S. Optimisation-de-forme, 97 5.2.1 Paramétrage d'un corps convexe, p.99

.. Discussion-duprobì-eme-en-dimension-supérieure...., 103 5.3.1 Développement en harmoniques sphériques 106 5.3.2 Corps de largeur constante et harmoniques sphériques, p.110

D. Sdp, Figure 5.3 ? Figure de gauche : exemple de corps convexe de largeur constante décrit par (5.3.19) avec N = 2. Figure de droite : représentation du corps convexe de surface minimale

]. T. Bibliographie1, F. J. Bayen, F. J. Bonnans, and . Silva, Characterization of local quadratic growth for strong minima in the optimal control of semi-linear elliptic equations. to appear in Trans, 2013.

T. Bayen and F. J. Silva, Characterization of local quadratic growth for strong minima in the optimal control of semi-linear parabolic equations, 2013.

T. Bayen and F. J. Silva, Weak and strong minima : from calculus of variation toward pde optimization. 1st IFAC Workshop on Control of Systems Modeled by Partial Differential Equations
URL : https://hal.archives-ouvertes.fr/hal-00913735

T. Bayen, M. Sebbah, and A. Rapaport, Minimal time control of the two tanks gradostat model under a cascade inputs constraint, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00798651

T. Bayen and F. Mairet, Optimization of the separation of two species in a chemostat, Automatica, vol.50, issue.4, 2013.
DOI : 10.1016/j.automatica.2014.02.024

URL : https://hal.archives-ouvertes.fr/hal-00817147

T. Bayen, F. Mairet, and M. Mazade, Optimal feeding strategy for the minimal time problem of a fed-batch bioreactor with mortality rate Accepted in Optimal Control and Application Methods, 2012.

T. Bayen, P. Gajardo, and F. Mairet, Optimal Synthesis for the Minimum Time Control Problems of Fed-Batch Bioprocesses for Growth Functions with Two Maxima, Journal of Optimization Theory and Applications, vol.11, issue.11, pp.521-553, 2013.
DOI : 10.1007/s10957-012-0225-0

URL : https://hal.archives-ouvertes.fr/hal-01088417

T. Bayen and F. Mairet, Minimal time control of fed-batch bioreactor with product inhibition, 2012 20th Mediterranean Conference on Control & Automation (MED), pp.1485-1496, 2013.
DOI : 10.1109/MED.2012.6265700

URL : https://hal.archives-ouvertes.fr/hal-00734225

T. Bayen and F. Mairet, Analysis of an optimal control problem connected to bioprocesses involving a saturated singular arc, Discrete and Continuous Dynamical Systems - Series B, vol.20, issue.1, 2013.
DOI : 10.3934/dcdsb.2015.20.39

URL : https://hal.archives-ouvertes.fr/hal-00879385

T. Bayen, F. Mairet, and M. Mazade, Fed-batch bioreactor with mortality rate, 9th IFAC Symposium on Nonlinear Control Systems (NOLCOS 2013), 2013.
DOI : 10.3182/20130904-3-FR-2041.00024

URL : https://hal.archives-ouvertes.fr/hal-00850367

T. Bayen, F. Mairet, and P. Gajardo, Minimal time control of fed-batch bioreactor with product inhibition, 2012 20th Mediterranean Conference on Control & Automation (MED)
DOI : 10.1109/MED.2012.6265700

URL : https://hal.archives-ouvertes.fr/hal-00734225

T. Bayen, F. Mairet, and M. Sebbah, Optimal control problem connected to a onedimensional kolmogorov equation, 2013.

T. Bayen, P. Mairet, F. Martinon, and M. Sebbah, Optimizing the anaerobic digestion of microalgae in a coupled process, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00754971

T. Bayen, F. Mairet, P. Martinon, and M. Mazade, Optimizing the anaerobic digestion of microalgae in a coupled process, European Control Conference 2013, pp.17-19, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00754971

T. Bayen, Analytical Parameterization of Rotors and Proof of a Goldberg Conjecture by Optimal Control Theory, SIAM Journal on Control and Optimization, vol.47, issue.6, pp.3007-3036, 2009.
DOI : 10.1137/070705325

