Free Access
Issue
ESAIM: COCV
Volume 27, 2021
Article Number 48
Number of page(s) 48
DOI https://doi.org/10.1051/cocv/2021036
Published online 04 June 2021
  1. G. Allaire, A. Münch and F. Periago, Long time behavior of a two-phase optimal design for the heat equation. SIAM J. Control Optim. 48 (2010) 5333–5356. [Google Scholar]
  2. B.D. Anderson and P.V. Kokotovic, Optimal control problems over large time intervals. Automatica 23 (1987) 355–363. [Google Scholar]
  3. S. Aniţa and D. Tataru, Null controllability for the dissipative semilinear heat equation. Appl. Math. Optim. 46 (2002) 97–105. [Google Scholar]
  4. V. Barbu, Nonlinear differential equations of monotone types in Banach spaces. Springer Science & Business Media (2010). [Google Scholar]
  5. L. Boccardo and G. Croce, Elliptic partial differential equations: existence and regularity of distributional solutions. De Gruyter Studies in Mathematics, De Gruyter (2013). [Google Scholar]
  6. H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext, Springer New York (2010). [Google Scholar]
  7. P. Cannarsa and C. Sinestrari, Vvol. 58 of Semiconcave functions, Hamilton-Jacobi equations, and optimal control. Springer Science & Business Media (2004). [Google Scholar]
  8. P. Cardaliaguet, J.-M. Lasry, P.-L. Lions and A. Porretta, Long time average of mean field games., Netw. Heterogen. Media 7 (2012). [Google Scholar]
  9. P. Cardaliaguet, J.-M. Lasry, P.-L. Lions and A. Porretta, Long time average of mean field games with a nonlocal coupling. SIAM J. Control Optim. 51 (2013) 3558–3591. [Google Scholar]
  10. D. Carlson, A. Haurie and A. Leizarowitz, Infinite Horizon Optimal Control: Deterministic and Stochastic Systems. Springer Berlin Heidelberg (2012). [Google Scholar]
  11. E. Casas, L.A. Fernandez and J. Yong, Optimal control of quasilinear parabolic equations. Proc. Roy. Soc. Edinburgh Sect. A: Math. 125 (1995) 545–565. [Google Scholar]
  12. E. Casas and M. Mateos, Optimal Control of Partial Differential Equations, Springer International Publishing, Cham (2017) pp. 3–59. [Google Scholar]
  13. E. Casas and F. Tröltzsch, Second order analysis for optimal control problems: improving results expected from abstract theory. SIAM J. Optim. 22 (2012) 261–279. [Google Scholar]
  14. T. Damm, L. Grüne, M. Stieler and K. Worthmann, An exponential turnpike theorem for dissipative discrete time optimal control problems. SIAM J. Control Optim. 52 (2014) 1935–1957. [Google Scholar]
  15. R. Dorfman, P. Samuelson and R. Solow, Linear Programming and Economic Analysis. Dover Books on Advanced Mathematics, Dover Publications (1958). [Google Scholar]
  16. J. Droniou, Intégration et Espaces de Sobolev à Valeurs Vectorielles., Université de Provence.. [Google Scholar]
  17. C. Esteve, D. Pighin, H. Kouhkouh and E. Zuazua, The turnpike property and the long-time behavior of the Hamilton-Jacobi equation.. [Google Scholar]
  18. L.C. Evans, Partial differential equations. Vol. 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, second ed. (2010). [Google Scholar]
  19. T. Faulwasser, K. Flaßamp, S. Ober-Blöbaum and K. Worthmann, Towards velocity turnpikes in optimal control of mechanical systems. In Proc. of 11th IFAC Symposium on Nonlinear Control Systems, NOLCOS 2019. IFAC-PapersOnLine 52 (2019) 490–495. [Google Scholar]
  20. E. Fernández-Cara and E. Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations. Ann. l’Institut Henri Poincaré (C) Non Linear Analysis 17 (2000) 583–616. [Google Scholar]
  21. L. Grüne and R. Guglielmi, Turnpike properties and strict dissipativity for discrete time linear quadratic optimal control problems. SIAM J. Control Optim. 56 (2018) 1282–1302. [Google Scholar]
  22. L. Grüne and M.A. Müller, On the relation between strict dissipativity and turnpike properties. Syst. Control Lett. 90 (2016) 45–53. [Google Scholar]
  23. L. Grüne, S. Pirkelmann and M. Stieler, Strict dissipativity implies turnpike behavior for time-varying discrete time optimal control problems. in Control Systems and Mathematical Methods in Economics: Essays in Honor of Vladimir M. Veliov. Springer (2018) 195–218. [Google Scholar]
  24. L. Grüne, M. Schaller and A. Schiela, Sensitivity analysis of optimal control for a class of parabolic PDEs motivated by model predictive control. SIAM J. Control Optim. 57 (2019) 2753–2774. [Google Scholar]
  25. L. Grüne, M. Schaller and A. Schiela, Exponential sensitivity and turnpike analysis for linear quadratic optimal control of general evolution equations. J. Differ. Equ. (2019). [Google Scholar]
  26. A. Haurie, Optimal control on an infinite time horizon: the turnpike approach. J. Math. Econ. 3 (1976) 81–102. [Google Scholar]
  27. V. Hernández-Santamaría, M. Lazar and E. Zuazua, Greedy optimal control for elliptic problems and its application to turnpike problems. Numer. Math. 141 (2019) 455–493. [Google Scholar]
