Banach空间中微分变分不等式系统的Bang-Bang准则

数学物理学报 ›› 2022, Vol. 42 ›› Issue (6): 1653-1670.

Banach空间中微分变分不等式系统的Bang-Bang准则

施翠云¹,宾茂君^2,^3,*()

¹ 桂林理工大学南宁分校, 南宁 530001
² 玉林师范学院广西高校复杂系统优化与大数据处理重点实验室, 广西玉林 537000
³ 玉林师范学院数学与统计学院, 广西玉林 537000

收稿日期:2021-10-24 出版日期:2022-12-26 发布日期:2022-12-16
通讯作者: 宾茂君 E-mail:bmj1999@163.com
基金资助:
广西自然科学基金(2020GXNSFAA159152);广西自然科学基金(2020GXNSFBA297142);广西自然科学基金(2021GXNSFAA220130);广西自然科学基金(2022GXNSFAA035617)

On the Bang-Bang Principle for Differential Variational Inequalities in Banach Spaces

Cuiyun Shi¹,Maojun Bin^2,^3,*()

¹ Guilin University of Technology at Nanning, Nanning 530001
² Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing, Yulin Normal University, Guangxi Yulin 537000
³ School of Mathematics and Statistics, Yulin Normal University, Guangxi Yulin 537000

Received:2021-10-24 Online:2022-12-26 Published:2022-12-16
Contact: Maojun Bin E-mail:bmj1999@163.com
Supported by:
the NSF of Guangxi(2020GXNSFAA159152);the NSF of Guangxi(2020GXNSFBA297142);the NSF of Guangxi(2021GXNSFAA220130);the NSF of Guangxi(2022GXNSFAA035617)

摘要/Abstract

摘要：

该文将讨论一类由半线性发展方程和广义变分不等式所组成的微分变分不等式系统.首先, 考虑广义变分不等式解集的性质.其次, 通过利用不动点定理和半群理论证明了微分变分不等式系统解的存在性.另外, 通过运用稠定性结果证明了微分变分不等式系统的Bang-Bang准则.同时, 运用一个障碍型抛物-椭圆系统来检验该文的主要结果.

关键词: 微分变分不等式, Bang-Bang准则, 稠定性, KKM映射

Abstract:

In this paper, we discuss a class of differential variational inequalities systems, which are obtained by semilinear evolution equations and generalized variational inequalities. At first, we consider the properties of solution set for generalized variational inequalities. Secondly, the existence results are shown by fixed point method for semilinear differential variational inequality. Our approaches are based on semigroup theory and fixed point theorem. Moreover, by using the density results, the nonlinear and infinite dimensional versions of the "bang-bang" principle for differential variational inequalities systems is derived. Also, an obstacle parabolic-elliptic system is given to illustrate the application of the obtained theory.

Key words: Differential variational inequalities, "Bang-Bang" principle, Density, KKM mapping

中图分类号:

O176.3

施翠云,宾茂君. Banach空间中微分变分不等式系统的Bang-Bang准则[J]. 数学物理学报, 2022, 42(6): 1653-1670.

Cuiyun Shi,Maojun Bin. On the Bang-Bang Principle for Differential Variational Inequalities in Banach Spaces[J]. Acta mathematica scientia,Series A, 2022, 42(6): 1653-1670.

