Optimization in Signal Processing and Communications, p.3388, 2010.

P. L. Combettes, J. Pesquet, H. Bauschke, R. Burachik, P. L. Combettes et al., Proximal splitting methods in signal processing, Fixed-Point Algorithms for Inverse Problems in Science and Engineering, pp.185-212
URL : https://hal.archives-ouvertes.fr/hal-00643807

H. Brézis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, 1973.

H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory. CMS books in Mathematics, 2011.
URL : https://hal.archives-ouvertes.fr/hal-01517477

E. Zeidler, Nonlinear Functional Analysis and its Applications, Part II : Monotone Operators, 1990.

G. J. Minty, Monotone (nonlinear) operators in Hilbert spaces, Duke Mathematical Journal, vol.29, p.341346, 1962.

R. T. Rockafellar, Maximal monotone relations and the second derivatives of nonsmooth functions, Ann. Inst. Henri Poincaré, vol.2, p.167184, 1985.

A. Haraux, Systèmes dynamiques dissipatifs et applications. RMA 17, 1991.

E. D. Sontag, Mathematical Control Theory, 1998.

H. Brézis, Analyse Fonctionnelle, 1983.

Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc, vol.73, p.591597, 1967.

R. E. Bruck, Asymptotic convergence of nonlinear contraction semigroups in Hilbert spaces, J. Funct. Anal, vol.18, pp.15-26, 1975.

F. Bonnans, Optimisation Continue, Mathématiques appliquées pour le Master/SMAI, Dunod, 2006.

A. S. Antipin, Minimization of convex functions on convex sets by means of dierential equations. Dierential Equations, vol.30, p.13651375, 1994.

J. Bolte, Continuous gradient projection method in Hilbert spaces, J. Optim. Theory Appl, vol.119, p.235259, 2003.

H. Attouch, L. Briceno, and P. L. Combettes, A parallel splitting method for coupled monotone inclusions, SIAM J. Control Optim, vol.48, p.32463270, 2010.

D. L. Zhu and P. Marcotte, Co-coercivity and its role in the convergence of iterative schemes for solving variational inequalities, J. Optim, vol.6, p.714726, 1996.

H. Attouch, P. Maingé, and P. Redont, A second-order dierential system with Hessiandriven damping ; application to non-elastic shock laws, Dier. Equ. Appl, vol.4, p.2765, 2012.

F. Alvarez, H. Attouch, J. Bolte, and P. Redont, A second-order gradient-like dissipative dynamical system with Hessian-driven damping. Application to optimization and mechanics, J. Math. Pures Appl, vol.81, p.747779, 2002.

B. Abbas and H. Attouch, Dynamical systems and forward-backward algorithms associated with the sum of a convex subdierential and a monotone cocoercive operator, Optimization, 2014.

B. Abbas, H. Attouch, and B. F. Svaiter, Newton-like dynamics and forward-backward methods for structured monotone inclusions in Hilbert spaces, J. Optim. Theory Appl, vol.161, issue.2, pp.331-360, 2014.
URL : https://hal.archives-ouvertes.fr/hal-02072808

A. S. Antipin, Minimization of convex functions on convex sets by means of dierential equations. Dierential Equations, vol.30, p.13651375, 1994.

H. Attouch, L. Briceno, and P. L. Combettes, A parallel splitting method for coupled monotone inclusions, SIAM J. Control Optim, vol.48, p.32463270, 2010.

H. Attouch, P. Redont, and B. F. Svaiter, Global convergence of a closed-loop regularized Newton method for solving monotone inclusions in Hilbert spaces, J. Optim. Theory Appl, vol.157, issue.3, p.624650, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00803194

H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory. CMS books in Mathematics, 2011.
URL : https://hal.archives-ouvertes.fr/hal-01517477

A. Beck and M. Teboulle, Gradient-based algorithms with applications in signal recovery problems, Convex Optimization in Signal Processing and Communications, p.3388, 2010.

J. Bolte, Continuous gradient projection method in Hilbert spaces, J. Optim. Theory Appl, vol.119, p.235259, 2003.

H. Brézis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, 1973.

H. Brézis, Analyse Fonctionnelle, 1983.

R. E. Bruck, Asymptotic convergence of nonlinear contraction semigroups in Hilbert spaces, J. Funct. Anal, vol.18, pp.15-26, 1975.

