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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.

MSC:
49J40 Variational inequalities
49K15 Optimality conditions for problems involving ordinary differential equations
46N10 Applications of functional analysis in optimization, convex analysis, mathematical programming, economics
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[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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