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Turnpike in optimal control of PDEs, ResNets, and beyond

Published online by Cambridge University Press:  09 June 2022

Borjan Geshkovski Open the ORCID record for Borjan Geshkovski [Opens in a new window]  and
Enrique Zuazua
Borjan Geshkovski
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA02139, USA E-mail: borjan@mit.edu
Enrique Zuazua
Affiliation:
Chair in Dynamics, Control, and Numerics, Alexander von Humboldt-Professorship, Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstrasse 11, 91052Erlangen, Germany, Chair of Computational Mathematics, Fundación Deusto, Av. de las Universidades 24, 48007 Bilbao, Basque Country, Spain, and Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049Madrid, Spain E-mail: enrique.zuazua@fau.de
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Abstract

The turnpike property in contemporary macroeconomics asserts that if an economic planner seeks to move an economy from one level of capital to another, then the most efficient path, as long as the planner has enough time, is to rapidly move stock to a level close to the optimal stationary or constant path, then allow for capital to develop along that path until the desired term is nearly reached, at which point the stock ought to be moved to the final target. Motivated in part by its nature as a resource allocation strategy, over the past decade, the turnpike property has also been shown to hold for several classes of partial differential equations arising in mechanics. When formalized mathematically, the turnpike theory corroborates insights from economics: for an optimal control problem set in a finite-time horizon, optimal controls and corresponding states are close (often exponentially) most of the time, except near the initial and final times, to the optimal control and the corresponding state for the associated stationary optimal control problem. In particular, the former are mostly constant over time. This fact provides a rigorous meaning to the asymptotic simplification that some optimal control problems appear to enjoy over long time intervals, allowing the consideration of the corresponding stationary problem for computing and applications. We review a slice of the theory developed over the past decade – the controllability of the underlying system is an important ingredient, and can even be used to devise simple turnpike-like strategies which are nearly optimal – and present several novel applications, including, among many others, the characterization of Hamilton–Jacobi–Bellman asymptotics, and stability estimates in deep learning via residual neural networks.

Type
Research Article
Information
Acta Numerica , Volume 31 , May 2022 , pp. 135 - 263
Copyright
© The Author(s), 2022. Published by Cambridge University Press

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Footnotes

Dedicated to the memory of Roland Glowinski

*

A major part of this work was completed while the author was affiliated with the Chair of Computational Mathematics, Fundación Deusto.

References

Aftalion, A. and Trélat, E. (2021), Pace and motor control optimization for a runner, J. Math. Biol. 83, 1–21.CrossRefGoogle ScholarPubMed
Agrachev, A. and Sarychev, A. (2021), Control on the manifolds of mappings with a view to the deep learning, J. Dyn. Control Syst. Available at doi:10.1007/s10883-021-09561-2.CrossRefGoogle Scholar
Agrachev, A., Barilari, D. and Boscain, U. (2019), A Comprehensive Introduction to Sub-Riemannian Geometry , Vol. 181 of Cambridge Studies in Advanced Mathematics, Cambridge University Press.Google Scholar
Agrachev, A., Baryshnikov, Y. and Sarychev, A. (2016), Ensemble controllability by Lie algebraic methods, ESAIM Control Optim. Calc. Var. 22, 921–938.CrossRefGoogle Scholar
Allahverdi, N., Pozo, A. and Zuazua, E. (2016), Numerical aspects of large-time optimal control of Burgers equation, ESAIM Math. Model. Numer. Anal. 50, 1371–1401.CrossRefGoogle Scholar
Allaire, G., Münch, A. and Periago, F. (2010), Long time behavior of a two-phase optimal design for the heat equation, SIAM J. Control Optim. 48, 5333–5356.CrossRefGoogle Scholar
Alonso, J. J. and Colonno, M. R. (2012), Multidisciplinary optimization with applications to sonic-boom minimization, Annu. Rev. Fluid Mech. 44, 505–526.CrossRefGoogle Scholar
Ammari, H., Asch, M., Bustos, L. G., Jugnon, V. and Kang, H. (2011), Transient wave imaging with limited-view data, SIAM J. Imag. Sci. 4, 1097–1121.CrossRefGoogle Scholar