J. E. Dennis and R. B. Schnabel, Numerical Methods for Unconstrained Minimization

. Prentice-hall, , 1983.

J. C. Dunn and D. P. Bertsekas, Ecient dynamic programming implementations of Newton's method for unconstrained optimal control problems, J. Optim. Theory Appl, vol.63, p.2338, 1999.

O. P. Ferreira and B. F. Svaiter, Kantorovich's theorem on Newton's method on Riemannian manifolds, Journal of Complexity, vol.18, p.304329, 2002.

O. Güler, On the convergence of the proximal point algorithm for convex minimization, SIAM J. Control Optim, vol.29, p.403419, 1991.

A. Haraux, Systèmes dynamiques dissipatifs et applications. RMA 17, 1991.

A. N. Iusem, B. F. Svaiter, and J. X. Da-cruz-neto, Central paths, generalized proximal point methods, and Cauchy trajectories in riemannian manifolds, SIAM J. Control Optim, vol.37, p.566588, 1989.

G. J. Minty, Monotone (nonlinear) operators in Hilbert spaces, Duke Mathematical Journal, vol.29, p.341346, 1962.

J. J. Moreau, Bounded variation in time, Topics in Nonsmooth Mechanics, p.176, 1988.
URL : https://hal.archives-ouvertes.fr/hal-01363799

J. Nocedal and S. Wright, Numerical Optimization. Springer Series in Operations Research, 1999.

Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc, vol.73, p.591597, 1967.

J. Peypouquet and S. Sorin, Evolution equations for maximal monotone operators : asymptotic analysis in continuous and discrete time, J. of Convex Analysis, vol.17, pp.1113-1163, 2010.

R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim, vol.14, p.877898, 1976.

R. T. Rockafellar, Maximal monotone relations and the second derivatives of nonsmooth functions, Ann. Inst. Henri Poincaré, vol.2, p.167184, 1985.

E. D. Sontag, Mathematical Control Theory, 1998.

M. V. Solodov and B. F. Svaiter, A globally convergent inexact Newton method for systems of monotone equations, Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, pp.355-369, 1999.

E. Zeidler, Nonlinear Functional Analysis and its Applications, Part II : Monotone Operators, 1990.

D. L. Zhu and P. Marcotte, Co-coercivity and its role in the convergence of iterative schemes for solving variational inequalities, J. Optim, vol.6, pp.714-726, 1996.

B. Abbas, H. Attouch, and B. F. Svaiter, -like dynamics and forward-backward methods for structured monotone inclusions in Hilbert spaces, J. Optim. Theory Appl, vol.161, issue.2, pp.331-360, 2014.

A. S. Antipin, Minimization of convex functions on convex sets by means of dierential equations, Dierential Equations, vol.30, p.32463270, 2010.

H. Attouch, J. Bolte, and B. F. Svaiter, Convergence of descent methods for semi-algebraic and tame problems : proximal algorithms, forward-backward splitting, and regularized Gauss-Seidel methods, Mathematical Programming, vol.137, issue.1, p.91129, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00790042

H. Attouch, L. M. Briceño-arias, and P. L. Combettes, A parallel splitting method for coupled monotone inclusions, SIAM J. Control Optim, vol.48, issue.5, p.32463270, 2010.

H. Attouch and R. Cominetti, A dynamical approach to convex minimization coupling approximation with the steepest descent method, J. Dierential Equations, vol.128, issue.2, p.519540, 1996.

H. Attouch and M. Czarnecki, Asymptotic behavior of coupled dynamical systems with multiscale aspects, J. Dierential Equations, vol.248, issue.6, pp.1315-1344, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00803408

H. Attouch, M. Czarnecki, and J. Peypouquet, Coupling forward-backward with penalty schemes and parallel splitting for constrained variational inequalities, SIAM J. Optim, vol.21, issue.4, p.12511274, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00803596

H. Attouch and X. Goudou, A continuous gradient-like dynamical approach to Paretooptimization in Hilbert spaces, Set-Valued and Variational Analysis, vol.22, p.189219, 2014.

