THEMES PRINCIPAUX DE RECHERCHE

Optimisation:

Systèmes de gradient. Méthodes proximales. Algorithmes forward-backward. Sélection asymptotique d'un optimum particulier. Prise en compte de la courbure.

Analyse non-linéaire:

Analyse convexe et non-lisse. Analyse multivoque. Régularisation et approximation. Conjugaison généralisée.

Systèmes dynamiques dissipatifs:

EDO, EDP, inclusions différentielles.

Mécanique non-régulière:

Problèmes de billard. Théorie des chocs. Frottement de Coulomb et stabilisation en temps fini.

ARTICLES PARUS OU A PARAITRE

• [1] H. Attouch, A. Cabot, P. Redont, The dynamics of elastic shocks via epigraphical regularization
  of a differential inclusion. Barrier and penalty approximations.
  Advances in Math. Sciences and Applications, 12 (2002), no. 1, 273-306.

• [2] A. Cabot, Motion with friction of a heavy particle on a manifold- Applications to Optimization.
  M2AN Mathematical Modelling Numerical Analysis, 36 (2002), no. 3, 505-516.

• [3] A. Cabot, M.-O. Czarnecki, Asymptotic control of pairs of oscillators coupled by a repulsion,
  with non isolated equilibria I: the regular case.
  SIAM Journal on Control and Optimization, 41 (2002), no. 4, 1254-1280.

• [4] A. Cabot, B. Mohammadi, Incomplete sensitivities and cost function reformulation
  leading to multi-criteria investigation of inverse problems.
  Optimal Control Applications and Methods, 24 (2003), no. 2, 73-84.

• [5] A. Cabot, Inertial gradient-like dynamical system controlled by a stabilizing term.
  Journal of Optimization- Theory and Applications, 120 (2004), no. 2, 275-303.

• [6] F. Alvarez, A. Cabot, Steepest descent with curvature dynamical system.
  Journal of Optimization- Theory and Applications, 120 (2004), no. 2, 247-273.

• [7] A. Cabot, The steepest descent dynamical system with control. Applications to constrained minimization.
  Control, Optimisation and Calculus of Variations, 10 (2004), no. 2, 243-258.

• [8] A. Cabot, Bounce law at the corners of convex billiards.
  Nonlinear Analysis, 57 (2004), no. 4, 597-614.

• [9] A. Cabot, Proximal point algorithm controlled by a slowly vanishing term. Applications to hierarchical minimization.
  SIAM Journal on Optimization, 15 (2005), no. 2, 555-572.

• [10] F. Alvarez, A. Cabot, On the asymptotic behavior of a system of steepest descent equations
  coupled by a vanishing mutual repulsion.
  Recent Advances in Optimization (edité par A. Seeger), Lectures Notes in Econom. and Math. Systems,
  Springer-Verlag (2006), 3-17.

• [11] S. Adly, H. Attouch, A. Cabot, Finite time stabilization of nonlinear oscillators subject to dry friction.
  Nonsmooth Mechanics and Analysis (edité par P. Alart, O. Maisonneuve et R.T. Rockafellar),
  Adv. in Math. and Mech., Kluwer (2006), 289-304.

• [12] B. Baji, A. Cabot, An inertial proximal algorithm with dry friction: finite convergence results.
  Set Valued Analysis, 14 (2006), no. 1, 1-23.

• [13] F. Alvarez, A. Cabot, Asymptotic selection of viscosity equilibria of semilinear evolution equations
  by the introduction of a slowly vanishing term.
  Discrete and Continuous Dynamical Systems, 15 (2006), no. 3, 921-938.

• [14] A. Cabot, A. Seeger, Multivalued exponentiation analysis. Part I: Maclaurin exponentials.
  Set Valued Analysis, 14 (2006), no. 4, 347-379.

• [15] A. Cabot, A. Seeger, Multivalued exponentiation analysis. Part II: Recursive exponentials.
  Set Valued Analysis, 14 (2006), no. 4, 381-411.

• [16] A. Cabot, L. Paoli, Asymptotics for some vibro-impact problem with linear dissipation term.
  Journal de Mathématiques Pures et Appliquées, 87 (2007), no. 3, 291-323.

• [17] B. Baji, A. Cabot, J.I. Diaz, Asymptotics for some nonlinear damped wave equation:
  finite time convergence versus exponential decay results.
  Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire, 24 (2007), no. 6, 1009-1028.

• [18] A. Cabot, Stabilization of oscillators subject to dry friction: finite time convergence versus exponential decay results.
  Transactions of the American Mathematical Society, 360 (2008), no. 1, 103-121.

• [19] A. Cabot, H. Engler, S. Gadat, On the long time behavior of second order differential equations
  with asymptotically small dissipation.
  Transactions of the American Mathematical Society, 361 (2009), 5983-6017.

• [20] A. Cabot, H. Engler, S. Gadat, Second order differential equations with asymptotically small dissipation
  and piecewise flat potentials.
  Electronic Journal of Differential Equations, 17 (2009), 33-38.

