半线性Riemann-Liouville分数阶发展方程反馈时间最优控制

数学物理学报 ›› 2024, Vol. 44 ›› Issue (3): 687-698.

半线性Riemann-Liouville分数阶发展方程反馈时间最优控制

宾茂君¹(),施翠云^2,^*()

1.广西应用数学中心(玉林师范学院) 广西玉林 537000
2.桂林理工大学南宁分校南宁 530001

收稿日期:2023-05-26 修回日期:2023-10-06 出版日期:2024-06-26 发布日期:2024-05-17
通讯作者: *施翠云，E-mail: shicuiyun99@163.com
作者简介:宾茂君，E-mail: bmj1999@163.com
基金资助:
广西自然科学基金(2021GXNSFAA220130);NSF of Guangxi Grants(2021GXNSFAA220130);广西自然科学基金(2022GXNSFAA035617);广西高校中青年教师科研基础能力提升项目

Time Optimal Control for Semilinear Riemann-Liouville Fractional Evolution Feedback Control Systems

Bin Maojun¹(),Shi Cuiyun^2,^*()

1. Center for Applied Mathematics of Guangxi, Yulin Normal University, Guangxi Yulin 537000
2. Campus of Nanning, Guilin University of Technology, Nanning 530001

Received:2023-05-26 Revised:2023-10-06 Online:2024-06-26 Published:2024-05-17
Supported by:
NSF of Guangxi Grants(2022GXNSFAA035617);Project of Guangxi Education

摘要/Abstract

摘要：

该文研究 Banach 空间中一类半线性 Riemann-Liouville. 首先, 利用不动点定理求解方程的温和解存在唯一性. 其次, 在半群是紧的前提下借助 Cesari 性质和 Fillipove 定理证明容许轨迹集的非空性. 在此基础上证明时间反馈最优控制问题的存在性结果. 最后, 给出实例来说明文章的主要结果.

关键词: 半线性分数阶微分方程, 容许轨迹, 容许控制, 反馈控制, 时间最优控制

Abstract:

In this article, we investigate a class of time feedback optimal control systems governed by Riemann-Liouville fractional semilinear differential equations in Banach space. At first, we discuss the existence and uniqueness of the mild solution for the equations by using fixed point theorem. Secondly, we show that the admissible trajectories set is nonempty involving the compactness of semigroup $ T(t)(t > 0) $ with the help of the Cesari roperty and the Fillippove theorem. Moreover, we also give an existence result for time eedback optimal control. In the end, an example is given to illustrate our main results.

Key words: Semilinear fractional differential equations, Admissible trajectories, Admissible control, Feedback control, Time optimal control

中图分类号:

O175.6

宾茂君, 施翠云. 半线性Riemann-Liouville分数阶发展方程反馈时间最优控制[J]. 数学物理学报, 2024, 44(3): 687-698.

Bin Maojun, Shi Cuiyun. Time Optimal Control for Semilinear Riemann-Liouville Fractional Evolution Feedback Control Systems[J]. Acta mathematica scientia,Series A, 2024, 44(3): 687-698.