  28. A. Ibañez, Optimal control of the Lotka–Volterra system: turnpike property and numerical simulations. J. Biol. Dyn. 11 (2017) 25–41. [Google Scholar]
  29. H. Kouhkouh, E. Zuazua, P. Carpentier and F. Santambrogio, Dynamic programming interpretation of turnpike and Hamilton-Jacobi-bellman equation (2018). Available online: http://bit.ly/2R7soRx. [Google Scholar]
  30. O.A. Ladyženskaja, V.A. Solonnikov and N.N. Ural’ceva, Linear and quasilinear equations of parabolic type. Translated from the Russian by S. Smith. Vol. 23 of Translations of Mathematical Monographs. American Mathematical Society, Providence, R.I. (1968). [Google Scholar]
  31. G. Lieberman, Second Order Parabolic Differential Equations. World Scientific (1996). [Google Scholar]
  32. N. Liviatan and P.A. Samuelson, Notes on turnpikes: stable and unstable. J. Econ. Theory 1 (1969) 454–475. [Google Scholar]
  33. L.W. McKenzie, Turnpike theorems for a generalized Leontief model. Econometrica (1963) 165–180. [Google Scholar]
  34. L.W. McKenzie, Turnpike theory. Econometrica (1976) 841–865. [Google Scholar]
  35. S. Mitter and J. Lions, Optimal Control of Systems Governed by Partial Differential Equations. Grundlehren der mathematischen Wissenschaften, Springer Berlin Heidelberg (1971). [Google Scholar]
  36. D. Pighin, Nonuniqueness of minimizers for semilinear optimal control problems. arXiv preprint arXiv:2002.04485 (2020). [Google Scholar]
  37. D. Pighin and N. Sakamoto, The turnpike with lack of observability. arXiv preprint arXiv:2007.14081 (2020). [Google Scholar]
  38. A. Porretta, On the turnpike property for mean field games. Min. Theory Appl. 3 (2018) 285–312. [Google Scholar]
  39. A. Porretta and E. Zuazua, Long time versus steady state optimal control. SIAM J. Control Optim. 51 (2013) 4242–4273. [Google Scholar]
  40. A. Porretta and E. Zuazua, Remarks on long time versus steady state optimal control. in Mathematical Paradigms of Climate Science. Springer (2016) 67–89. [Google Scholar]
  41. M. Protter and H. Weinberger, Maximum Principles in Differential Equations. Springer, New York (2012). [Google Scholar]
  42. A. Rapaport and P. Cartigny, Turnpike theorems by a value function approach. ESAIM: COCV 10 (2004) 123–141. [EDP Sciences] [Google Scholar]
  43. A. Rapaport and P. Cartigny, Competition between most rapid approach paths: necessary and sufficient conditions. J. Optim. Theory Appl. 124 (2005) 1–27. [Google Scholar]
  44. J.P. Raymond and H. Zidani, Hamiltonian Pontryagin’s principles for control problems governed by semilinear parabolic equations. Appl. Math. Optim. 39 (1999) 143–177. [Google Scholar]
  45. R. Rockafellar, Saddle points of Hamiltonian systems in convex problems of Lagrange. J. Optim. Theory Appl. 12 (1973) 367–390. [Google Scholar]
  46. H. Royden and P. Fitzpatrick, Real Analysis (Classic Version). Math Classics, Pearson Education (2017). [Google Scholar]
  47. P.A. Samuelson, The general saddlepoint property of optimal-control motions. J. Econ. Theory 5 (1972) 102–120. [Google Scholar]
  48. J. Simon, Compact sets in the space Lp(0, T; B). Ann. Math. Pura ed Appl. (1986). [Google Scholar]
  49. E. Trélat and C. Zhang, Integral and measure-turnpike properties for infinite-dimensional optimal control systems. Math. Control Signals Syst. 30 (2018) 3. [Google Scholar]
  50. E. Trélat, C. Zhang and E. Zuazua, Steady-state and periodic exponential turnpike property for optimal control problems in Hilbert spaces. SIAM J. Control Optim. 56 (2018) 1222–1252. [Google Scholar]
  51. E. Trélat and E. Zuazua, The turnpike property in finite-dimensional nonlinear optimal control. J. Differ. Equ. 258 (2015) 81–114. [Google Scholar]
  52. F. Tröltzsch, Optimal Control of Partial Differential Equations: Theory, Methods, and Applications, Graduate studies in mathematics. [Google Scholar]
  53. J. Von Neumann A model of general economic equilibrium’, review of economic studies, xiii, 1-9 (translation ofueber ein oekonomisches gleichungssystem und eine verallgemeinerung des brouwerschen fixpunksatzes’, ergebnisse eines mathematischen kolloquiums, 1937, 8, 73-83). Rev. Econ. Stud. 67 (1945) 76–84. [Google Scholar]
  54. R. Wilde and P. Kokotovic, A dichotomy in linear control theory. IEEE Trans. Autom. Control 17 (1972) 382–383. [Google Scholar]
  55. Z. Wu, J. Yin and C. Wang, Elliptic & Parabolic Equations. World Scientific (2006). [Google Scholar]
  56. S. Zamorano, Turnpike property for two-dimensional Navier-Stokes equations. J. Math. Fluid Mech. 20 (2018) 869–888. [Google Scholar]
  57. A.J. Zaslavski, Vol. 80 of Turnpike properties in the calculus of variations and optimal control. Springer Science & Business Media (2006). [Google Scholar]
  58. E. Zuazua, Large time control and turnpike properties for wave equations. Annu. Rev. Control (2017). [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.