Abstract
Control problems play a key role in many fields of Engineering, Economics and Sciences. This applies, in particular, to climate sciences where, often times, relevant problems are formulated in long time scales. The problem of the possible asymptotic simplification (as time tends to infinity) then emerges naturally. More precisely, assuming, for instance, that the free dynamics under consideration stabilizes towards a steady state solution, the following question arises: Do time averages of optimal controls and trajectories converge to the steady optimal controls and states as the time-horizon tends to infinity?This question is very closely related to the so-called turnpike property stating that, often times, the optimal trajectory joining two points that are far apart, consists in, departing from the point of origin, rapidly getting close to the steady-state (the turnpike) to stay there most of the time, to quit it only very close to the final destination and time.In this paper we focus on the semilinear heat equation. We prove some partial results and enumerate a number of interesting topics of future research, indicating also some connections with shape design and inverse problems theory.
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References
Allaire, G., Münch, A., Periago, F.: Long time behavior of a two-phase optimal design for the heat equation. SIAM J. Control. Optim. 48, 5333–5356 (2010)
Anderson, B.D.O., Kokotovic, P.V.: Optimal control problems over large time intervals. Autom. J. IFAC 23, 355–363 (1987)
Cardaliaguet, P., Lasry, J-M., Lions, P.-L., Porretta, A.: Long time average of mean field games. Netw. Heterog. Media 7, 279–301 (2012)
Cardaliaguet, P., Lasry, J-M., Lions, P.-L., Porretta, A.: Long time average of mean field games in case of nonlocal coupling. SIAM J. Control. Optim. 51, 3558–3591 (2013)
Casas, E., Mateos, M.: Optimal Control for Partial Differential Equations. Proccedings of Escuela Hispano Francesa 2016. Oviedo, Spain (to appear)
Casas, E., Tröltzsch, F.: Second order analysis for optimal control problems: improving results expected from abstract theory. SIAM J. Optim. 22, 261–279 (2012)
Chichilnisky, G.: What is Sustainable Development? Man-Made Climate Change, pp. 42–82. Physica-Verlag HD, Heidelberg/New York (1999)
Choulli, M.: Une introduction aux problèmes inverses elliptiques et paraboliques. Mathematiques & Applications, vol. 65. Springer, Berlin (2009)
Damm, T., Grüne, L., Stielerz, M., Worthmann, K.: An exponential turnpike theorem for dissipative discrete time optimal control problems. SIAM J. Control. Optim. 52, 1935–1957 (2014)
Grüne, L.: Economic receding horizon control without terminal. Autom. J. IFAC 49, 725–734 (2013)
Isakov, V.: Inverse Problems for Partial Differential Equations, 2nd edn. Applied Mathematical Sciences, vol. 127. Springer, New York (2006)
Jameson, A.: Optimization methods in computational fluid dynamics (with Ou, K.). In: Blockley, R., Shyy, W. (eds.) Encyclopedia of Aerospace Engineering. John Wiley & Sons, Hoboken (2010)
Jendoubi, M.A.: A simple unified approach to some convergence theorems of L. Simon. J. Funct. Anal. 153, 187–202 (1998)
Nodet, M., Bonan, B., Ozenda O., Ritz, C.: Data Assimilation in Glaciology. Advanced Data Assimilation for Geosciences. Les Houches, France (2012)
Porretta A., Zuazua, E.: Long time versus steady state optimal control. SIAM J. Control. Optim. 51, 4242–4273 (2013)
Privat, Y., Trélat, E., Zuazua, E.: Optimal location of controllers for the one-dimensional wave equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 30, 1097–1126 (2013)
Privat, Y., Trélat, E., Zuazua, E.: Optimal observation of the one-dimensional wave equation. J. Fourier Anal. Appl. 19, 514–544 (2013)
Privat, Y., Trélat, E., Zuazua, E.: Complexity and regularity of maximal energy domains for the wave equation with fixed initial data. Discret. Cont. Dyn. Syst. 35, 6133–6153 (2015)
Privat, Y., Trélat, E., Zuazua, E.: Optimal shape and location of sensors and controllers for parabolic equations with random initial data. Arch. Ration. Mech. Anal. 216, 921–981 (2015)
Trélat, E., Zuazua, E.: The turnpike property in finite-dimensional nonlinear optimal control. J. Differ. Equ. 258, 81–114 (2015)
Tröltzsch, F.: Optimal Control of Partial Differential Equations. Theory, Methods and Applications. Graduate Studies in Mathematics, vol. 112. American Mathematical Society, Providence (2010)
Zaslavski, A.J.: Turnpike properties in the calculus of variations and optimal control. Nonconvex Optimization and its Applications, vol. 80. Springer, New York (2006)
Zuazua, E.: Propagation, observation, and control of waves approximated by finite difference methods. SIAM Rev. 47 (2), 197–243 (2005)
Zuazua, E.: Controllability and observability of partial differential equations: some results and open problems. In: Dafermos, C.M., Feireisl E. (eds.) Handbook of Differential Equations: Evolutionary Equations, vol. 3, pp. 527–621. Elsevier Science, Amsterdam/Boston (2006)
Acknowledgements
Enrique Zuazua was partially supported by the Advanced Grant
NUMERIWAVES/FP7-246775 of the European Research Council Executive Agency, the FA9550-14-1-0214 of the EOARD-AFOSR, FA9550-15-1-0027 of AFOSR, the MTM2011-29306 and MTM2014-52347 Grants of the MINECO, and a Humboldt Award at the University of Erlangen-Nürnberg. This work was done while the second author was visiting the Laboratoire Jacques Louis Lions with the support of the Paris City Hall “Research in Paris” program.
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Porretta, A., Zuazua, E. (2016). Remarks on Long Time Versus Steady State Optimal Control. In: Ancona, F., Cannarsa, P., Jones, C., Portaluri, A. (eds) Mathematical Paradigms of Climate Science. Springer INdAM Series, vol 15. Springer, Cham. https://doi.org/10.1007/978-3-319-39092-5_5
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