Backhoff, J., Conforti, G., Gentil, I. and Léonard, C. (2020), The mean field Schrödinger problem: Ergodic behavior, entropy estimates and functional inequalities, Probab . Theory Related Fields 178, 475–530.CrossRefGoogle Scholar
Bardos, C., Lebeau, G. and Rauch, J. (1992), Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim. 30, 1024–1065.CrossRefGoogle Scholar
Barles, G. and Souganidis, P. E. (2000), On the large time behavior of solutions of Hamilton–Jacobi equations, SIAM J. Math. Anal. 31, 925–939.CrossRefGoogle Scholar
Barles, G., Ley, O., Nguyen, T.-T. and Phan, T. V. (2019), Large time behavior of unbounded solutions of first-order Hamilton–Jacobi equations in ${\mathbb{R}}^n$ , Asymptot. Anal. 112, 1–22.Google Scholar
Barron, A. R., Cohen, A., Dahmen, W. and DeVore, R. A. (2008), Approximation and learning by greedy algorithms, Ann. Statist. 36, 64–94.CrossRefGoogle Scholar
Beauchard, K., Coron, J.-M. and Rouchon, P. (2010), Controllability issues for continuous-spectrum systems and ensemble controllability of Bloch equations, Comm. Math. Phys. 296, 525–557.CrossRefGoogle Scholar
Benning, M., Celledoni, E., Ehrhardt, M. J., Owren, B. and Schönlieb, C.-B. (2019), Deep learning as optimal control problems: Models and numerical methods, J. Comput. Dyn. 6, 171–198.CrossRefGoogle Scholar
Benzi, M., Golub, G. H. and Liesen, J. (2005), Numerical solution of saddle point problems, in Acta Numerica, Vol. 14, Cambridge University Press, pp. 1–137.Google Scholar
Bertsekas, D. (2019), Reinforcement Learning and Optimal Control, Athena Scientific.Google Scholar
Bertsekas, D. (2021), Lessons from AlphaZero for optimal, model predictive, and adaptive control. Available at arXiv:2108.10315.Google Scholar
Betts, J. T. (2010), Practical Methods for Optimal Control and Estimation using Nonlinear Programming, SIAM.CrossRefGoogle Scholar
Brogliato, B., Lozano, R., Maschke, B. and Egeland, O. (2007), Dissipative Systems Analysis and Control: Theory and Applications, Communications and Control Engineering Series, second edition, Springer.CrossRefGoogle Scholar
Buckdahn, R., Quincampoix, M. and Renault, J. (2015), On representation formulas for long run averaging optimal control problem, J. Differential Equations 259, 5554–5581.CrossRefGoogle Scholar
Burq, N. and Gérard, P. (1997), Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes, C.R. Acad. Sci. Ser. I Math. 325, 749–752.Google Scholar
Cannarsa, P., Beauchard, K. and Guglielmi, R. (2013), Null controllability of Grushin-type operators in dimension two, J. Eur. Math. Soc. 16, 67–101.Google Scholar
Cardaliaguet, P. (2010), Notes on mean field games. Notes from P. L. Lions’ lectures at the Collège de France.Google Scholar
Cardaliaguet, P. and Porretta, A. (2019), Long time behavior of the master equation in mean field game theory, Anal. PDE 12, 1397–1453.CrossRefGoogle Scholar
Cardaliaguet, P. and Porretta, A. (2020), An introduction to Mean Field Game Theory, in Mean Field Games (Cardaliaguet, P. and Porretta, A., eds), Vol. 2281 of Lecture Notes in Mathematics, Springer, pp. 1–158.CrossRefGoogle Scholar
Cardaliaguet, P., Lasry, J.-M., Lions, P.-L. and Porretta, A. (2012), Long time average of mean field games, Netw . Heterog. Media 7, 279.CrossRefGoogle Scholar
Cardaliaguet, P., Lasry, J.-M., Lions, P.-L. and Porretta, A. (2013), Long time average of mean field games with a nonlocal coupling, SIAM J. Control Optim. 51, 3558–3591.CrossRefGoogle Scholar
Carmichael, N. and Quinn, M. D. (1985), Fixed Point Methods in Nonlinear Control, Springer.CrossRefGoogle Scholar
Casas, E. and Mateos, M. (2002), Second order optimality conditions for semilinear elliptic control problems with finitely many state constraints, SIAM J. Control Optim. 40, 1431–1454.CrossRefGoogle Scholar
Casas, E. and Tröltzsch, F. (2002), Second-order necessary and sufficient optimality conditions for optimization problems and applications to control theory, SIAM J. Optim. 13, 406–431.CrossRefGoogle Scholar
Castro, C., Lozano, C., Palacios, F. and Zuazua, E. (2007), Systematic continuous adjoint approach to viscous aerodynamic design on unstructured grids, AIAA J. 45, 2125–2139.CrossRefGoogle Scholar
Cazenave, T. and Haraux, A. (1998), An Introduction to Semilinear Evolution Equations, Vol. 13 of Oxford Lecture Series in Mathematics and its Applications, The Clarendon Press, Oxford University Press.Google Scholar
Celledoni, E., Ehrhardt, M. J., Etmann, C., McLachlan, R. I., Owren, B., Schönlieb, C.-B. and Sherry, F. (2021), Structure-preserving deep learning, European J. Appl. Math. 32, 888–936.CrossRefGoogle Scholar
Chen, T. Q., Rubanova, Y., Bettencourt, J. and Duvenaud, D. K. (2018), Neural ordinary differential equations, in Advances in Neural Information Processing Systems 31 (Bengio, S. et al., eds), Curran Associates, pp. 6571–6583.Google Scholar