H. Attouch and P. E. Maingé, Asymptotic behavior of second order dissipative evolution equations combining potential with non-potential eects, ESAIM Control Optim. Calc. of Var, vol.17, issue.3, pp.836-857, 2011.

H. Attouch, J. Peypouquet, and P. Redont, A dynamical approach to an inertial forwardbackward algorithm for convex minimization, SIAM J. Optim, vol.24, issue.1, p.232256, 2014.
URL : https://hal.archives-ouvertes.fr/hal-02072843

H. Attouch, J. Peypouquet, and P. Redont, Forward-backward and backward-forward algorithms for structured monotone inclusions, working paper, 2014.

H. Attouch and M. Théra, A general duality principle for the sum of two operators, J. Convex Analysis, vol.3, p.124, 1996.

J. Baillon, Un exemple concernant le comportement asymptotique de la solution du problème du dt + ??(u) 0, J. Funct. Anal, vol.28, p.369376, 1978.

J. Baillon and H. Brézis, Une remarque sur le comportement asymptotique des semigroupes non linéaires, Houston J. Math, vol.2, p.57, 1976.

J. Baillon and R. Cominetti, A convergence result for non-autonomous subgradient evolution equations and its application to the steepest descent exponential penalty trajectory in linear programming, J. Funct. Anal, vol.187, p.263273, 2001.

J. Baillon and G. Haddad, Quelques propriétés des opérateurs angle-bornés et ncycliquement monotones, Israël J. Math, vol.26, p.137150, 1977.

H. H. Bauschke and P. L. Combettes, Convex analysis and monotone operator theory in Hilbert spaces, 2011.
URL : https://hal.archives-ouvertes.fr/hal-01517477

A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM Journal on Imaging Sciences, vol.2, issue.1, p.183202, 2009.

W. Bian and X. Xue, Asymptotic behavior analysis on multivalued evolution inclusion with projection in Hilbert space, Optimization, 2013.

J. Bolte, Continuous gradient projection method in Hilbert spaces, J. Optim. Theory Appl, vol.119, issue.2, p.235259, 2003.

H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, 1973.

R. E. Bruck, Asymptotic convergence of nonlinear contraction semigroups in Hilbert spaces, J. Funct. Anal, vol.18, p.1526, 1975.

A. Cabot, The steepest descent dynamical system with control. Applications to constrained minimization, ESAIM Control Optim. Calc. Var, vol.10, p.243258, 2004.
URL : https://hal.archives-ouvertes.fr/hal-00068880

P. L. Combettes, Solving monotone inclusions via compositions of nonexpansive averaged operators, Optimization, vol.53, p.475504, 2004.
URL : https://hal.archives-ouvertes.fr/hal-00017830

P. L. Combettes and S. A. Hirstoaga, Approximating curves for nonexpansive and monotone operators, Journal of Convex Analysis, vol.13, issue.3-4, p.633646, 2006.
URL : https://hal.archives-ouvertes.fr/hal-00017642

A. Daniilidis, Gradient dynamical systems, tame optimization and applications, Lecture Notes, 2009.

S. A. Hirstoaga, Approximation et résolution de problèmes d'équilibre, de point xe et d'inclusion monotone, 2006.

P. L. Lions and B. Mercier, Splitting algorithms for the sum of two nonlinear operators, SIAM Journal on Numerical Analysis, vol.16, p.964979, 1979.

Y. E. Nesterov, A method for solving the convex programming problem with convergence rate O(1/k 2 ), Dokl. Akad. Nauk SSSR, vol.269, issue.3, p.543547, 1983.

Z. , Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc, vol.73, p.591597, 1967.

, Let us list some interesting questions to be examined in the future : 1. Examine the discrete, algorithmical version, and the corresponding asymptotic selection property for the forward-backward algorithm

, Study the case where ?(t) depends on t in an open-loop form

, Study the case where the Levenberg-Marquart regularization term is given in a closedloop form, ?(t) = ?( ?(t) ) as in

B. Abbas and H. Attouch, Dynamical systems and forward-backward algorithms associated with the sum of a convex subdierential and a monotone cocoercive operator, Optimization, 2014.