参考文献 33

[1]	Barbu V. The time-optimal control problem for parabolic variational inequalities. Appl Math Optim, 1984, 11: 1-22
[2]	Berkovitz L D. Optimal Control Theory. New York: Springer-Verlag, 1974
[3]	Bin M J. Time optimal control for semilinear fractional evolution feedback control systems. Optimization, 2019, 68(4): 819-832
[4]	Bin M J, Deng H Y, Li Y X, Zhao J. Properties of the set of admissible "state control" part for a class of fractional semilinear evolution control systems. Fract Calc Appl Anal, 2021, 24(4): 1275-1298
[5]	Bin M J, Liu Z H. On the "bang-bang" principle for nonlinear evolution hemivariational inequalities control systems. J Math Anal Appl, 2019, 480(1): 123364
[6]	Bin M J, Liu Z H. Relaxation in nonconvex optimal control for nonlinear evolution hemivariational inequalities. Nonlinear Analysis: Real World Applications, 2019, 50: 613-632
[7]	Dixon J, McKee S. Weakly singular discrete gronwall inequalities. Z Angew Math Mech, 1986, 68(11): 535-544
[8]	Fattorini H O. Time-Optimal control of solutions of operational differential equations. SIAM J Control, 1964, 2: 54-59
[9]	Glockle W G, Nonnenmacher T F. A fractional calculus approach of self-similar protein dynamics. Biophys J, 1995, 68: 46-53
[10]	Henry D. Geometric Theory of Semilinear Parabolic Equations. Berlin-New York: Springer-Verlag, 1981
[11]	Himmelberg C J. Measurable relations. Fund Math, 1975, 87: 53-72
[12]	Hu S, Papageorgiou N S. Handbook of Multivalued Analysis:Volume I Theory. Dordrecht: Kluwer Academic Publishers, 1997
[13]	Jeong J M, Kim J R, Roh H H. Optimal control problems for semilinear evolution equations. J Korean Math Soc, 2008, 45(3): 757-769
[14]	Jeong J M, Son S J. Time optimal control of semilinear control systems involving time delays. J Optim Theory Appl, 2015, 165: 793-811
[15]	Kilbas A A, Srivastava H M, Trujillo J J. Theory and Applications of Fractional Differential Equations. Netherlands: North-Holland, 2006
[16]	Kumar S, Sukavanam N. Approximate controllability of fractional order semilinear systems with bounded delay. J Diff Equ, 2012, 252(11): 6163-6174
[17]	LaSalle J P. The time optimal control problem// Cesari L, Lasalle J, Lefschetz S. Contributions to the Theory of Nonlinear Oscillations. Princeton: Princeton University Press, 1960, 1(5): 1-24
[18]	Li X, Yong J. Optimal Control Theory for Infinite Dimensional Systems. Boston: Birkhäuser, 1995
[19]	Lightbourne J H, Rankin S M. A partial functional differential equation of Sobolev type. J Math Anal Appl, 1983, 93: 328-337
[20]	Liu X Y, Liu Z H, Fu X. Relaxation in nonconvex optimal control problems described by fractional differential equations. J Math Anal Appl, 2014, 409(1): 446-458
[21]	Liu Z H, Bin M J. Approximate controllability of impulsive Riemann-Liouville fractional equations in Banach spaces. J Int Equat Appl, 2014, 26(4): 527-551
[22]	Liu Z H, Li X W. Approximate controllability for a class of hemivariational inequalities. Nonlinear Anal RWA, 2015, 22: 581-591
[23]	Liu Z H, Li X W. Approximate controllability of fractional evolution systems with Riemann-Liouville fractional derivatives. SIAM J Control Optim, 2015, 53(4): 1920-1933
[24]	Liu Z H, Zeng S D, Bai Y R. Maximum principles for multi-term space-time variable-order fractional diffusion equations and their applications. Fractional Calculus & Applied Analysis, 2016, 19(1): 188-211
[25]	Lu L, Liu Z H. Existence and controllability results for stochastic fractional evolution hemivariational inequalities. Appl Math Comput, 2015, 268: 1164-1176
[26]	Mainardi F, Paradisi P, Gorenflo R. Probability distributions generated by fractional diffusion equations//Kertesz J, Kondor I, eds. Dordrecht: Kluwer Academic Publisher, 2000
[27]	Podlubny I. Fractional Differential Equations. San Diego: Academic Press, 1999
[28]	Wang J R, Fečkan M, Zhou Y. Approximate controllability of sobolev type fractional evolution systems with nonlocal conditions. Evolution Equations and Control Theory, 2017, 6(3): 471-486
[29]	Wang J R, Fečkan M, Zhou Y. Fractional order differential switched systems with coupled nonlocal initial and impulsive conditions. Bulletin des Sciences Mathématiques, 2017, 141: 727-746
[30]	Wang J R, Zhou Y, Wei W. Optimal feedback control for semilinear fractional evolution equations in Banach spaces. Syst Contr Letter, 2012, 61: 472-476
[31]	Warga J. Optimal Control of Differential and Functional Equations. New York: Academic Press, 1972
[32]	Yong J M. Time optimal controls for semilinear distributed parameter systems-existence theory and necessary conditions. Kodai Mathematical Journal, 1991, 14(2): 239-253
[33]	Zhou Y, Jiao F. Existence of mild solutions for fractional neutral evolution equations. Comput Math Appl, 2010, 59: 1063-1077