Christof, C. and Hafemeyer, D. (2022), On the nonuniqueness and instability of solutions of tracking-type optimal control problems, Math. Control Relat. Fields 12, 421–431.CrossRefGoogle Scholar
Cirant, M. and Porretta, A. (2021), Long time behavior and turnpike solutions in mildly non-monotone mean field games, ESAIM Control Optim. Calc. Var. 27, 86.CrossRefGoogle Scholar
Cohen, A. and DeVore, R. (2015), Approximation of high-dimensional parametric PDEs, in Acta Numerica, Vol. 24, Cambridge University Press, pp. 1–159.Google Scholar
Coron, J.-M. (2007), Control and Nonlinearity, Vol. 136 of Mathematical Surveys and Monographs, American Mathematical Society.Google Scholar
Cuchiero, C., Larsson, M. and Teichmann, J. (2020), Deep neural networks, generic universal interpolation, and controlled ODEs, SIAM J. Math. Data Sci. 2, 901–919.CrossRefGoogle Scholar
Cybenko, G. (1989), Approximation by superpositions of a sigmoidal function, Math. Control Signals Systems 2, 303–314.CrossRefGoogle Scholar
Dáger, R. and Zuazua, E. (2006), Wave Propagation, Observation and Control in 1- $d$ Flexible Multi-structures, Vol. 50 of Mathématiques & Applications, Springer.Google Scholar
Damm, T., Grüne, L., Stieler, M. and Worthmann, K. (2014), An exponential turnpike theorem for dissipative discrete time optimal control problems, SIAM J. Control Optim. 52, 1935–1957.CrossRefGoogle Scholar
Datko, R. (1972), Uniform asymptotic stability of evolutionary processes in a Banach space, SIAM J. Math. Anal. 3, 428–445.CrossRefGoogle Scholar
Dean, S., Mania, H., Matni, N., Recht, B. and Tu, S. (2020), On the sample complexity of the linear quadratic regulator, Found. Comput. Math. 20, 633–679.CrossRefGoogle Scholar
DeVore, R., Hanin, B. and Petrova, G. (2021), Neural network approximation, in Acta Numerica, Vol. 30, Cambridge University Press, pp. 327–444.Google Scholar
DeVore, R., Petrova, G. and Wojtaszczyk, P. (2013), Greedy algorithms for reduced bases in Banach spaces, Constr. Approx. 37, 455–466.CrossRefGoogle Scholar
Dorfman, R., Samuelson, P. A. and Solow, R. M. (1958), Linear Programming and Economic Analysis, Courier Corporation.Google Scholar
Dupont, E., Doucet, A. and Teh, Y. W. (2019), Augmented neural ODEs, in Advances in Neural Information Processing Systems 32 (Wallach, H. et al., eds), Curran Associates, pp. 3134–3144.Google Scholar
E, W. (2017), A proposal on machine learning via dynamical systems, Commun. Math. Stat. 5, 1–11.CrossRefGoogle Scholar
Esteve, C. and Zuazua, E. (2020), The inverse problem for Hamilton–Jacobi equations and semiconcave envelopes, SIAM J. Math. Anal. 52, 5627–5657.CrossRefGoogle Scholar
Esteve, C., Kouhkouh, H., Pighin, D. and Zuazua, E. (2020), The turnpike property and the long-time behavior of the Hamilton-Jacobi equation. Available at arXiv:2006.10430.Google Scholar
Esteve-Yagüe, C. and Geshkovski, B. (2021), Sparse approximation in learning via neural ODEs. Available at arXiv:2102.13566.Google Scholar
Esteve-Yagüe, C., Geshkovski, B., Pighin, D. and Zuazua, E. (2020), Large-time asymptotics in deep learning. Available at arXiv:2008.02491.Google Scholar
Esteve-Yagüe, C., Geshkovski, B., Pighin, D. and Zuazua, E. (2022), Turnpike in Lipschitz-nonlinear optimal control, Nonlinearity 35, 1652.CrossRefGoogle Scholar
Evans, L. C. (1998), Partial Differential Equations, Vol. 19 of Graduate Studies in Mathematics, American Mathematical Society.Google Scholar
Faulwasser, T. and Grüne, L. (2022), Turnpike properties in optimal control: An overview of discrete-time and continuous-time results, in Numerical Control: Part A (Trélat, E. and Zuazua, E., eds), Vol. 23 of Handbook of Numerical Analysis, Elsevier, pp. 367–400.CrossRefGoogle Scholar
Faulwasser, T., Flaßkamp, K., Ober-Blöbaum, S. and Worthmann, K. (2019), Towards velocity turnpikes in optimal control of mechanical systems, IFAC-PapersOnLine 52, 490–495.CrossRefGoogle Scholar
Faulwasser, T., Flaßkamp, K., Ober-Blöbaum, S. and Worthmann, K. (2021a), A dissipativity characterization of velocity turnpikes in optimal control problems for mechanical systems, IFAC-PapersOnLine 54, 624–629.CrossRefGoogle Scholar
Faulwasser, T., Hempel, A.-J. and Streif, S. (2021b), On the turnpike to design of deep neural nets: Explicit depth bounds. Available at arXiv:2101.03000.Google Scholar
Faulwasser, T., Korda, M., Jones, C. N. and Bonvin, D. (2017), On turnpike and dissipativity properties of continuous-time optimal control problems, Automatica 81, 297–304.CrossRefGoogle Scholar
