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Infinite horizon problems in the calculus of variations. The role of transformations with an application to the brachistochrone problem. (English) Zbl 1419.49018
Summary: In this paper we consider a class of infinite horizon variational problems resulting from a transformation of singular variational problems. Herein we assume that the objective is convex. The problem setting implies a weighted Sobolev space as state space. For this class of problems we establish necessary optimality conditions in form of a Pontryagin type maximum principle. A duality concept of convex analysis is provided and used to establish sufficient optimality conditions. We apply the theoretical results proven to the problem of the brachistochrone.

49J40 Variational inequalities
49K15 Optimality conditions for problems involving ordinary differential equations
46N10 Applications of functional analysis in optimization, convex analysis, mathematical programming, economics
Full Text: DOI
[1] Aseev, SM; Kryazhimskii, AV, The Pontryagin Maximum Principle and optimal economic growth problems, Proc. Steklov. Inst. Math., 257, 1-255, (2007) · Zbl 1215.49001
[2] Aseev, SM; Veliov, VM, Maximum principle for problems with dominating discount, Dyn. Contin. Discret. Impuls. Syst. Ser. B, 19, 43-63, (2012) · Zbl 1266.49003
[3] Aubin, JP; Clarke, FH, Shadow prices and duality for a class of optimal control problems, SIAM J. Conrol Optim., 17, 567-586, (1979) · Zbl 0439.49018
[4] Balder, EJ, An existence result for optimal economic growth problems, J. Math. Anal. Appl., 95, 195-213, (1983) · Zbl 0517.49002
[5] Balder, E.J.: The Brachistochrone problem made elementary. http://www.math.uu.nl/people/balder/ (2002)
[6] Carlson, D.A., Haurie, A.B., Leizarowitz, A.: Infinite Horizon Optimal Control. Springer, New York (1991) · Zbl 0758.49001
[7] Clarke, F.: Functional Analysis, Calculus of Variations and Optimal Control. Springer, New York (2013) · Zbl 1277.49001
[8] Coleman, R.: A Detailed Analysis of the Brachistochrone Problem. arXiv:1001.2181v2[math.OC] (2012)
[9] Dunford, N., Schwartz, J.T.: Linear Operators. Part I: General Theory. Wiley-Interscience, New York (1988) · Zbl 0635.47001
[10] Elstrodt, J.: Maß und Integrationstheorie. Springer, Berlin (1996) · Zbl 0861.28001
[11] Grass, D., Caulkins, J.P., Feichtinger, G., Tragler, G., Behrens, D.A.: Optimal Control of Nonlinear Processes. Springer, Berlin (2008) · Zbl 1149.49001
[12] Feichtinger, G., Hartl, R.F.: Optimale Kontrolle ökonomischer Prozesse. de Gruyter, Berlin (1986) · Zbl 0612.90001
[13] Garg, D.; Hager, WW; Rao, AV, Pseudospectral methods for solving infinite-horizon optimal control problems, Automatica, 47, 829-837, (2011) · Zbl 1215.49040
[14] Halkin, H., Necessary conditions for optimal control problems with infinite horizons, Econometrica, 42, 267-272, (1979) · Zbl 0301.90009
[15] Goldstine, H.H.: A History of the Calculus of Variations. Springer, New York (1980) · Zbl 0452.49002
[16] Hall, AC: The Analysis and Synthesis of Linear Servomechanism. The Technology Press, M.I.T., Cambridge (1943)
[17] Ioffe, A.D., Tichomirow, V.M.: Theorie der Extremalaufgaben. VEB Deutscher Verlag der Wissenschaften, Berlin (1979)
[18] Ito, K.; Kunisch, K., Receding horizon optimal control for infinite dimensional systems, ESAIM: Control Optim. Calc. Var., 8, 741-760, (2010) · Zbl 1066.49020
[19] Kalman, RE, Contribution to the theory of optimal control, Bol. Soc. Matem. Mex., 5, 102-119, (1960)
[20] Kosmol, P., Bemerkungen zur Brachistochrone, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 54, 91-94, (1984) · Zbl 0585.49001
[21] Kufner, A.: Weighted Sobolev Spaces. Wiley, Chichester (1985) · Zbl 0567.46009
[22] Lykina, V., An existence theorem for a class of infinite horizon optimal control problems, J. Optim. Theory Appl., 69, 50-73, (2016) · Zbl 1343.49002
[23] Lykina, V.; Pickenhain, S.; Wagner, M., Different interpretations of the improper integral objective in an infinite horizon control problem, Math. Anal. Appl., 340, 498-510, (2008) · Zbl 1141.49003
[24] Magill, MJP, Pricing infinite horizon programs, J. Math. Anal. Appl., 88, 398-421, (1982) · Zbl 0492.90011
[25] Michel, P., On the transversality condition in infinite horizon optimal problems, Econometrica, 50, 975-985, (1982) · Zbl 0483.90026
[26] Pickenhain, S.: On adequate transversality conditions for infinite horizon optimal control problems - a famous example of Halkin. In: Crespo Cuaresma, J., Palokangas, T., Tarasyev, A. (eds.) Dynamic Systems, Economic Growth, and the Environment, pp. 3-22. Springer, Berlin (Dynamic Modelling and Econometrics in Economics and Finance 12) (2010)
[27] Pickenhain, S.: Hilbert space treatment of optimal control problems with Infinite Horizon. In: Bock, H.G., Phu, H.X., Rannacher, R., Schloeder, J.P. (eds.) Modelling, Simulation and Optimization of Complex Processes - HPSC 2012, pp. 169-182. Springer, Berlin (2014)
[28] Pickenhain, S., Infinite horizon optimal control problems in the light of convex analysis in Hilbert spaces, J. Set-Valued Var. Anal., 23, 169-189, (2015) · Zbl 1311.49084
[29] Pickenhain, S.; Burtchen, A.; Kolo, K.; Lykina, V., An indirect pseudospectral method for linear-quadratic infinite horizon optimal control problems, Optimization, 65, 609-633, (2016) · Zbl 1334.49104
[30] Pickenhain, S.; Lykina, V.; Wagner, M., On the lower semicontinuity of functionals involving Lebesgue or improper Riemann integrals in infinite horizon optimal control problems, Control. Cybern., 37, 451-468, (2008) · Zbl 1236.49028
[31] Lykina, V.; Pickenhain, S., Weighted functional spaces in infinite horizon optimal control problems: a systematic analysis of hidden opportunities and advantages, J. Math. Anal. Appl., 454, 195-218, (2017) · Zbl 1367.49018
[32] Ramsey, FP, A mathematical theory of savings, Econ. J., 152, 543-559, (1928)
[33] Sontag, E.D.: Mathematical Control Theory. Springer, Berlin (1990) · Zbl 0703.93001
[34] Sussmann, H.J., Willems, J.C.: 300 years of optimal control: from the brachistochrone to the maximum principle. IEEE Control. Syst. Mag. 32-44 (1997)
[35] Troutman, J.L.: Variational Calculus and Optimal Control, Optimization with Elementary Convexity. Springer, Berlin (1996) · Zbl 0865.49001
[36] Wiener, N.: Extrapolation, Interpolation and Smoothing of Stationary Time Series. MIT Press, Cambridge (1949) · Zbl 0036.09705
[37] Yosida, K.: Functional Analysis. Springer, New York (1974) · Zbl 0286.46002
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