پیوندهای مفید
آمار
| تعداد نشریات | 21 |
| تعداد شمارهها | 687 |
| تعداد مقالات | 9,985 |
| تعداد مشاهده مقاله | 70,957,405 |
| تعداد دریافت فایل اصل مقاله | 62,422,654 |
Generalizations of the Hilbert-Weierstrass theorem and Tonelli-Morrey theorem: The regularity of solutions of differential equations and optimal control problems | ||
| International Journal of Nonlinear Analysis and Applications | ||
| مقاله 10، دوره 16، شماره 8، آبان 2025، صفحه 103-119 اصل مقاله (430.27 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22075/ijnaa.2022.27413.3731 | ||
| نویسنده | ||
| Saman Khoramian* | ||
| Faculty of Mathematics and Computer, Kharazmi University, Tehran, Iran | ||
| تاریخ دریافت: 25 تیر 1401، تاریخ بازنگری: 25 مهر 1401، تاریخ پذیرش: 12 آبان 1401 | ||
| چکیده | ||
| One of the basic problems in the “Calculus of Variations” is the minimization of the following functional: $$F(x)=\int_a^b f(t,x(t),x'(t)) dt,$$ over a class of functions $x$ defined on the interval $[a,b]$. According to a regularity theorem, solutions to this fundamental problem are found in a smaller class of more regular functions. However, they were originally considered to belong to a larger class. In this context, two theorems attributed to “Hilbert-Weierstrass” and “Tonelli-Morrey” are two classical studies of the regularity of discussion for the solutions to this problem. As higher-order differential equations and higher-order optimal control problems become more prevalent in the literature, regularity issues for these problems should receive more attention. Therefore, a generalization of the above regularity theorems is presented here, namely the regularity of solutions to the following functional $$F(x)=\int_a^b f(t,x(t),x'(t),\dots,x^{(n-1)}(t)) dt$$ where $n \geq 2$. It is expected that this extension will be helpful in discussing the regularity of higher-order differential equations and optimal control problems. | ||
| کلیدواژهها | ||
| Boundary value problems؛ Classical solution؛ Regularity؛ Weak solution | ||
| مراجع | ||
|
[1] T. Barbu, V. Barbu, V. Biga, and D. Coca, A PDE variational approach to image denoising and restoration, Nonlinear Anal.: Real World Appl. 10 (2009), no. 3, 1351–1361. [2] A.M. Bloch and P.E. Crouch, On the equivalence of higher order variational problems and optimal control problems, Proc. 35th Conf. Decis. Control, Kobe, Japan, 1996, pp. 1648–1653. [3] A. Chambolle, R.A. Devore, N.-Y. Lee, and B.J. Lucier, Nonlinear wavelet image processing: Variational problems, compression, and noise removal through wavelet shrinkage, IEEE Trans. Image Process. 7 (1998), no. 3, 319–335. [4] F. Clarke, Functional Analysis, Calculus of Variations and Optimal Control, Vol. 264. Springer, 2013. [5] I. Daubechies, M. Defrise, and C. De Mol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Comm. Pure Appl. Math. 57 (2004), 1413–1457. [6] R. Devore and B. Lucier, Fast wavelet techniques for near optimal image processing, IEEE Military Commun. Conf. Records, San Diego, Oct. 11-14, 1992, IEEE Press, Piscataway, NJ, 1992, pp. 1129–1135. [7] P. Drabek and J. Milota, Methods of Nonlinear Analysis: Applications to Differential Equations, Springer, 2nd edition, 2013. [8] P. du Bois-Reymond, Erlauterungen zu den anfangsgrunden der variationsrechnung, Math. Ann. 15 (1879), 283–314. [9] H.H. Goldstine, A History of the Calculus of Variations from