URL : https://hal.archives-ouvertes.fr/hal-00798653

T. Bayen and J. B. Hiriart-urruty, Objets convexes de largeur constante (en 2d) ou d'epaisseur constante (en 3d) : du neuf avec du vieux, Ann. Sci. Math. Quebec, vol.36, pp.17-42, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00798652

T. Bayen, T. Lachand-robert, and E. Oudet, Analytic Parametrization of Three-Dimensional Bodies of Constant Width, Archive for Rational Mechanics and Analysis, vol.12, issue.2, pp.225-249, 2007.
DOI : 10.1007/s00205-007-0060-x

URL : https://hal.archives-ouvertes.fr/hal-00385121

T. Bayen and D. Henrion, Semidefinite programming for optimizing convex bodies under width constraints. Optimization Methods and Software, pp.27-61073, 2012.
DOI : 10.1080/10556788.2010.547580

URL : https://hal.archives-ouvertes.fr/hal-00495031

T. Bayen, Parametrization of a convex optimization problem by optimal control theory and proof of a Goldberg conjecture, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference, 2009.
DOI : 10.1109/CDC.2009.5400248

T. Bayen, Y. Chitour, F. Jean, and P. Mason, Asymptotic analysis of an optimal control problem connected to the human locomotion, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference, 2009.
DOI : 10.1109/CDC.2009.5400873

L. S. Pontryagin, V. G. Boltyanskiy, R. V. Gamkrelidze, and E. F. Mishchenko, Mathematical theory of optimal processes, 1964.

A. Milyutin and N. Osmolovski?-i, Calculus of Variations and Optimal Control, Systems and Control : Foundations and Applications, 1998.

J. F. Bonnans and N. P. , Osmolovski? ?. Second-order analysis of optimal control problems with control and initial-final state constraints, Journal of Convex Analysis, vol.17, issue.3, pp.885-913, 2010.

J. F. Bonnans and N. P. , Osmolovski? ?. Characterization of a local quadratic growth of the hamiltonian for control constrained optimal control problems. Dynamics of Continuous, Discrete and Impulsive Systems, special issue dedicated to the memory of, Professor Arie Leizarowitz, vol.19, issue.12, pp.1-16, 2011.

J. F. Bonnans, X. Dupuis, and L. Pfeiffer, Second-Order Necessary Conditions in Pontryagin Form for Optimal Control Problems, SIAM Journal on Control and Optimization, vol.52, issue.6, 2013.
DOI : 10.1137/130923452

URL : https://hal.archives-ouvertes.fr/hal-00825273

J. F. Bonnans, X. Dupuis, and L. Pfeiffer, Second-order sufficient conditions for strong solutions to optimal control problems, ESAIM-COCV, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00825260

U. E. Raitums, The maximum principle in optimal control problems for an elliptic equation (in Russian), Zeitschrift f??r Analysis und ihre Anwendungen, vol.5, issue.4, pp.291-306, 1986.
DOI : 10.4171/ZAA/200

J. F. Bonnans and E. Casas, MAXIMUM PRINCIPLES IN THE OPTIMAL CONTROL OF SEMILINEAR ELLIPTIC SYSTEMS, 5th IFAC Conference on Distributed Parameter Systems, pp.33-1274, 1989.
DOI : 10.1016/B978-0-08-037036-1.50076-8

J. F. Bonnans and E. Casas, Un principe de Pontryagine pour le contr??le des syst??mes semilin??aires elliptiques, Journal of Differential Equations, vol.90, issue.2, pp.288-303, 1991.
DOI : 10.1016/0022-0396(91)90149-4

E. Casas, F. Troeltzsch, and A. Unger, Second Order Sufficient Optimality Conditions for a Nonlinear Elliptic Boundary Control Problem, Zeitschrift f??r Analysis und ihre Anwendungen, vol.15, issue.3, pp.687-707, 1996.
DOI : 10.4171/ZAA/723

J. F. Bonnans, Second-Order Analysis for Control Constrained Optimal Control Problems of Semilinear Elliptic Systems, Applied Mathematics and Optimization, vol.38, issue.3, pp.38-3303, 1998.
DOI : 10.1007/s002459900093

