Infinite Horizon Optimal Control Problems in the Light of Convex Analysis in Hilbert Spaces

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• 作者：Sabine Pickenhain (1)
  
• 关键词：Infinite horizon ; Optimal control ; Pontryagin鈥檚 Maximum Principle ; Existence theorem ; Weighted sobolev spaces ; Linear ; quadratic regulator ; 46E35 ; 49K15 ; 49N10
• 刊名：Set-Valued and Variational Analysis
• 出版年：2015
• 出版时间：March 2015
• 年：2015
• 卷：23
• 期：1
• 页码：169-189
• 全文大小：399 KB
• 参考文献：1. Aseev, S.M., Kryazhimskii, A.V.: The Pontryagin Maximum Principle and optimal economic growth problems. Proc. Steklov Inst. Math. 257, 1鈥?55
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  5. Carlson, D.A., Haurie, A.B.: Infinite Horizon Optimal Control. Springer-Verlag, New York, Berlin, Heidelberg (1991) CrossRef
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  11. Ioffe, A.D., Tichomirow, V.M.: Theorie der Extremalaufgaben. VEB Deutscher Verlag der Wissenschaften, Berlin (1979)
  12. Kalman, R.E.: Contribution to the theory of optimal control. Bol. Soc. Matem. Mex, 5 (1960)
  13. Kufner, A.: Weighted Sobolev Spaces. John Wiley & Sons, Chichester, etc (1985)
  14. Letov, A.M.: Analytic controller design I, II. Autom. Remote Contr. 21, 303鈥?06 (1960)
  15. Lykina, V. Beitr盲ge zur Theorie der Optimalsteuerungsprobleme mit unendlichem Zeithorizont, Dissertation. BTU Cottbus, http://opus.kobv.de/btu/volltexte/2010/1861/pdf/dissertationLykina.pdf (2010)
  16. Lykina, V., Pickenhain, S., Wagner, M.: On a resource allocation model with infinite horizon, vol. 204, pp 595鈥?01, Appl. Math. Comput. (2008)
  17. Pickenhain, S., Lykina, V.: Sufficiency conditions for infinite horizon optimal control problems. In Recent Advances in Optimization. In: Seeger, A. (ed.) Lecture Notes in Economics and Mathematical Systems, vol. 563, pp 217鈥?32. Springer, Berlin, etc (2006)
  18. Pickenhain, S.: On adequate transversality conditions for infinite horizon optimal control problems 鈥?a famous example of Halkin. In: Crespo Cuaresma, J., Palokangas, T., Tarasyev, A (eds.) Dynamic Systems, Economic Growth, and the Environment. Dynamic Modeling and Econometrics in Economics and Finance, vol. 12, pp 3鈥?2. Springer, Berlin etc (2010)
  19. Pickenhain, S. Hilbert Space Treatment of Optimal Control Problems with Infinite Horizon. Preprint Reihe Mathematik M-01/2012, BTU Cottbus. (accepted in Modelling, Simulation and Optimization of Complex Processes, Springer). http://www.math.tu-cottbus.de/INSTITUT/lsopt/publication/preprint/pickenh/M_01_2012.pdf (2012)
  20. Yosida, K.: Functional Analysis. Springer-Verlag, New York (1974) CrossRef
  
• 作者单位：Sabine Pickenhain (1)
  
  1. Brandenburg University of Technology, Cottbus-Senftenberg, Germany
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  Geometry
  
• 出版者：Springer Netherlands
• ISSN：1877-0541

文摘

In this paper a class of linear-quadratic infinite horizon optimal control problems is considered. Problems of this type are not only of practical interest. They also appear as an approximation of nonlinear problems. The key idea is to introduce weighted Sobolev spaces as state space and weighted Lebesgue spaces as control spaces into the problem setting. We investigate the question of existence of an optimal solution in these spaces and establish a Pontryagin type Maximum Principle as a necessary optimality condition including transversality conditions.