M. Ahmadi, G. Valmorbida, and A. Papachristodoulou, A convex approach to hydrodynamic analysis, 54th IEEE Conference on Decision and Control (CDC), pp.7262-7267, 2015.

M. Ahmadi, G. Valmorbida, and A. Papachristodoulou, Dissipation inequalities for the analysis of a class of PDEs, Automatica, vol.66, pp.163-171, 2016.

J. C. Del-alamo and J. Jiménez, Linear energy amplification in turbulent channels, Journal of Fluid Mechanics, vol.559, pp.205-213, 2006.

A. Alexakis and C. R. Doering, Energy and enstrophy dissipation in steady state 2d turbulence, Physics Letters A, vol.359, issue.6, pp.652-657, 2006.

B. Bamieh and M. Dahleh, Energy amplification in channel flows with stochastic excitation, Physics of Fluids, vol.13, issue.11, pp.3258-3269, 2001.

K. M. Bobba, B. Bamieh, and J. C. Doyle, Highly optimized transitions to turbulence, Proceedings of the 41st IEEE Conference on, vol.4, pp.4559-4562, 2002.

D. Bovet and P. Crescenzi, Introduction to the Theory of Complexity, 1994.

S. Boyd, L. Vandenberghe, K. J. Bullock, R. E. Cooper, and F. H. Abernathy, Convex optimization, Structural similarity in radial correlations and spectra of longitudinal velocity fluctuations in pipe flow, vol.88, pp.585-608, 1978.

K. M. Butler and B. F. Farrell, Three-dimensional optimal perturbations in viscous shear flow, Physics of Fluids A: Fluid Dynamics, vol.4, issue.8, pp.1637-1650, 1992.

S. Chernyshenko, Relationship between the methods of bounding time averages, 2017.

S. Chernyshenko, P. Goulart, D. Huang, and A. Papachristodoulou, Polynomial sum of squares in fluid dynamics: a review with a look ahead, Royal Society of London. Philosophical Transactions A. Mathematical, Physical and Engineering Sciences, vol.372, 2014.

G. Chesi, A. Tesi, A. Vicino, and R. Genesio, On convexification of some minimum distance problems, 5th European Control Conference, 1999.

S. Childress, R. R. Kerswell, and A. D. Gilbert, Bounds on dissipation for Navier-Stokes flow with Kolmogorov forcing, Physica D: Nonlinear Phenomena, vol.158, issue.1, pp.105-128, 2001.

M. D. Choi, T. Y. Lam, and B. Reznick, Sums of squares of real polynomials, Symposia in Pure Mathematics, vol.58, pp.103-126, 1995.

R. F. Curtain and H. J. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, Texts in Applied Mathematics, vol.21, 1995.

L. De-branges, The Stone-Weierstrass theorem, Proceedings of the American Mathematical Society, vol.10, issue.5, pp.822-824, 1959.

C. R. Doering and P. Constantin, Variational bounds on energy dissipation in incompressible flows: Shear flow, Phys. Rev. E, vol.49, pp.4087-4099, 1994.

C. R. Doering and C. Foias, Energy dissipation in body-forced turbulence, Journal of Fluid Mechanics, vol.467, pp.289-306, 2002.

C. R. Doering and J. D. Gibbon, Applied Analysis of the Navier-Stokes Equations, Cambridge Texts in Applied Mathematics, vol.12, 1995.

P. G. Drazin and W. H. Reid, Hydrodynamic Stability, 1981.

G. Fantuzzi, Bounds for Rayleigh-Bénard convection between free-slip boundaries with an imposed heat flux, Journal of Fluid Mechanics, vol.837, 2018.

B. F. Farrell, . Ioannou, and J. Petros, Stochastic forcing of the linearized Navier-Stokes equations, Physics of Fluids A, vol.5, issue.11, pp.2600-2609, 1993.

A. Gahlawat and M. M. Peet, A convex sum-of-squares approach to analysis, state feedback and output feedback control of parabolic PDEs, IEEE Transactions on Automatic Control, vol.62, issue.4, pp.1636-1651, 2017.

D. F. Gayme, B. J. Mckeon, B. Bamieh, A. Papachristodoulou, and J. C. Doyle, Amplification and nonlinear mechanisms in plane Couette flow, Physics of Fluids, vol.23, issue.6, 2011.

