×

Calculus of variations and optimal control for generalized functions. (English) Zbl 1483.49024

The authors introduce a framework for the calculus of variations and the theory of optimal control for a class of generalized functions, called generalized smooth functions (GSF, for short), which extend Sobolev-Schwartz distributions and Colombeau generalized functions. In a certain sense the article is a follow up of A. Lecke et al. [Adv. Nonlinear Anal. 8, 779–808 (2019; Zbl 1448.49008)]. Firstly, after introducing some basic concepts of nonstandard analysis, the authors make use of fundamental results on GSF; it is presented, in a detailed way, the calculus of variations approach, which follows the traditional path, that is, initially proving, in this general context, the Fundamental Lemma, Euler-Lagrange equations, D’ Alembert principle, du Bois-Reymond optimality condition and Noether’s theorem in Lagrangian formalism among other results. Secondly, it is handled the theory of optimal control, that includes a version of Pontryagin maximum principle, and Noether’s theorem in Hamiltonian formalism in this GSF setting. Some examples and applications of the theory afore presented close the article providing a study of a singularly variable length pendulum, oscillations damped by two media and Pais-Uhlenbeck oscillator with singular frequencies.

MSC:

49K15 Optimality conditions for problems involving ordinary differential equations
46F30 Generalized functions for nonlinear analysis (Rosinger, Colombeau, nonstandard, etc.)

Citations:

Zbl 1448.49008

Software:

Mathematica
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Abakumova, V. A.; Kaparulin, D. S.; Lyakhovich, S. L., Conservation laws and stability of higher derivative extended Chern-Simons, J. Phys.: Conf. Ser., 1337, Article 012001 pp. (2019)
[2] Aragona, J.; Garcia, A. R.G.; Juriaans, S. O., Generalized solutions of a nonlinear parabolic equation with generalized functions as initial data, Nonlinear Anal. TMA, 71, 11, 5187-5207 (2009) · Zbl 1182.46032
[3] Benci, V.; Luperi Baglini, L.; Squassina, M., Generalized solutions of variational problems and applications, Adv. Nonlinear Anal., 9, 124-147 (2020) · Zbl 1414.26051
[4] Bessis, D. N.; Ledyaev, Yu. S.; Vinter, R. B., Dualization of the Euler and Hamiltonian inclusions, Nonlinear Anal. TMA, 43, 7, 861-882 (2001) · Zbl 1004.49016
[5] Biolek, Z.; Biolek, D.; Biolkova, V., Lagrangian for circuits with higher-order elements, Entropy, 21, 11, 1059 (2019)
[6] Brogliato, B., Nonsmooth Mechanics. Models, Dynamics and Control (1999), Springer Verlag: Springer Verlag London · Zbl 0917.73002
[7] Cheng, C.-W.; Mizel, V. J., On the Lavrentiev phenomenon for autonomous second-order integrands, Arch. Ration. Mech. Anal., 126, 21-33 (1994) · Zbl 0812.49011
[8] Colombeau, J. F., New Generalized Functions and Multiplication of Distributions (1984), North-Holland: North-Holland Amsterdam · Zbl 0761.46021
[9] Colombeau, J. F., (Multiplication of Distributions; A Tool in Mathematics, Numerical Engineering and Theoretical Physics. Multiplication of Distributions; A Tool in Mathematics, Numerical Engineering and Theoretical Physics, Lecture Notes in Mathematics, vol. 1532 (1992), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0519.46045
[10] Colombeau, J. F., Mathematical problems on generalized functions and the canonical Hamiltonian formalism (2007), see https://arxiv.org/abs/0708.3425
[11] (Cresson, J., Fractional Calculus in Analysis, Dynamics and Optimal Control. Fractional Calculus in Analysis, Dynamics and Optimal Control, Mathematics Research Developments (2014), Nova Publishers: Nova Publishers New York)
[12] Csörnyei, M.; Kirchheim, B.; O’Neil, T. C.; Preiss, D.; Winter, S., Universal singular sets in the calculus of variations, Arch. Ration. Mech. Anal., 190, 371-424 (2008) · Zbl 1218.49049
[13] Davie, A. M., Singular minimisers in the calculus of variations in one dimension, Arch. Ration. Mech. Anal., 101, 2, 161-177 (1988) · Zbl 0656.49005
[14] Dirac, P. A.M., The physical interpretation of the quantum dynamics, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 113, 621-641 (1926) · JFM 53.0846.01
[15] Djukić, D. S., Noether’s theorem for optimum control systems, Internat. J. Control, 1, 18, 667-672 (1973) · Zbl 0281.49009
[16] Erlacher, E.; Grosser, M., Ordinary differential equations in algebras of generalized functions, (Molahajloo, S.; Pilipović, S.; Toft, J.; Wong, M. W., Pseudo-Differential Operators, Generalized Functions and Asymptotics. Pseudo-Differential Operators, Generalized Functions and Asymptotics, Operator Theory: Advances and Applications, vol. 231 (2013)), 253-270 · Zbl 1275.46025
[17] Frederico, G. S.F.; Odzijewicz, T.; Torres, D. F.M., Noether’s theorem for nonsmooth extremals of variational problems with time delay, Appl. Anal., 93, 1, 153-170 (2014) · Zbl 1287.49018
[18] Frederico, G. S.F.; Torres, D. F.M., Conservation laws for invariant functionals containing compositions, Appl. Anal., 86, 9, 1117-1126 (2007), arXiv:0704.0949 · Zbl 1190.49027
[19] Frederico, G. S.F.; Torres, D. F.M., Non-conservative Noether’s theorem for fractional action-like variational problems with intrinsic and observer times, Int. J. Ecol. Econ. Stat., 9, F07, 74-82 (2007)
[20] Frederico, G. S.F.; Torres, D. F.M., A non-differentiable quantum variational embedding in presence of time delays, Int. J. Differ. Equ., 8, 1, 49-62 (2013)
[21] Gelfand, I. M.; Fomin, S. V., Calculus of Variations (2000), Dover Publications · Zbl 0964.49001
[22] Giordano, P.; Kunzinger, M., (Oberguggenberger, M.; Toft, J.; Vindas, J.; Wahlberg, P., Inverse Function Theorems for Generalized Smooth Functions. Invited Paper for the Special Issue ISAAC - Dedicated to Prof. Stevan Pilipovic for his 65 Birthday. Inverse Function Theorems for Generalized Smooth Functions. Invited Paper for the Special Issue ISAAC - Dedicated to Prof. Stevan Pilipovic for his 65 Birthday, Springer series Operator Theory: Advances and Applications (2016), Birkhaeuser: Birkhaeuser Basel)