Fernández-Cara, E., Guerrero, S., Imanuvilov, O. Y. and Puel, J.-P. (2004), Local exact controllability of the Navier–Stokes system, J. Math. Pures Appl. 83, 1501–1542.CrossRefGoogle Scholar
Fleming, W. H. and McEneaney, W. M. (1995), Risk-sensitive control on an infinite time horizon, SIAM J. Control Optim. 33, 1881–1915.CrossRefGoogle Scholar
Fujita, Y., Ishii, H. and Loreti, P. (2006), Asymptotic solutions of Hamilton-Jacobi equations in Euclidean $n$ space, I ndiana Univ. Math. J. 55, 1671–1700.CrossRefGoogle Scholar
Fursikov, A. V. and Imanuvilov, O. (1996), Controllability of Evolution Equations, Vol. 34 of Lecture Notes Series, Research Institute of Mathematics, Global Analysis Research Center, Seoul National University.Google Scholar
Gad-el-Hak, M. (2007), Flow Control: Passive, Active, and Reactive Flow Management, Cambridge University Press.Google Scholar
Garcia, C. E., Prett, D. M. and Morari, M. (1989), Model predictive control: Theory and practice: A survey, Automatica 25, 335–348.CrossRefGoogle Scholar
Geshkovski, B. (2020), Null-controllability of perturbed porous medium gas flow, ESAIM Control Optim. Calc. Var. 26, 85.CrossRefGoogle Scholar
Geshkovski, B. and Zuazua, E. (2021), Optimal actuator design via Brunovsky’s normal form. Available at arXiv:2108.05629.Google Scholar
Ghil, M. and Malanotte-Rizzoli, P. (1991), Data assimilation in meteorology and oceanography, Adv . Geophys. 33, 141–266.Google Scholar
Glowinski, R. and Lions, J.-L. (1995), Exact and approximate controllability for distributed parameter systems, in Acta Numerica, Vol. 4, Cambridge University Press, pp. 159–328.Google Scholar
Grathwohl, W., Chen, R. T. Q., Bettencourt, J., Sutskever, I. and Duvenaud, D. (2019), FFJORD: Free-form continuous dynamics for scalable reversible generative models, in International Conference on Learning Representations (ICLR 2019).Google Scholar
Grüne, L. and Guglielmi, R. (2018), Turnpike properties and strict dissipativity for discrete time linear quadratic optimal control problems, SIAM J. Control Optim. 56, 1282–1302.CrossRefGoogle Scholar
Grüne, L. and Guglielmi, R. (2021), On the relation between turnpike properties and dissipativity for continuous time linear quadratic optimal control problems, Math. Control Relat. Fields 11, 169–188.CrossRefGoogle Scholar
Grüne, L. and Müller, M. A. (2016), On the relation between strict dissipativity and turnpike properties, Systems Control Lett. 90, 45–53.CrossRefGoogle Scholar
Grüne, L. and Pannek, J. (2017), Nonlinear model predictive control, in Nonlinear Model Predictive Control: Theory and Algorithms, Communications and Control Engineering, Springer, pp. 45–69.Google Scholar
Grüne, L., Kellett, C. M. and Weller, S. R. (2017), On the relation between turnpike properties for finite and infinite horizon optimal control problems, J. Optim. Theory Appl. 173, 727–745.CrossRefGoogle Scholar
Grüne, L., Pirkelmann, S. and Stieler, M. (2018), Strict dissipativity implies turnpike behavior for time-varying discrete time optimal control problems, in Control Systems and Mathematical Methods in Economics (Feichtinger, G. et al., eds), Vol. 687 of Lecture Notes in Economics and Mathematical Systems, Springer, pp. 195–218.CrossRefGoogle Scholar
Grüne, L., Schaller, M. and Schiela, A. (2019), Sensitivity analysis of optimal control for a class of parabolic PDEs motivated by model predictive control, SIAM J. Control Optim. 57, 2753–2774.CrossRefGoogle Scholar
Grüne, L., Schaller, M. and Schiela, A. (2020), Exponential sensitivity and turnpike analysis for linear quadratic optimal control of general evolution equations, J. Differential Equations 268, 7311–7341.CrossRefGoogle Scholar
Grüne, L., Schaller, M. and Schiela, A. (2021), Abstract nonlinear sensitivity and turnpike analysis and an application to semilinear parabolic PDEs, ESAIM Control Optim. Calc. Var. 27, 56.CrossRefGoogle Scholar
Grüne, L., Schaller, M. and Schiela, A. (2022), Efficient Model Predictive Control for parabolic PDEs with goal oriented error estimation, SIAM J. Sci. Comput. 44, A471–A500.CrossRefGoogle Scholar
Gueye, M. (2014), Exact boundary controllability of 1-D parabolic and hyperbolic degenerate equations, SIAM J. Control Optim. 52, 2037–2054.CrossRefGoogle Scholar
Gugat, M. (2021), On the turnpike property with interior decay for optimal control problems, Math. Control Signals Systems 33, 237–258.CrossRefGoogle Scholar
Gugat, M. and Hante, F. M. (2019), On the turnpike phenomenon for optimal boundary control problems with hyperbolic systems, SIAM J. Control Optim. 57, 264–289.CrossRefGoogle Scholar