URL : https://hal.archives-ouvertes.fr/inria-00073680

J. F. Bonnans and H. Zidani, Optimal Control Problems with Partially Polyhedric Constraints, SIAM Journal on Control and Optimization, vol.37, issue.6, pp.1726-1741, 1999.
DOI : 10.1137/S0363012998333724

URL : https://hal.archives-ouvertes.fr/inria-00073340

E. Casas and F. Troeltzsch, Second-Order Necessary Optimality Conditions for Some State-Constrained Control Problems of Semilinear Elliptic Equations, Applied Mathematics and Optimization, vol.39, issue.2, pp.211-228, 1999.
DOI : 10.1007/s002459900104

E. Casas and F. Troeltzsch, Second-Order Necessary and Sufficient Optimality Conditions for Optimization Problems and Applications to Control Theory, SIAM Journal on Optimization, vol.13, issue.2, pp.406-431, 2002.
DOI : 10.1137/S1052623400367698

E. Casas and F. Troeltzsch, First- and Second-Order Optimality Conditions for a Class of Optimal Control Problems with Quasilinear Elliptic Equations, SIAM Journal on Control and Optimization, vol.48, issue.2, pp.48-2688, 2009.
DOI : 10.1137/080720048

E. Casas, J. C. De-los-reyes, and F. Troeltzsch, Sufficient Second-Order Optimality Conditions for Semilinear Control Problems with Pointwise State Constraints, SIAM Journal on Optimization, vol.19, issue.2, pp.19-2616, 2008.
DOI : 10.1137/07068240X

X. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems, Birkhäuser, 1994. Systems and Control : Foundations and Applications
DOI : 10.1007/978-1-4612-4260-4

H. Goldberg and F. Troeltzsch, Second-Order Sufficient Optimality Conditions for a Class of Nonlinear Parabolic Boundary Control Problems, SIAM Journal on Control and Optimization, vol.31, issue.4, pp.1007-1025, 1993.
DOI : 10.1137/0331045

J. P. Raymond and F. Troeltzsch, Second order sufficient optimality conditions for nonlinear parabolic control problems with state constraints. Discrete and Continuous Dynamical Systems, pp.431-450, 2000.

A. Rösch and F. Troeltzsch, Sufficient Second-Order Optimality Conditions for a Parabolic Optimal Control Problem with Pointwise Control-State Constraints, SIAM Journal on Control and Optimization, vol.42, issue.1, pp.138-154, 2003.
DOI : 10.1137/S0363012902403262

R. A. Adams, Sobolev spaces, 1975.

L. C. Evans, Partial differential equations, Amer. Math Soc. Graduate Studies in Mathematics, vol.19, 1998.

D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, 1983.

G. Stampacchia, Le probl??me de Dirichlet pour les ??quations elliptiques du second ordre ?? coefficients discontinus, Annales de l???institut Fourier, vol.15, issue.1, pp.189-258, 1965.
DOI : 10.5802/aif.204

URL : http://archive.numdam.org/article/AIF_1965__15_1_189_0.pdf

J. F. Bonnans and F. J. Silva, Asymptotic Expansion for the Solutions of Control Constrained Semilinear Elliptic Problems with Interior Penalties, SIAM Journal on Control and Optimization, vol.49, issue.6, pp.2494-2517, 2011.
DOI : 10.1137/090778602

R. T. Rockafellar, Integral functionals, normal integrands and measurable selections, Nonlinear operators and the calculus of variations, pp.157-207, 1975.
DOI : 10.1090/S0002-9947-1971-0282283-0

J. F. Bonnans, Pontryagin's principle for the optimal control of semilinear elliptic systems with state constraints, [1991] Proceedings of the 30th IEEE Conference on Decision and Control, 1991.
DOI : 10.1109/CDC.1991.261763

J. F. Bonnans and A. Shapiro, Perturbation analysis of optimization problems, 2000.
DOI : 10.1007/978-1-4612-1394-9

F. Tröltzsch, Optimal Control of Partial Differential Equations -Theory, Methods and Applications, Graduate Studies in Mathematics, vol.112, 2010.

A. Hoffman, On approximate solutions of systems of linear inequalities, Journal of Research of the National Bureau of Standards, Section B, Mathematical Sciences, vol.49, pp.263-265, 1952.