[23] Giordano, P.; Kunzinger, M., A convenient notion of compact sets for generalized functions, Proc. Edinb. Math. Soc., 61, 1, 57-92 (2018) · Zbl 1400.46036
[24] P. Giordano, M. Kunzinger, H. Vernaeve, A Grothendieck topos of generalized functions I: basic theory. Preprint. See: http://www.mat.univie.ac.at/giordap7/ToposI.pdf. · Zbl 1328.46037
[25] Giordano, P.; Kunzinger, M.; Vernaeve, H., Strongly internal sets and generalized smooth functions, J. Math. Anal. Appl., 422, 1, 56-71 (2015) · Zbl 1328.46037
[26] Giordano, P.; Luperi Baglini, L., Asymptotic gauges: Generalization of Colombeau type algebras, Math. Nachr., 289, 2-3, 247-274 (2016) · Zbl 1380.46034
[27] Giunashvili, Z., Bott connection and generalized functions on Poisson manifold (2003), See https://arxiv.org/abs/math/0301364
[28] Gogodze, I. K., Symmetry in problems of optimal control (in Russian), (Proc. of Extended Sessions of Seminar of the Vekua Institute of Applied Mathematics, Vol. 3 (1988), Tbilisi University: Tbilisi University Tbilisi), 39-42, no 3
[29] Gratwick, R.; Preiss, D., A one-dimensional variational problem with continuous Lagrangian and singular minimizer, Arch. Ration. Mech. Anal., 202, 177-211 (2011) · Zbl 1266.70028
[30] Graves, L. M., Discontinuous solutions in the calculus of variations, Bull. Amer. Math. Soc., 36, 831-846 (1930) · JFM 56.0432.03
[31] Grosser, M.; Kunzinger, M.; Oberguggenberger, M.; Steinbauer, R., Geometric Theory of Generalized Functions (2001), Kluwer: Kluwer Dordrecht
[32] Hawking, S. W.; Hertog, T., Living with ghosts, Phys. Rev. D, 65, Article 103515 pp. (2002)
[33] Hestenes, M. R., Calculus of Variations and Optimal Control Theory (1966), John Wiley & Sons · Zbl 0173.35703
[34] Katz, M. G.; Tall, D., A Cauchy-Dirac delta function, Found. Sci. (2012), See http://arxiv.org/abs/1206.0119
[35] Konjik, S.; Kunzinger, M.; Oberguggenberger, M., Foundations of the calculus of variations in generalized function algebras, Acta Appl. Math., 103, 2, 169-199 (2008) · Zbl 1146.49021
[36] Kuhn, S., The derivative à la Carathéodory, Amer. Math. Monthly, 98, 1, 40-44 (1991) · Zbl 0761.26002
[37] Kunzinger, M.; Oberguggenberger, M.; Steinbauer, R.; Vickers, J. A., Generalized flows and singular ODEs on differentiable manifolds, Acta Appl. Math., 80, 2, 221-241 (2004) · Zbl 1060.46030
[38] Künzle, A. F., Singular Hamiltonian systems and symplectic capacities, (Singularities and Differential Equations Banach Center Publications, Vol. 33 (1996)), 171-187, See http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.bwnjournal-article-bcpv33z1p171bwm · Zbl 0855.58027
[39] Lakshminarayanan, V.; Ghatak, A. K.; Thygarajan, K., Lagrangian Optics (2002), Springer Science Business Media, LLC
[40] Laugwitz, D., Definite values of infinite sums: aspects of the foundations of infinitesimal analysis around 1820, Arch. Hist. Exact Sci., 39, 3, 195-245 (1989) · Zbl 0766.01011
[41] Lazo, M. J.; Krumreich, C. E., The action principle for dissipative systems, J. Math. Phys., 55, Article 122902 pp. (2014) · Zbl 1308.70032
[42] Lecke, A.; Luperi Baglini, L.; Giordano, P., The classical theory of calculus of variations for generalized functions, Adv. Nonlinear Anal., 779-808 (2019) · Zbl 1448.49008
[43] Lerman, E.; Montgomery, R.; Sjamaar, R., Examples of singular reduction, (Symplectic Geometry. Symplectic Geometry, London Math. Soc. Lecture Note Ser., vol. 192 (1993), Cambridge Univ. Press: Cambridge Univ. Press Cambridge) · Zbl 0812.58034
[44] Li, F.; Hua, Q.; Zhang, S., New periodic solutions of singular Hamiltonian systems with fixed energies, J. Inequal. Appl., 400 (2014), See http://dx.doi.org/10.1186/1029-242X-2014-400 · Zbl 1332.35010
[45] Lim, C. C., On singular Hamiltonians: the existence of quasi-periodic solutions and nonlinear stability, Bull. Amer. Math. Soc. (N.S.), 20, 1, 35-40 (1989) · Zbl 0699.34042
[46] L. Luperi Baglini, P. Giordano, A Grothendieck topos of generalized functions II: ODE. See http://www.mat.univie.ac.at/giordap7/ToposII.pdf/www.mat.univie.ac.at/giordap7/ToposII.pdf. · Zbl 1373.46035
[47] Luperi Baglini, L.; Giordano, P., The category of Colombeau algebras, Mon.hefte Math. (2017) · Zbl 1373.46035
[48] Mannheim, P. D.; Davidson, A., Dirac quantization of the Pais-Uhlenbeck fourth order oscillator, Phys. Rev. A, 71, Article 042110 pp. (2005) · Zbl 1227.81222