the 17th through the 19th Century, Vol. 5. Springer Science & Business Media, 2012. [10] T. Gyulov, G. Morosanu, and S. Tersian, Existence for a semilinear sixth-order ODE, Math. Anal. Appl. 321 (2006), no. 1, 86–98. [11] S. Khoramian, An iterative thresholding algorithm for linear inverse problems with multi-constraint and its application, Appl. Comput. Harmon. Anal. 32 (2012), 109-130. [12] S. Krantz and H. Parks, Implicit Function Theorem: History, Theory, and Applications, Springer, 2003. [13] C.T. Lee, Some remarks on the fifth-order KdV equations, Math. Anal. Appl. 425 (2015), no. 1, 281–294. [14] Z. Liu, J.S. Ume, D.R. Anderson, and S.M. Kang, Twin monotone positive solutions to a singular nonlinear third-order differential equation, Math. Anal. Appl. 334 (2007), no. 1, 299–313. [15] C.B. Morrey, Multiple Integrals in the Calculus of Variations, Springer Science & Business Media, 2009. [16] G.F. Simmons and J.S. Robertson, Differential Equations with Applications and Historical Notes, 2nd edition, McGraw-Hill, 1991. [17] M. Tenenbaum and H. Pollard, Ordinary Differential Equations, Dover Publications, 1985. [18] L. Tonelli, Fondamenti di Calcolo Delle Variazioni, Zanichelli, Bologna, 1921. [19] S. Treanta and C. Udriste, Optimal control problems with higher order ODEs constraints, Balkan J. Geom. Appl. 18 (2013), no. 1, 71–86. [20] F. Wu, Existence of eventually positive solutions of fourth order quasilinear differential equations, Math. Anal. Appl. 389 (2012), no. 1, 632–646. [21] M. Younis, D. Singh, and A.A.N. Abdou, A fixed point approach for tuning circuit problem in dislocated b-metric spaces, Math. Meth. Appl. Sci. 45 (2020), no. 4, 2234–2253. [22] M. Younis, A. Stretenovic, and S. Radenovic, Some critical remarks on “Some new fixed point results in rectangular metric spaces with an application to fractional-order functional differential equations”, Nonlinear Anal. Modell. Control 27 (2022), no. 1, 163–178. [23] M. Younis, D. Singh, I. Altun, and V. Chauhan, Graphical structure of extended b-metric spaces: An application to the transverse oscillations of a homogeneous bar, Int. J. Nonlinear Sci. Numer. Simul. 23 (2022), no. 7-8, 1239–1252. [24] M. Younis, D. Singh, S. Radenovic, and M. Imdad, Convergence theorems for generalized contractions and applications, Filomat 34 (2020), no. 3,945–964. [25] J. Zhu, Optimization of Power System Operation, 2nd ed., Wiley-IEEE Press, 2015. [26] J.A. Momoh, Smart Grid: Fundamentals of Design and Analysis, Wiley-IEEE Press, 2012. [27] H. Singh, Introduction to Smart Grid: Concepts, Technologies, and Evolution, Wiley, 2018. [28] F. Li, W. Qiao, H. Sun, H. Wan, J. Wang, Y. Xia, Z. Xu, and P. Zhang, Smart transmission grid: Vision and framework, IEEE Trans. Smart Grid 1 (2010), no. 2, 168–177. [29] S.P. Timoshenko and J.M. Gere, Theory of Elastic Stability, 2nd ed., McGraw-Hill, 1961. [30] J.M. Gere and S.P. Timoshenko, Mechanics of Materials, 4th ed., PWS Publishing Company, 1997. [31] A.C. Ugural, Stresses in Beams, Plates, and Shells, 3rd ed., CRC Press, 2009. [32] L. Meirovitch, Fundamentals of Vibrations, McGraw-Hill, 2001. [33] S.S. Rao, Mechanical Vibrations, 5th ed., Prentice Hall, 2011. [34] S. Timoshenko and D.H. Young, Vibration Problems in Engineering, 3rd ed., D. Van Nostrand Company, 1955. | ||
|
آمار تعداد مشاهده مقاله: 1,251 تعداد دریافت فایل اصل مقاله: 421 |
||