Gugat, M., Schuster, M. and Zuazua, E. (2021), The finite-time turnpike phenomenon for optimal control problems: Stabilization by non-smooth tracking terms, in Stabilization of Distributed Parameter Systems: Design Methods and Applications (Sklyar, G. and Zuyev, A., eds), Vol. 2 of SEMA SIMAI Springer Series, Springer, pp. 17–41.CrossRefGoogle Scholar
Haber, E. and Ruthotto, L. (2017), Stable architectures for deep neural networks, Inverse Problems 34, 014004.CrossRefGoogle Scholar
Han, Z.-J. and Zuazua, E. (2021), Slow decay and turnpike for infinite-horizon hyperbolic LQ problems. Available at arXiv:2108.10240.Google Scholar
Haraux, A. (1989), Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps, Port. Math. 46, 245–258.Google Scholar
Haurie, A. (1976), Optimal control on an infinite time horizon: The turnpike approach, J. Math. Econom. 3, 81–102.CrossRefGoogle Scholar
He, K., Zhang, X., Ren, S. and Sun, J. (2016), Deep residual learning for image recognition, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 770–778.Google Scholar
He, X., Mo, Z., Wang, P., Liu, Y., Yang, M. and Cheng, J. (2019), ODE-inspired network design for single image super-resolution, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp. 1732–1741.Google Scholar
Hébrard, P. and Henrot, A. (2005), A spillover phenomenon in the optimal location of actuators, SIAM J. Control Optim. 44, 349–366.CrossRefGoogle Scholar
Hegoburu, N., Magal, P. and Tucsnak, M. (2018), Controllability with positivity constraints of the Lotka–McKendrick system, SIAM J. Control Optim. 56, 723–750.CrossRefGoogle Scholar
Heiland, J. and Zuazua, E. (2021), Classical system theory revisited for turnpike in standard state space systems and impulse controllable descriptor systems, SIAM J. Control Optim. 59, 3600–3624.CrossRefGoogle Scholar
Hernández-Santamaría, V., Lazar, M. and Zuazua, E. (2019), Greedy optimal control for elliptic problems and its application to turnpike problems, Numer. Math. 141, 455–493.CrossRefGoogle Scholar
Hinze, M., Pinnau, R., Ulbrich, M. and Ulbrich, S. (2008), Optimization with PDE Constraints, Vol. 23 of Mathematical Modelling: Theory and Applications, Springer.Google Scholar
Holmes, P., Lumley, J. L., Berkooz, G. and Rowley, C. W. (2012), Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge University Press.CrossRefGoogle Scholar
Ishii, H. (2006), Asymptotic solutions for large time of Hamilton–Jacobi equations, in Proceedings of the International Congress of Mathematicians 3, European Mathematical Society, pp. 213–227.Google Scholar
Ishii, H. (2008), Asymptotic solutions for large time of Hamilton–Jacobi equations in Euclidean $n$ space, Ann. Inst. H. Poincaré Anal. Non Linéaire 25, 231–266.CrossRefGoogle Scholar
Ito, K. and Kunisch, K. (2008), Lagrange Multiplier Approach to Variational Problems and Applications, SIAM.CrossRefGoogle Scholar
Jameson, A. (1988), Aerodynamic design via control theory, J. Sci. Comput. 3, 233–260.CrossRefGoogle Scholar
Jameson, A. and Ou, K. (2010), Optimization methods in computational fluid dynamics, in Encyclopedia of Aerospace Engineering, Wiley.Google Scholar
Jean, F. and Prandi, D. (2015), Complexity of control-affine motion planning, SIAM J. Control Optim. 53, 816–844.CrossRefGoogle Scholar
Joly, R. and Laurent, C. (2014), A note on the semiglobal controllability of the semilinear wave equation, SIAM J. Control Optim. 52, 439–450.CrossRefGoogle Scholar
Kellett, C. M., Weller, S. R., Faulwasser, T., Grüne, L. and Semmler, W. (2019), Feedback, dynamics, and optimal control in climate economics, Annu. Rev. Control 47, 7–20.CrossRefGoogle Scholar
Kidger, P., Morrill, J., Foster, J. and Lyons, T. (2020), Neural controlled differential equations for irregular time series, in Advances in Neural Information Processing Systems 33 (Larochelle, H. et al., eds), Curran Associates, pp. 6696–6707.Google Scholar
Komornik, V. (1997), Rapid boundary stabilization of linear distributed systems, SIAM J. Control Optim. 35, 1591–1613.CrossRefGoogle Scholar
Krizhevsky, A., Sutskever, I. and Hinton, G. E. (2012), ImageNet classification with deep convolutional neural networks, in Advances in Neural Information Processing Systems 25 (Pereira, F. et al., eds), Curran Associates, pp. 1097–1105.Google Scholar
Lance, G., Trélat, E. and Zuazua, E. (2020), Shape turnpike for linear parabolic PDE models, Systems Control Lett. 142, 104733.CrossRefGoogle Scholar
Lasry, J.-M. and Lions, P.-L. (2007), Mean field games, Japan. J. Math. 2, 229–260.CrossRefGoogle Scholar
Lazar, M. and Zuazua, E. (2016), Greedy controllability of finite dimensional linear systems, Automatica 74, 327–340.CrossRefGoogle Scholar