[49] Marsden, J. E., Generalized Hamiltonian mechanics, Arch. Ration. Mech. Anal., 28, 4, 323-361 (1968) · Zbl 0155.51302
[50] Marsden, J. E., Hamiltonian one parameter groups. A mathematical exposition of infinite dimensional Hamiltonian systems with applications in classical and quantum mechanics, Arch. Ration. Mech. Anal., 28, 5, 362-396 (1968) · Zbl 0159.54801
[51] Marsden, J. E., Non-smooth geodesic flows and classical mechanics, Canad. Math. Bull., 12, 209-212 (1969) · Zbl 0177.27601
[52] Mazaheri, H.; Hosseinzadeh, A.; Ahmadian, M. T., Nonlinear oscillation analysis of a pendulum wrapping on a cylinder, Sci. Iran. Trans. B, 19, 2, 335-340 (2012)
[53] Mordukhovich, B. S.; Sarabi, M. E., Generalized differentiation of piecewise linear functions in second-order variational analysis, Nonlinear Anal. TMA, 132, 240-273 (2016) · Zbl 1329.49024
[54] Mukhammadiev, A.; Tiwari, D.; Apaaboah, G.; Giordano, P., Supremum, Infimum and hyperlimits of Colombeau generalized numbers (2020), Article in preparation. See http://www.mat.univie.ac.at/giordap7/Hyperlim.pdf · Zbl 1483.46042
[55] Oberguggenberger, M., Generalized functions in nonlinear models - A survey, Nonlinear Anal. TMA, 47, 8, 5029-5040 (2001) · Zbl 1042.46510
[56] Oberguggenberger, M.; Vernaeve, H., Internal sets and internal functions in Colombeau theory, J. Math. Anal. Appl., 341, 649-659 (2008) · Zbl 1173.46024
[57] Pais, A.; Uhlenbeck, G. E., On field theories with non-localized action, Phys. Rev., 79, 145-165 (1950) · Zbl 0040.13203
[58] Parker, P. E., Distributional geometry, J. Math. Phys., 20, 1423 (1979) · Zbl 0442.53064
[59] Pontryagin, L. S.; Boltyanskii, V. G.; Gamkrelidze, R. V.; Mishchenko, E. F., Selected Works. Vol. 4. The Mathematical Theory of Optimal Processes (1986), Gordon & Breach: Gordon & Breach New York, Translated from the Russian by K. N. Trirogoff, Translation edited by L. W. Neustadt, Reprint of the 1962 English translation
[60] Rapoport, A.; Rom-Kedar, V.; Turaev, D., Approximating multi-dimensional Hamiltonian flows by billiards, Comm. Math. Phys., 272, 567-600 (2007) · Zbl 1129.37031
[61] Robinson, A., Function theory on some nonarchimedean fields, Amer. Math. Monthly, 80, 6, 87-109 (1973), Part II: Papers in the Foundations of Mathematics · Zbl 0269.26020
[62] Sage, A. P., Optimum Systems Control (1968), Prentice-Hall, Inc.: Prentice-Hall, Inc. Englewood Cliffs, N.J. · Zbl 0192.51502
[63] Shvartsman, I. A., Finite-dimensional approximations in the derivation of necessary optimality conditions in nonsmooth constrained optimal control, Nonlinear Anal. TMA, 63, 5-7, e1665-e1672 (2005) · Zbl 1224.49023
[64] Stojanović, M., Extension of Colombeau algebra to derivatives of arbitrary order D \(\alpha, \alpha\in \operatorname{R} + \bigcup \{ 0 \} \). Application to ODEs and PDEs with entire and fractional derivatives, Nonlinear Anal. TMA, 71, 11, 5458-5475 (2009) · Zbl 1186.46045
[65] Sychev, M. A., Another theorem of classical solvability ‘in small’ for one-dimensional variational problems, Arch. Ration. Mech. Anal., 202, 269-294 (2011) · Zbl 1257.49039
[66] Tanaka, K., A prescribed energy problem for a singular Hamiltonian system with a weak force, J. Funct. Anal., 113, 2, 351-390 (1993) · Zbl 0771.70014
[67] Tuckey, C., (Nonstandard Methods in the Calculus of Variations. Nonstandard Methods in the Calculus of Variations, Pitman Research Notes in Mathematics Series, vol. 297 (1993), Longman Scientific & Technical: Longman Scientific & Technical Harlow) · Zbl 0794.49001
[68] Turaev, D.; Rom-Kedar, V., On smooth Hamiltonian flows limited to ergodic billiards, (Benkadda, S.; Zaslavsky, G. M., Chaos, Kinetics and Nonlinear Dynamics in Fluids and Plasmas. Chaos, Kinetics and Nonlinear Dynamics in Fluids and Plasmas, Lecture Notes in Physics, vol. 511 (1998), Springer: Springer Berlin, Heidelberg) · Zbl 0924.58062
[69] Vickers, J. A., Distributional geometry in general relativity, J. Geom. Phys., 62, 692-705 (2012) · Zbl 1242.53098
[70] von Neumann, J., (Taub, A. H., Method in the Physical Sciences. Collected Works Vol. VI. Theory of Games, Astrophysics, Hydro-Dynamics and Meteorology (1961), Pergamon Press: Pergamon Press Oxford)
[71] See https://reference.wolfram.com/language/ref/NDSolve.html, Wolfram Research, Inc., Mathematica, Champaign, IL.
[72] See https://mathworld.wolfram.com/HeavisideStepFunction.html and https://mathworld.wolfram.com/DeltaFunction.html.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.