D. F. Gayme, B. J. Mckeon, A. Papachristodoulou, B. Bamieh, and J. C. Doyle, A streamwise constant model of turbulence in plane Couette flow 665, pp.99-119, 2010.

D. Goluskin and G. Fantuzzi, Bounds on mean energy in the kuramoto-sivashinsky equation computed using semidefinite programming, 2018.

P. J. Goulart and S. Chernyshenko, Global stability analysis of fluid flows using sumof-squares, Physica D: Nonlinear Phenomena, vol.241, issue.6, pp.692-704, 2012.

M. Grant, S. Boyd, and Y. Ye, CVX: Matlab software for disciplined convex programming, 2008.

S. Grossmann, The onset of shear flow turbulence, Rev. Mod. Phys, vol.72, pp.603-618, 2000.

L. H. Gustavsson, Energy growth of three-dimensional disturbances in plane Poiseuille flow, Journal of Fluid Mechanics, vol.224, pp.241-260, 1991.

L. H. Gustavsson, Energy growth of three-dimensional disturbances in plane poiseuille flow, Journal of Fluid Mechanics, vol.224, pp.241-260, 1991.

P. H. Heins, B. Jones, . Ll, and A. S. Sharma, Passivity-based output-feedback control of turbulent channel flow, Automatica, vol.69, pp.348-355, 2016.

D. J. Hill and P. J. Moylan, Dissipative dynamical systems: basic input-output and state properties, Journal of the Franklin Institute, vol.309, issue.5, pp.327-357, 1980.

D. Huang, S. Chernyshenko, P. Goulart, D. Lasagna, O. Tutty et al., , 2015.

, Sum-of-squares of polynomials approach to nonlinear stability of fluid flows: an example of application, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol.471, 2183.

N. Hutchins and I. Marusic, Evidence of very long meandering features in the logarithmic region of turbulent boundary layers, vol.579, pp.1-28, 2007.

D. D. Joseph, Stability of fluid motions, 1976.

D. D. Joseph and W. Hung, Contributions to the nonlinear theory of stability of viscous flow in pipes and between rotating cylinders, Archive for Rational Mechanics and Analysis, vol.44, issue.1, pp.1-22, 1971.

M. R. Jovanovi?, Modeling, analysis, and control of spatially distributed systems, 2004.

M. R. Jovanovi? and B. Bamieh, Componentwise energy amplification in channel flows, Journal of Fluid Mechanics, vol.534, pp.145-183, 2005.

H. K. Khalil, Noninear Systems, 1996.

K. J. Kim and R. Adrian, , vol.11, pp.417-422, 1999.

S. J. Kline, W. C. Reynolds, F. A. Schraub, and P. W. Runstadler, The structure of turbulent boundary layers, vol.30, pp.741-773, 1967.

J. B. Lasserre, Moments, Positive Polynomials and Their Applications, 2009.

J. Löfberg, YALMIP : A toolbox for modeling and optimization in MATLAB, Proceedings of the CACSD Conference, 2004.

G. Lumer and R. S. Phillips, Dissipative operators in a banach space, Pacific J. Math, vol.11, issue.2, pp.679-698, 1961.

B. J. Mckeon and A. S. Sharma, A critical-layer framework for turbulent pipe flow, Journal of Fluid Mechanics, vol.658, pp.336-382, 2010.

R. Moarref, M. R. Jovanovic, J. A. Tropp, A. S. Sharma, and B. J. Mckeon, A low-order decomposition of turbulent channel flow via resolvent analysis and convex optimization, Physics of Fluids, vol.26, issue.5, p.51701, 2014.

T. S. Motzkin, The arithmetic-geometric inequality, 1967 Inequalities Symposium, pp.205-224, 1965.

Y. Nesterov and A. Nemirovskii, Interior-point Polynomial Algorithms in Convex Programming, Society for Industrial and Applied Mathematics, 1994.

A. Papachristodoulou, J. Anderson, G. Valmorbida, S. Prajna, P. Seiler et al., SOSTOOLS: Sum of squares optimization toolbox for MATLAB, 2013.

P. Parrilo, Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization, 2000.

L. E. Payne and H. F. Weinberger, An optimal Poincare inequality for convex domains, Archive for Rational Mechanics and Analysis, vol.5, issue.1, pp.286-292, 1960.