Le Balc’h, K. (2020), Local controllability of reaction–diffusion systems around nonnegative stationary states, ESAIM Control Optim. Calc. Var. 26, 55.CrossRefGoogle Scholar
Lebeau, G. (1996), Équation des ondes amorties, in Algebraic and Geometric Methods in Mathematical Physics (Kaciveli, 1993) (de Monvel, A. B. and Marchenko, V., eds), Vol. 19 of Mathematical Physics Studies, Kluwer Academic, pp. 73–109.CrossRefGoogle Scholar
Lebeau, G. and Robbiano, L. (1995), Contrôle exact de l’équation de la chaleur, Comm . Partial Differential Equations 20, 335–356.CrossRefGoogle Scholar
LeCun, Y. (1988), A theoretical framework for back-propagation, in Proceedings of the 1988 Connectionist Models Summer School (Touretzky, D. et al., eds), Morgan Kaufmann, pp. 21–28.Google Scholar
LeCun, Y., Cortes, C. and Burges, C. (2010), MNIST handwritten digit database, ATT Labs [Online]. Available at http://yann.lecun.com/exdb/mnist.Google Scholar
Lee, E. B. and Markus, L. (1967), Foundations of Optimal Control Theory, Wiley.Google Scholar
Li, Q., Chen, L., Tai, C. and E, W. (2017), Maximum principle based algorithms for deep learning, J. Mach. Learn. Res. 18, 5998–6026.Google Scholar
Li, Q., Lin, T. and Shen, Z. (2019), Deep learning via dynamical systems: An approximation perspective. Available at arXiv:1912.10382.Google Scholar
Liard, T. and Zuazua, E. (2021), Initial data identification for the one-dimensional Burgers equation, IEEE Trans. Automat. Control. Available at doi:10.1109/TAC.2021.3096921.CrossRefGoogle Scholar
Lions, J.-L. (1971), Optimal Control of Systems Governed by Partial Differential Equations, Vol. 170 of Grundlehren der mathematischen Wissenschaften, Springer.CrossRefGoogle Scholar
Lions, J.-L. (1988a), Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués, Tome 1, Contrôlabilité Exacte, Masson.Google Scholar
Lions, J.-L. (1988b), Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev. 30, 1–68.CrossRefGoogle Scholar
Lissy, P. and Moreau, C. (2021), State-constrained controllability of linear reaction–diffusion systems, ESAIM Control Optim. Calc. Var. 27, 70.CrossRefGoogle Scholar
Liu, H. and Markowich, P. (2020), Selection dynamics for deep neural networks, J. Differential Equations 269, 11540–11574.CrossRefGoogle Scholar
Lohéac, J. and Zuazua, E. (2016), From averaged to simultaneous controllability, Ann. Fac. Sci. Toulouse Math. 25, 785–828.CrossRefGoogle Scholar
Lohéac, J., Trélat, E. and Zuazua, E. (2017), Minimal controllability time for the heat equation under unilateral state or control constraints, Math. Models Methods Appl. Sci. 27, 1587–1644.CrossRefGoogle Scholar
Lou, H. and Wang, W. (2019), Turnpike properties of optimal relaxed control problems, ESAIM Control Optim. Calc. Var. 25, 74.CrossRefGoogle Scholar
Macià, F. (2021), Observability results related to fractional Schrödinger operators, Vietnam J. Math. 49, 919–936.CrossRefGoogle Scholar
Maity, D., Tucsnak, M. and Zuazua, E. (2019), Controllability and positivity constraints in population dynamics with age structuring and diffusion, J. Math. Pures Appl. 129, 153–179.CrossRefGoogle Scholar
Mazari, I. and Ruiz-Balet, D. (2020), Quantitative stability for eigenvalues of Schrödinger operator, quantitative bathtub principle & application to the turnpike property for a bilinear optimal control problem. Available at arXiv:2010.10798.Google Scholar
Mazari, I., Ruiz-Balet, D. and Zuazua, E. (2020), Constrained control of gene-flow models. Available at arXiv:2005.09236.Google Scholar
McKenzie, L. W. (1976), Turnpike theory, Econometrica 44, 841–865.CrossRefGoogle Scholar
Mohammadi, B. and Pironneau, O. (2010), Applied Shape Optimization for Fluids, Oxford University Press.Google Scholar
Münch, A. and Zuazua, E. (2010), Numerical approximation of null controls for the heat equation: Ill-posedness and remedies, Inverse Problems 26, 085018.CrossRefGoogle Scholar
Nordhaus, W. D. (1992), An optimal transition path for controlling greenhouse gases, Science 258, 1315–1319.CrossRefGoogle ScholarPubMed
Papamakarios, G., Nalisnick, E., Rezende, D. J., Mohamed, S. and Lakshminarayanan, B. (2021), Normalizing flows for probabilistic modeling and inference, J. Mach. Learn. Res. 22, 1–64.Google Scholar
Pighin, D. (2020), Nonuniqueness of minimizers for semilinear optimal control problems. Available at arXiv:2002.04485.Google Scholar
Pighin, D. (2021), The turnpike property in semilinear control, ESAIM Control Optim. Calc. Var. 27, 48.CrossRefGoogle Scholar
Pighin, D. and Sakamoto, N. (2020), The turnpike with lack of observability. Available at arXiv:2007.14081.Google Scholar