• [21] A. Cabot, Asymptotics for a gradient system with memory term.
  Proceedings of the American Mathematical Society, 137 (2009), 3013-3024.

• [22] B. Baji, A. Cabot, On some curvature-dependent steplength for the gradient method.
  Journal of Convex Analysis, 17 (2010), 765-780.

• [23] A. Cabot, P. Frankel, Asymptotics for some proximal-like method involving inertia and memory aspects.
  Set Valued and Variational Analysis, 19 (2011), 59-74.

• [24] H. Attouch, A. Cabot, P. Frankel, J. Peypouquet, Alternating proximal algorithms for linearly constrained variational inequalities. Application to domain decomposition for PDE's.
  Nonlinear Analysis: Theory, Methods & Applications, 74 (2011), 7455-7473.

• [25] A. Cabot, P. Frankel, Alternating proximal algorithms with asymptotically vanishing coupling. Application to domain decomposition for PDE's.
  Optimization, 61 (2012), 307-325.

• [26] A. Cabot, P. Frankel, Asymptotics for some semilinear hyperbolic equations with non-autonomous damping.
  Journal of Differential Equations, 252 (2012), 294-322.

• [27] A. Cabot, L. Thibault, Inclusion of subdifferentials, linear well-conditioning and steepest descent equation.
  SIAM J. Optim., 23 (2013), 552-575.

• [28] A. Cabot, L. Thibault, Sequential formulae for the normal cone to sublevel sets.
  Transactions of the American Mathematical Society, 366 (2014), 6591-6628.

• [29] A. Cabot, A. Seeger, Multivalued exponentiation analysis. Part III: Forward exponentials.
  Set-Valued and Variational Analysis, 22 (2014), 617-638.

• [30] A. Cabot, A. Jourani, L. Thibault, Envelopes for sets and functions: regularization and generalized conjugacy.
  Mathematika, 63 (2017), 383-432.

• [31] H. Attouch, A. Cabot, Asymptotic stabilization of inertial gradient dynamics with time-dependent viscosity.
  Journal of Differential Equations, 263 (2017), 5412-5458.

• [32] H. Attouch, A. Cabot, M.-O. Czarnecki, Asymptotic behavior of nonautonomous monotone and subgradient evolution equations.
  Transactions of the American Mathematical Society, 370 (2018), 755-790.

• [33] H. Attouch, A. Cabot, Convergence rates of inertial forward-backward algorithms.
  SIAM Journal on Optimization, 28 (2018), 849-874.

• [34] H. Attouch, A. Cabot, Convergence of damped inertial dynamics governed by regularized maximally monotone operators.
  Journal of Differential Equations, 264 (2018), 7138-7182.

• [35] H. Attouch, A. Cabot, Z. Chbani, H. Riahi, Rate of convergence of inertial gradient dynamics with time-dependent viscous damping coefficient.
  Evolution Equations and Control Theory, 7 (2018), 353-371.

• [36] H. Attouch, A. Cabot, Z. Chbani, H. Riahi, Inertial forward-backward algorithms with perturbations. Application to Tikhonov regularization.
  Journal of Optimization- Theory and Applications, 179 (2018), 1-36.

• [37] A. Cabot, A. Jourani, L. Thibault, D. Zagrodny, The attainment set of φ-envelope and genericity properties.
  A paraître dans Mathematica Scandinavica.

• [38] A. Cabot, A. Jourani, L. Thibault, Envelopes for sets and functions II: generalized polarity and conjugacy.
  A paraître dans Journal of Nonlinear and Convex Analysis.

• [39] H. Attouch, A. Cabot, Convergence of a relaxed inertial forward-backward algorithm for structured monotone inclusions.
  Accepté dans Applied Mathematics and Optimization.

ENCADREMENT DOCTORAL

• Encadrement de la thèse de B. Baji. Soutenance en février 2009 à l'Université Montpellier II:
  "Quelques contributions à des systèmes dynamiques et algorithmes issus de la mécanique non-régulière et de l'optimisation".

• Co-encadrement avec H. Attouch de la thèse de P. Frankel. Soutenance en septembre 2011 à l'Université Montpellier II:
  "Comportement asymptotique de systèmes dynamiques discrets et continus en Optimisation et EDP".

• Rapporteur et membre du jury de thèse de T. Reytier. Soutenance en février 2012 à l'Université Toulouse III:
  "Modelling fatigue spectra of aircraft structures under gust loads".

• Membre du jury de thèse de I. Kecis. Soutenance en juin 2014 à l'Université Montpellier II:
  "Fonctions non convexes inférieurement s-régulières".

• Membre du jury de thèse de T. Zakaryan. Soutenance en décembre 2014 à l'Université de Bourgogne:
  "Sur les ensembles et fonctions prox-réguliers et leurs applications".

• Membre du jury de thèse de E. Vilches. Soutenance en juillet 2017 à l'Université de Bourgogne:
  "Differential inclusions involving normal cones of nonregular sets in Hilbert spaces".