J. F. Bonnans, Optimal control of a semilinear parabolic equation with singular arcs, Optimization Methods and Software, vol.37, issue.2
DOI : 10.1007/s10107-004-0559-y

URL : https://hal.archives-ouvertes.fr/hal-00740698

J. F. Bonnans, X. Dupuis, and L. Pfeiffer, Second-Order Necessary Conditions in Pontryagin Form for Optimal Control Problems, SIAM Journal on Control and Optimization, vol.52, issue.6, 2013.
DOI : 10.1137/130923452

URL : https://hal.archives-ouvertes.fr/hal-00825273

J. F. Bonnans, X. Dupuis, and L. Pfeiffer, Second-order sufficient conditions for strong solutions to optimal control problems, ESAIM-COCV
URL : https://hal.archives-ouvertes.fr/hal-00825260

T. Aubin, Un théorème de compacité, CRAS, vol.256, pp.5042-5044, 1963.

J. Simon, Compact sets in L p (0, T ; B) CXLVI, Annali Mat. Pura appl. (IV), pp.65-96, 1987.

O. V. Besov and V. P. , Il'in, and S.M. Nikol'skii. Integral representations of functions and imbedding theorems, 1979.

M. Valadier and C. Castaing, Convex analysis and measurable multifunctions, Lecture Notes in Math, issue.580, 1977.

H. L. Smith and P. Waltman, The Theory of the Chemostat, Dynamics of Microbial Competition, Cambridge Studies in Mathematical Biology, vol.13, 1995.

U. Boscain and B. Piccoli, Optimal syntheses for control systems on 2- D manifolds, of Mathématiques & Applications (Berlin) [Mathematics & Applications, 2004.

D. A. Carlson and A. Haurie, Infinite horizon optimal control, of Lecture Notes in Economics and Mathematical Systems Theory and applications, 1987.

P. Cardaliaguet, On the Regularity of Semipermeable Surfaces in Control Theory with Application to the Optimal Exit-Time Problem (Part I), SIAM Journal on Control and Optimization, vol.35, issue.5, pp.1638-1652, 1997.
DOI : 10.1137/S0363012995287295

P. Cardaliaguet, On the Regularity of Semipermeable Surfaces in Control Theory with Application to the Optimal Exit-Time Problem (Part I), SIAM Journal on Control and Optimization, vol.35, issue.5, pp.1653-1671, 1997.
DOI : 10.1137/S0363012995287295

M. Quincampoix, Differential Inclusions and Target Problems, SIAM Journal on Control and Optimization, vol.30, issue.2, pp.324-335, 1992.
DOI : 10.1137/0330020

B. Bonnard and M. Chyba, Singular Trajectories and their role in Control Theory, 2002.

J. Moreno, Optimal time control of bioreactors for the wastewater treatment, Optimal Control Applications and Methods, vol.20, issue.3, pp.145-164, 1999.
DOI : 10.1002/(SICI)1099-1514(199905/06)20:3<145::AID-OCA651>3.0.CO;2-J

D. Dochain and A. Rapaport, Minimal time control of fed-batch processes with growth functions having several maxima, IEEE Trans. Automat. Control, issue.11, pp.562671-2676, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00859560

B. Bonnard, M. Chyba, and D. Sugny, Time-Minimal Control of Dissipative Two-Level Quantum Systems: The Generic Case, IEEE Transactions on Automatic Control, vol.54, issue.11, pp.2598-2610, 2009.
DOI : 10.1109/TAC.2009.2031212

URL : https://hal.archives-ouvertes.fr/hal-00508378

H. Schaettler and U. Ledzewicz, Anti-angiogenic therapy in cancer treatment as an optimal control problem, SIAM J. on Control and Optimization, vol.46, issue.3, pp.1052-1079, 2007.

H. Schaettler and M. Jankovic, A synthesis of time-optimal controls in the presence of saturated singular arcs, Forum Mathematicum, vol.5, issue.5, pp.203-241, 2009.
DOI : 10.1515/form.1993.5.203

P. Gajardo, H. Ramírez, C. , and A. Rapaport, Minimal Time Sequential Batch Reactors with Bounded and Impulse Controls for One or More Species, SIAM Journal on Control and Optimization, vol.47, issue.6, pp.2827-2856, 2008.
DOI : 10.1137/070695204

URL : https://hal.archives-ouvertes.fr/hal-00857823

C. Italo and . Dolcetta, Hamilton-Jacobi-Bellman equations and optimal control In Variational calculus, optimal control and applications (Trassenheide, Internat. Ser. Numer. Math, vol.124, pp.121-132, 1996.