J. Peixinho, T. Mullin, S. Prajna, A. Papachristodoulou, and F. Wu, Nonlinear control synthesis by sum of squares optimization: A Lyapunov-based approach, Control Conference, vol.96, pp.157-165, 2004.

G. Pujals, M. Gar?ia-villalba, C. Cossu, and S. Depardon, A note on optimal transient growth in turbulent channel flows, Physics of Fluids, vol.21, issue.1, 2009.
URL : https://hal.archives-ouvertes.fr/hal-01021124

S. C. Reddy and D. S. Henningson, Energy growth in viscous channel flows, Journal of Fluid Mechanics, vol.252, pp.209-238, 1993.

O. Reynolds, An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous and the law of resistance in parallel channels, Philos. Trans, issue.51, p.935, 1883.

B. Reznick, Some concrete aspects of Hilbert's 17th problem, Contemporary mathematics, vol.253, pp.251-272, 2000.

B. Rollin, Y. Dubief, and C. R. Doering, Variations on Kolmogorov flow: turbulent energy dissipation and mean flow profiles, Journal of Fluid Mechanics, vol.670, pp.204-213, 2011.

S. G. Rubin and P. K. Khosla, Polynomial interpolation methods for viscous flow calculations, Journal of Computational Physics, vol.24, issue.3, pp.217-244, 1977.

A. Van-der-schaft, L2-gain and passivity techniques in nonlinear control, 2017.

P. J. Schmid, Nonmodal stability theory, Annual Review of Fluid Mechanics, vol.39, issue.1, pp.129-162, 2007.
URL : https://hal.archives-ouvertes.fr/hal-01023333

P. J. Schmid and D. S. Henningson, Optimal energy density growth in Hagen-Poiseuille flow, Journal of Fluid Mechanics, vol.277, pp.197-225, 1994.

P. J. Schmid and D. S. Henningson, Stability and transition in shear flows, 2001.

J. Serrin, On the stability of viscous fluid motions, Arch. Ration. Mech. Anal, vol.3, pp.1-13, 1959.

A. S. Sharma, J. F. Morrison, B. J. Mckeon, D. J. Limebeer, W. H. Koberg et al., Relaminarisation of re? = 100 channel flow with globally stabilising linear feedback control, Physics of Fluids, vol.23, issue.12, p.125105, 2011.

S. Shivakumar and M. Peet, Computing input-output properties of coupled PDE systems, 2018.

E. D. Sontag, Input to state stability: Basic concepts and results, Nonlinear and Optimal Control Theory, vol.1932, pp.163-220, 2008.

E. D. Sontag, Input to state stability, Encyclopedia of Systems and Control, pp.1-14, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00639491

J. F. Sturm, Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones, 1998.

W. Tang, C. P. Caulfield, and W. R. Young, Bounds on dissipation in stress-driven flow, Journal of Fluid Mechanics, vol.510, pp.333-352, 2004.

G. I. Taylor, Stability of a viscous liquid contained between two rotating cylinders, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol.223, pp.289-343, 1923.

N. Tillmark and P. H. Alfredsson, Experiments on transition in plane Couette flow, Journal of Fluid Mechanics, vol.235, pp.89-102, 1992.

L. N. Trefethen, A. E. Trefethen, S. C. Reddy, and T. A. Driscoll, Hydrodynamic stability without eigenvalues, Science, vol.261, issue.5121, pp.578-584, 1993.

J. C. Vassilicos, Dissipation in turbulent flows, Annual Review of Fluid Mechanics, vol.47, issue.1, pp.95-114, 2015.

R. Vazquez and M. Krstic, Control of turbulent and magnetohydrodynamic channel flows: boundary stabilization and state estimation, 2008.

D. Weihs, On the polynomial approximation of boundary-layer flow profiles, Applied Scientific Research, vol.31, issue.4, pp.253-266, 1975.

J. C. Willems, Dissipative dynamical systems part I: General theory. Archive for, Rational Mechanics and Analysis, vol.45, issue.5, pp.321-351, 1972.

J. C. Willems, Dissipative dynamical systems, European Journal of Control, vol.13, issue.23, pp.134-151, 2007.