F. Alvarez and A. Cabot, Asymptotic selection of viscosity equilibria of semilinear evolution equations by the introduction of a slowly vanishing term, Discrete and Continuous Dynamical Systems, vol.15, pp.921-938, 2006.

H. Attouch, Viscosity solutions of minimization problems, SIAM J. Optimization, vol.6, pp.769-806, 1996.

H. Attouch, G. Buttazzo, and G. Michaille, Variational analysis in Sobolev and BV spaces, Society for Industrial and Applied Mathematics (SIAM), vol.6
URL : https://hal.archives-ouvertes.fr/hal-02072798

H. Attouch and M. Czarnecki, Asymptotic behavior of coupled dynamical systems with multiscale aspects, Journal of Dierential Equations, vol.248, pp.1315-1344, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00803408

H. Attouch, J. Peypouquet, and P. Redont, Backward-forward algorithms for structured monotone inclusions in Hilbert spaces
URL : https://hal.archives-ouvertes.fr/hal-02072613

H. Attouch, P. Redont, and B. F. Svaiter, Global convergence of a closed-loop regularized Newton method for solving monotone inclusions in Hilbert spaces, J. Optim. Theory Appl, vol.157, issue.3, p.624650, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00803194

J. B. Baillon and R. Cominetti, A convergence result for nonautonomous subgradient evolution equations and its application to the steepest descent exponential penalty trajectory in linear programming, Journal of Functional Analysis, vol.187, p.263273, 2001.

H. H. Bauschke and P. L. Combettes, Convex analysis and monotone operator theory in Hilbert spaces, 2011.
URL : https://hal.archives-ouvertes.fr/hal-01517477

H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, 1973.

F. E. Browder, Nonlinear Operators and Nonlinear Equations of Evolution in Banach Spaces, Proc. Sympos. Pure Math, vol.18, issue.2, 1976.

A. Cabot, The steepest descent dynamical system with control. Applications to constrained minimization, ESAIM : Control, Optimization and Calculus of Variations, vol.10, pp.243-258, 2004.
URL : https://hal.archives-ouvertes.fr/hal-00068880

H. Furuya, K. Miyashiba, and N. Kenmochi, Asymptotic behavior of solutions to a class of nonlinear evolution equations, Journal of Dierential Equations, vol.62, p.7394, 1986.

A. Haraux, Systèmes dynamiques dissipatifs et applications. RMA 17, 1991.

S. Hirstoaga, Approximation et résolution de problèmes d'équilibre, de point xe et d'inclusion monotone, UPMC Paris, vol.6, 2006.

S. Reich, Nonlinear evolution equations and nonlinear ergodic theorems, Nonlinear Anal, vol.1, p.319330, 1976.

R. Rockafellar, R. , and J. Wets, Variational Analysis, Grundlehren der mathematischen Wissenschaften, vol.317, 1998.

E. D. Sontag, Mathematical Control Theory, 1998.

B. Abbas and H. Attouch, Dynamical systems and forward-backward algorithms associated with the sum of a convex subdierential and a monotone cocoercive operator, Optimization, 2014.

S. Adly, R. Cibulka, and H. V. Ngai, Newton's method for solving inclusions using set-valued approximations, SIAM Journal on Optimization, vol.25, issue.1, p.159184, 2015.

H. Attouch, P. Redont, and B. F. Svaiter, Global convergence of a closed-loop regularized Newton method for solving monotone inclusions in Hilbert spaces, J. Optim. Theory Appl, vol.157, issue.3, p.624650, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00803194

H. Attouch, P. Maingé, and P. Redont, A second-order dierential system with Hessiandriven damping ; application to non-elastic shock laws, Dier. Equ. Appl, vol.4, p.2765, 2012.

R. I. Boµ and E. R. Csetnek, A forward-backward dynamical approach to the minimization of the sum of a nonsmooth convex with a smooth nonconvex function, 2015.

R. I. Boµ and E. R. Csetnek, Convergence rates for forward-backward dynamical systems associated with strongly monotone inclusions, 2015.