Pighin, D. and Zuazua, E. (2018), Controllability under positivity constraints of semilinear heat equations, Math. Control Relat. Fields 8, 935–964.CrossRefGoogle Scholar
Pighin, D. and Zuazua, E. (2019), Controllability under positivity constraints of multi-d wave equations, in Trends in Control Theory and Partial Differential Equations (Alabau-Boussouira, F. et al., eds), Vol. 32 of Springer INdAM Series, Springer, pp. 195–232.CrossRefGoogle Scholar
Pinkus, A. (1999), Approximation theory of the MLP model in neural networks, in Acta Numerica, Vol. 8, Cambridge University Press, pp. 143–195.Google Scholar
Porretta, A. and Zuazua, E. (2013), Long time versus steady state optimal control, SIAM J. Control Optim. 51, 4242–4273.CrossRefGoogle Scholar
Porretta, A. and Zuazua, E. (2016), Remarks on long time versus steady state optimal control, in Mathematical Paradigms of Climate Science (Ancona, F. et al., eds), Vol. 15 of Springer INdAM Series, Springer, pp. 67–89.CrossRefGoogle Scholar
Pouchol, C., Trélat, E. and Zuazua, E. (2019), Phase portrait control for 1D monostable and bistable reaction–diffusion equations, Nonlinearity 32, 884–909.CrossRefGoogle Scholar
Prandi, D. (2014), Hölder equivalence of the value function for control-affine systems, ESAIM Control Optim. Calc. Var. 20, 1224–1248.CrossRefGoogle Scholar
Privat, Y., Trélat, E. and Zuazua, E. (2015), Optimal shape and location of sensors for parabolic equations with random initial data, Arch. Ration. Mech. Anal. 216, 921–981.CrossRefGoogle Scholar
Privat, Y., Trélat, E. and Zuazua, E. (2016), Optimal observability of the multi-dimensional wave and Schrödinger equations in quantum ergodic domains, J. European Math. Soc. 18, 1043–1111.CrossRefGoogle Scholar
Quincampoix, M. and Renault, J. (2011), On the existence of a limit value in some nonexpansive optimal control problems, SIAM J. Control Optim. 49, 2118–2132.CrossRefGoogle Scholar
Ramsey, F. P. (1928), A mathematical theory of saving, Econom. J. 38, 543–559.Google Scholar
Rapaport, A. and Cartigny, P. (2004), Turnpike theorems by a value function approach, ESAIM Control Optim. Calc. Var. 10, 123–141.CrossRefGoogle Scholar
Recht, B. (2019), A tour of reinforcement learning: The view from continuous control, Annu. Rev. Control Robot. Auton. Syst. 2, 253–279.CrossRefGoogle Scholar
Renault, J. and Venel, X. (2017), Long-term values in Markov decision processes and repeated games, and a new distance for probability spaces, Math. Oper. Res. 42, 349–376.CrossRefGoogle Scholar
Rosset, S., Zhu, J. and Hastie, T. (2004), Boosting as a regularized path to a maximum margin classifier, J. Mach. Learn. Res. 5, 941–973.Google Scholar
Ruiz-Balet, D., Affili, E. and Zuazua, E. (2021), Interpolation and approximation via momentum ResNets and neural ODEs. Available at arXiv:2110.08761.Google Scholar
Ruiz-Balet, D. and Zuazua, E. (2020), Control under constraints for multi-dimensional reaction–diffusion monostable and bistable equations, J. Math. Pures Appl. 143, 345–375.CrossRefGoogle Scholar
Ruiz-Balet, D. and Zuazua, E. (2021), Neural ODE control for classification, approximation and transport. Available at arXiv:2104.05278.Google Scholar
Sakamoto, N., Pighin, D. and Zuazua, E. (2019), The turnpike property in nonlinear optimal control: A geometric approach, in 2019 IEEE 58th Conference on Decision and Control (CDC), IEEE, pp. 2422–2427.CrossRefGoogle Scholar
Samuelson, P. A. (1965), A catenary turnpike theorem involving consumption and the golden rule, Amer. Econom. Rev. 55, 486–496.Google Scholar
Samuelson, P. A. (1976), The periodic turnpike theorem, Nonlinear Analysis: Theory, Methods & Applications 1, 3–13.CrossRefGoogle Scholar
Sander, M. E., Ablin, P., Blondel, M. and Peyré, G. (2021), Momentum residual neural networks, in Proceedings of the 38th International Conference on Machine Learning, Vol. 139 of Proceedings of Machine Learning Research, PMLR, pp. 9276–9287.Google Scholar
Seidman, T. I. (1987), Invariance of the reachable set under nonlinear perturbations, SIAM J. Control Optim. 25, 1173–1191.CrossRefGoogle Scholar
Sontag, E. D. and Qiao, Y. (1999), Further results on controllability of recurrent neural networks, Systems Control Lett. 36, 121–129.CrossRefGoogle Scholar
Sontag, E. D. and Sussmann, H. (1997), Complete controllability of continuous-time recurrent neural networks, Systems Control Lett. 30, 177–183.CrossRefGoogle Scholar
Stuart, A. M. (2010), Inverse problems: A Bayesian perspective, in Acta Numerica, Vol. 19, Cambridge University Press, pp. 451–559.Google Scholar
Trélat, E. (2005), Contrôle Optimal: Théorie & Applications, Mathématiques Concrètes, Vuibert, Paris.Google Scholar