参考文献 47

1	Aubin J P , Cellina A . Differential Inclusions: Set-Valued Maps and Viability Theory. Berlin: Springer-Verlag, 1984
2	Avgerinos E P , Papageorgiou N S . Differential variational inequalities in $R.N$. Indian Journal of Pure and Applied Mathematics, 1997, 28, 475- 498
3	Bin M J . Time optimal control for semilinear fractional evolution feedback control systems. Optimization, 2019, 68 (4): 819- 832 doi: 10.1080/02331934.2018.1552956
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 doi: 10.1515/fca-2021-0055
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 doi: 10.1016/j.jmaa.2019.07.054
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 doi: 10.1016/j.nonrwa.2019.05.013
7	Bohnenblust H F, Karlin S. On a theorem of ville//Kuhn H W, Tucker A W. Contributions to the Theory of Games, Vol I. Princeton, NJ: Princeton University Press, 1950: 155-160
8	Bressan A . Differential inclusions with non-closed, non-convex right hand side. Diff Integral Equ, 1990, 3, 633- 638
9	Chen X J , Wang Z Y . Differential variational inequality approach to dynamic games with shared constraints. Math Program Ser A, 2014, 146, 379- 408 doi: 10.1007/s10107-013-0689-1
10	Chen Y Q , O'Regan D , Agarwal R P . Anti-periodic solutions for evolution equations associated with monotone type mappings. Appl Math Lett, 2010, 23 (11): 1320- 1325 doi: 10.1016/j.aml.2010.06.022
11	Deimling K . Multivalued Differential Equations. Berlin: De Gruyter, 1992
12	Fan K . Some properties of convex sets related to fixed point theorems. Math Ann, 1984, 266, 519- 537 doi: 10.1007/BF01458545
13	Friesz T L , Bernstein D , Suo Z , Tobin R . Dynamic network user equilibrium with state-dependent time lags. Networks and Spatial Economics, 2001, 1 (3/4): 319- 347 doi: 10.1023/A:1012896228490
14	Friesz T L , Rigdon M A , Mookherjee R . Differential variational inequalities and shipper dynamic oligopolistic network competition. Transportation Research Part B, 2006, 40, 480- 503 doi: 10.1016/j.trb.2005.07.002
15	Fryszkowski A . Continuous selections for a class of nonconvex multivalued maps. Studia Math, 1983, 76, 163- 174 doi: 10.4064/sm-76-2-163-174
16	Himmelberg C J . Measurable relations. Fund Math, 1975, 87, 53- 72 doi: 10.4064/fm-87-1-53-72
17	Hu S , Papageorgiou N S . Handbook of Multivalued Analysis, Volume I: Theory. New York: Springer, 1997
18	Kamemskii M , Obukhovskii V , Zecca P . Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Space. Berlin: Water de Gruyter, 2001
19	Ke T D , Loi N V , Obukhovskii V . Decay solutions for a class of fractional differential variational inequalities. Fract Calc Appl Anal, 2015, 18, 531- 553 doi: 10.1515/fca-2015-0033
20	Li X S , Huang N J , Donal O R . A class of impulsive differential variational inequalities in finite dimensional spaces. J Franklin Inst, 2016, 353 (13): 3151- 3175 doi: 10.1016/j.jfranklin.2016.06.011
21	Liu X Y , Liu Z H . On the "bang-bang" principle for a class of fractional semilinear evolution inclusions. Proc Royal Soc Edinburgh, 2014, 44 (2): 333- 349
22	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 doi: 10.1016/j.jmaa.2013.07.032
23	Liu Z H , Bin M J , Liu X Y . On the "bang-bang" principle for a class of Riemann-Liouville fractional semilinear evolution inclusions. Mathematica Slovaca, 2016, 66 (6): 1329- 1344 doi: 10.1515/ms-2016-0226
24	Lightbourne J H , Rankin S M . A partial functional differential equation of Sobolev type. J Math Anal Appl, 1983, 93, 328- 337 doi: 10.1016/0022-247X(83)90178-6
25	Liu Z H , Migórski S , Zeng S D . Partial differential variational inequalities involving nonlocal boundary conditions in Banach spaces. J Differ Equ, 2017, 263 (7): 3989- 4006 doi: 10.1016/j.jde.2017.05.010