E. Trélat, B. Bonnard, and J. Caillau, Second order optimality conditions in the smooth case and applications in optimal control, ESAIM Control Optim. Calc. Var, vol.13, issue.2, pp.207-236, 2007.

F. Rampazzo and A. Bressan, Impulsive control systems with commutative vector fields, Journal of Optim. Theory and Applicationss, vol.71, issue.2, pp.67-83, 1991.

P. Cartigny and A. Rapaport, Turnpike theorems in infinite horizon by a value function approach, ESAIM Control, Optimization and Calculus of Variations (COCV), vol.10, pp.123-141, 2004.

F. Zanolin, Continuation theorems for the periodic problem via the translation operator, Rend. Sem. Mat. Univ. Pol. Torino, vol.54, issue.1, 1996.

L. Cesari, Optimization?Theory and Applications, Problems with Ordinary Differential Equations, Applications of Mathematics, vol.17, 1983.

A. Akhmetzhanov, O. Bernard, F. Grognard, and P. Masci, Optimization of a photobioreactor biomass production using natural light, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00516705

K. Mischaikow, H. L. Smith, and H. Thieme, Asymptotically autonomous semiflows: chain recurrence and Lyapunov functions, Transactions of the American Mathematical Society, vol.347, issue.5, pp.1669-1685, 1995.
DOI : 10.1090/S0002-9947-1995-1290727-7

H. M. Robbins, A Generalized Legendre-Clebsch Condition for the Singular Cases of Optimal Control, IBM Journal of Research and Development, vol.11, issue.4, pp.361-372, 1967.
DOI : 10.1147/rd.114.0361

R. Howard, Convex bodies of constant width and constant brightness, Advances in Mathematics, vol.204, issue.1, pp.241-261, 2006.
DOI : 10.1016/j.aim.2005.05.015

T. Lachand-robert and M. A. Peletier, Newton's Problem of the Body of Minimal Resistance in the Class of Convex Developable Functions, Mathematische Nachrichten, vol.173, issue.1, pp.153-176, 2001.
DOI : 10.1002/1522-2616(200106)226:1<153::AID-MANA153>3.0.CO;2-2

T. Lachand-robert and M. A. Peletier, An example of non-convex minimization and an application to Newton's problem of the body of least resistance, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, vol.18, issue.2, pp.179-198, 2001.
DOI : 10.1016/S0294-1449(00)00062-7

R. Schneider, Convex bodies : the Brunn-Minkowski theory, volume 44 of Encyclopedia of Mathematics and its Applications, 1993.

D. E. Mcclure and R. A. Vitale, Polygonal approximation of plane convex bodies, Journal of Mathematical Analysis and Applications, vol.51, issue.2, pp.326-358, 1975.
DOI : 10.1016/0022-247X(75)90125-0

B. Dumitrescu, Positive trigonometric polynomials and signal processing applications . Signals and Communication Technology, 2007.
DOI : 10.1007/978-3-319-53688-0

J. Löfberg, YALMIP : a toolbox for modeling and optimization in MATLAB, 2004 IEEE International Conference on Robotics and Automation (IEEE Cat. No.04CH37508), 2004.
DOI : 10.1109/CACSD.2004.1393890

M. Ko?vara, M. D. Sting, J. Henrion, and . Löfberg, Solving polynomial static output feedback problems with penbmi, Proc. of joint IEEE Conf. on Decision and Control and Europ. Control Conf, 2005.

J. Bernard and L. , Moments, positive polynomials and their applications, volume 1 of Imperial College Press Optimization Series, 2010.

G. Carlier, On a theorem of Alexandrov, J. Nonlinear Convex Anal, vol.5, issue.1, pp.49-58, 2004.

B. Juttler, Z. Sir, and J. Gravesen, Curves and surfaces represented by polynomial support functions, Theoretical Computer Sciences, vol.392, pp.141-157, 2008.