Trélat, E. (2012), Optimal control and applications to aerospace: Some results and challenges, J. Optim. Theory Appl. 154, 713–758.CrossRefGoogle Scholar
Trélat, E. (2020), Linear turnpike theorem. Available at arXiv:2010.13605.Google Scholar
Trélat, E. and Zhang, C. (2018), Integral and measure-turnpike properties for infinite-dimensional optimal control systems, Math. Control Signals Systems 30, 1–34.CrossRefGoogle Scholar
Trélat, E. and Zuazua, E. (2015), The turnpike property in finite-dimensional nonlinear optimal control, J. Differential Equations 258, 81–114.CrossRefGoogle Scholar
Trélat, E., Wang, G. and Xu, Y. (2019), Characterization by observability inequalities of controllability and stabilization properties, Pure Appl. Anal. 2, 93–122.CrossRefGoogle Scholar
Trélat, E., Zhang, C. and Zuazua, E. (2018a), Optimal shape design for 2D heat equations in large time, Pure Appl. Functional Anal. 3, 255–269.Google Scholar
Trélat, E., Zhang, C. and Zuazua, E. (2018b), Steady-state and periodic exponential turnpike property for optimal control problems in Hilbert spaces, SIAM J. Control Optim. 56, 1222–1252.CrossRefGoogle Scholar
Tröltzsch, F. (2010), Optimal Control of Partial Differential Equations: Theory, Methods, and Applications, Vol. 112 of Graduate Studies in Mathematics, American Mathematical Society.Google Scholar
Tucsnak, M. and Weiss, G. (2000), Simultaneous exact controllability and some applications, SIAM J. Control Optim. 38, 1408–1427.CrossRefGoogle Scholar
Tucsnak, M. and Weiss, G. (2009), Observation and Control for Operator Semigroups, Springer.CrossRefGoogle Scholar
Valein, J. and Zuazua, E. (2009), Stabilization of the wave equation on 1-D networks, SIAM J. Control Optim. 48, 2771–2797.CrossRefGoogle Scholar
Vinter, R. B. (2010), Optimal Control, Springer.CrossRefGoogle Scholar
von Neumann, J. (1937), Über ein ökonomisches Gleichungssystem und eine Verallgemeinerung des brouwerschen Fixpunktsatzes, Erge . Math. Kolloq. 8, 73–83.Google Scholar
Von Stryk, O. and Bulirsch, R. (1992), Direct and indirect methods for trajectory optimization, Ann. Oper. Res. 37, 357–373.CrossRefGoogle Scholar
Warma, M. and Zamorano, S. (2021), Exponential turnpike property for fractional parabolic equations with non-zero exterior data, ESAIM Control Optim. Calc. Var. 27, 1.CrossRefGoogle Scholar
Wiener, N. (1949), Cybernetics or Control and Communication in the Animal and the Machine, MIT Press.Google Scholar
Willems, J. C. (1972), Dissipative dynamical systems, part I: General theory, Arch . Ration. Mech. Anal. 45, 321–351.CrossRefGoogle Scholar
Zabczyk, J. (2020), Mathematical Control Theory, Springer.CrossRefGoogle Scholar
Zamorano, S. (2018), Turnpike property for two-dimensional Navier–Stokes equations, J. Math. Fluid Mech. 20, 869–888.CrossRefGoogle Scholar
Zanon, M., Grüne, L. and Diehl, M. (2016), Periodic optimal control, dissipativity and MPC, IEEE Trans. Automat. Control 62, 2943–2949.CrossRefGoogle Scholar
Zaslavski, A. J. (2005), Turnpike Properties in the Calculus of Variations and Optimal Control, Vol. 80 of Nonconvex Optimization and its Applications, Springer.Google Scholar
Zaslavski, A. J. (2007), Turnpike results for discrete-time optimal control systems arising in economic dynamics, Nonlinear Anal. Theory Methods Appl. 67, 2024–2049.CrossRefGoogle Scholar
Zaslavski, A. J. (2015), Turnpike Theory of Continuous-Time Linear Optimal Control Problems, Vol. 104 of Springer Optimization and its Applications, Springer.CrossRefGoogle Scholar
Zhang, C., Bengio, S., Hardt, M., Recht, B. and Vinyals, O. (2021), Understanding deep learning (still) requires rethinking generalization, Commun. Assoc. Comput. Mach. 64, 107–115.Google Scholar
Zhang, X. (2000), Exact controllability of semilinear evolution systems and its application, J. Optim. Theory Appl. 107, 415–432.CrossRefGoogle Scholar
Zhang, X. and Zuazua, E. (2004), Problem 5.5: Exact controllability of the semilinear wave equation, in Unsolved Problems in Mathematical Systems and Control Theory (Blondel, V. D. and Megretski, A., eds), Princeton Unversity Press.Google Scholar
Zuazua, E. (1993), Exact controllability for semilinear wave equations in one space dimension, Ann. Inst. H. Poincaré Anal. Non Linéaire 10, 109–129.CrossRefGoogle Scholar
Zuazua, E. (2005), Propagation, observation, and control of waves approximated by finite difference methods, SIAM Rev. 47, 197–243.CrossRefGoogle Scholar
Zuazua, E. (2017), Large time control and turnpike properties for wave equations, Annu. Rev. Control 44, 199–210.CrossRefGoogle Scholar