26	Liu Z H , Zeng S D . Differential variational inequalities in infinite Banach spaces. Acta Math Sci, 2017, 37B (1): 26- 32
27	Liu Z H , Zeng S D . Penalty method for a class of differential variational inequalities. Applicable Analysis, 2021, 100 (7): 1574- 1589 doi: 10.1080/00036811.2019.1652736
28	Liu Z H , Zeng S D , Motreanu D . Evolutionary problems driven by variational inequalities. J Diff Equ, 2016, 260, 6787- 6799 doi: 10.1016/j.jde.2016.01.012
29	Lu L , Liu Z H , Jiang W , Luo J . Solvability and optimal controls for semilinear fractional evolution hemivariational inequalities. Math Methods Appl Sci, 2016, 39, 5452- 5464 doi: 10.1002/mma.3930
30	Lu L , Liu Z H , Obukhovskii V . Second order differential variational inequalities involving anti-periodic boundary value conditions. Journal of Mathematical Analysis and Applications, 2019, 473 (2): 846- 865 doi: 10.1016/j.jmaa.2018.12.072
31	Melanz D , Jayakumar P , Negrut D . Experimental validation of a differential variational inequality-based approach for handling friction and contact in vehicle/granular-terrain interactio. J Terramechanics, 2016, 65, 1- 13 doi: 10.1016/j.jterra.2016.01.004
32	Migórski S , Ochal A , Sofonea M . Nonlinear Inclusions and Hemivariational Inequalities: Models and Analysis of Contact Problems. New York: Springer, 2013
33	Pang J S , Stewart D E . Differential variational inequalities. Math Program, 2008, 113, 345- 424 doi: 10.1007/s10107-006-0052-x
34	Papageorgiou N S . On the "bang-bang" principle for nonlinear evolution inclusions. Aequationes Math, 1993, 45, 267- 280 doi: 10.1007/BF01855884
35	Pazy A . Semigroups of Linear Operators and Applications to Partial Differential Equations. New York: Springer-Verlag, 1983
36	Royden H L . Real Analysis. New Jersey: Prentice-Hall, Englewood Cliffs, 1988
37	Suslov S I . Nonlinear bang-bang principle in Rⁿ. Mat Zametki, 1991, 49 (5): 110- 116
38	Tolstonogov A A . Extremal selections of multivalued mappings and the "bang-bang" principle for evolution inclusions. Dokl Akad Nauk SSSR, 1991, 317, 589- 593
39	Tolstonogov A A . Relaxation in non-convex control problems described by first-order evolution equations. Math Sb, 1999, 190 (11): 135- 160 doi: 10.4213/sm441
40	Tolstonogov A A . Properties of the set of admissible "state-control" pairs for first-order evolution control systems. Izvestiya: Math, 2001, 65 (3): 617- 640 doi: 10.1070/IM2001v065n03ABEH000343
41	Tolstonogov A A . Relaxation in nonconvex optimal control problems with subdifferential operators. J Math Sci, 2007, 140 (6): 850- 872 doi: 10.1007/s10958-007-0021-9
42	Tolstonogov A A . Relaxation in control systems of subdifferential type. Izvestiya Math, 2006, 70 (1): 121- 152 doi: 10.1070/IM2006v070n01ABEH002306
43	Tolstonogov A A , Tolstonogov D A . On the "bang-bang" principle for nonlinear evolution inclusions. Nonlinear Differ Equa Appl, 1999, 6, 101- 118 doi: 10.1007/s000300050067
44	Tolstonogov A A , Tolstonogov D A . L_p-continuous extreme selectors of multifunctions with decomposable values: Existence theorems. Set-Valued Anal, 1996, 4, 173- 203 doi: 10.1007/BF00425964
45	Tolstonogov A A , Tolstonogov D A . L_p-continuous extreme selectors of multifunctions with decomposable values: Relaxation theorems. Set-Valued Anal, 1996, 4, 237- 269 doi: 10.1007/BF00419367
46	Zeidler E. . Nonlinear Functional Analysisand Its Applications. Ⅱ/B: Nonlinear Monotone Operators. Berlin: Springer-Verlag, 1990
47	Zhu Q J . On the solution set of differential inclusions in Banach space. J Differ Equ, 1991, 93 (2): 213- 237 doi: 10.1016/0022-0396(91)90011-W

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