多解椭圆型方程的最优控制问题
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摘要
本文主要研究了几类多解椭圆型方程的最优控制问题．首先，在凸性条件下考虑了半线性多解椭圆型方程的最优边界控制问题，证明了变分不等式．其次，在非凸情形下首次采用松弛控制方法研究了具有两点边值条件的多解常微分方程以及一类只有两个解的半线性椭圆型方程的最优控制问题，得到了Pontryagin's最大值原理．最后讨论了一类半线性椭圆方程的最优控制的存在性及不存在性问题．
全文共分为三部分内容：
在第一部分即本文的第二章和第三章中，我们分别讨论了带有线性边界条件以及非线性边界条件的半线性椭圆型方程的最优边界控制问题．在凸性条件下，通过构造相应的惩罚问题并取极限，我们以变分不等式的形式给出了原始问题最优对所满足的必要条件．
在第二部分即本文的第四章中，我们考虑了一类具有两点边值条件的多解常微分方程的最优控制问题．由于控制区域及指标泛函可能非凸，那么相应的惩罚问题可能无解．在这样的情形下，我们引入了新的工具即松弛控制理论．通过构造松弛惩罚问题再取极限，从而得到了原问题最优对所满足的Pontryagin’s最大值原理．
第三部分包含了本文的第五章，本章分为两节．第一节，在非凸情形下我们致力于一类只有两个解的半线性椭圆型方程的最优控制问题，通过对状态方程解性质的分析并建立适当的能量估计，我们证明了松弛控制方法对本问题的适用性并得到了最优对的Pontryagin's最大值原理．在第二节，采用松弛控制方法我们证明了一类最优控制问题的存在性及不存在性结果．
This dissertation investigates mainly the optimal control problems for severalclasses of multi-solution elliptic type equations.Firstly,under the assumption of con-vexity,we consider an optimal boundary control problem for semilinear multi-solutionelliptic type equations and prove the varional inequality.Secondly,in the non-convexcase,using the relaxed controls we study an optimal control problem for a multi-solutionordinary differential equation with two-point boundary value condition and that for aclass of semilinear elliptic type equations which admit exactly two solutions.We obtainthe Pontryagin's maximum principle.Finally we discuss the existence and nonexistenceof an optimal control problem governed by a class of semilinear elliptic equations.
This dissertation consists of three parts.
In the first part,i.e.Chapter 2 and Chapter 3 of this dissertation,we discussoptimal boundary control problems for a semilinear elliptic type equation with linearboundary condition and nonlinear boundary condition,respectively.Under the convexcase,by constructing the corresponding penalizing problem and passing to the limit,we give the necessary conditions for the optimal pairs of the original problem in theform of variational inequality.
In the second part,i.e.Chapter 4 of this dissertation,we consider an optimalcontrol problem for a class of multi-solution ordinary differential equations with two-point boundary value condition.Since the control domain and the objective functionalmay be non-convex,the corresponding penalizing problem may have no solution.Inthis case,we introduce a new tool,i.e.relaxed control theory.By constructing relaxedpenalizing problem and then passing to the limit,we obtain the Pontryagin's maximumprinciple for the optimal pairs of the original problem.
The third part consists of Chapter 5 and this chapter consists of two sections.Inthe first section,under the non-convex case we are devoted to investigate an optimalcontrol problem for a class of semilinear elliptic type equations which admit exactly twosolutions.By analyzing the property of a soltuion of the state equation and establishing the proper energy estimate,we prove the relaxed control method is fit for our problemand gain the Pontryagin's maximum principle.In the second section,by using relaxedcontrols we prove the existence and nonexistence results of a class of optimal controlproblems.
全文共分为三部分内容：
在第一部分即本文的第二章和第三章中，我们分别讨论了带有线性边界条件以及非线性边界条件的半线性椭圆型方程的最优边界控制问题．在凸性条件下，通过构造相应的惩罚问题并取极限，我们以变分不等式的形式给出了原始问题最优对所满足的必要条件．
在第二部分即本文的第四章中，我们考虑了一类具有两点边值条件的多解常微分方程的最优控制问题．由于控制区域及指标泛函可能非凸，那么相应的惩罚问题可能无解．在这样的情形下，我们引入了新的工具即松弛控制理论．通过构造松弛惩罚问题再取极限，从而得到了原问题最优对所满足的Pontryagin’s最大值原理．
第三部分包含了本文的第五章，本章分为两节．第一节，在非凸情形下我们致力于一类只有两个解的半线性椭圆型方程的最优控制问题，通过对状态方程解性质的分析并建立适当的能量估计，我们证明了松弛控制方法对本问题的适用性并得到了最优对的Pontryagin's最大值原理．在第二节，采用松弛控制方法我们证明了一类最优控制问题的存在性及不存在性结果．
This dissertation investigates mainly the optimal control problems for severalclasses of multi-solution elliptic type equations.Firstly,under the assumption of con-vexity,we consider an optimal boundary control problem for semilinear multi-solutionelliptic type equations and prove the varional inequality.Secondly,in the non-convexcase,using the relaxed controls we study an optimal control problem for a multi-solutionordinary differential equation with two-point boundary value condition and that for aclass of semilinear elliptic type equations which admit exactly two solutions.We obtainthe Pontryagin's maximum principle.Finally we discuss the existence and nonexistenceof an optimal control problem governed by a class of semilinear elliptic equations.
This dissertation consists of three parts.
In the first part,i.e.Chapter 2 and Chapter 3 of this dissertation,we discussoptimal boundary control problems for a semilinear elliptic type equation with linearboundary condition and nonlinear boundary condition,respectively.Under the convexcase,by constructing the corresponding penalizing problem and passing to the limit,we give the necessary conditions for the optimal pairs of the original problem in theform of variational inequality.
In the second part,i.e.Chapter 4 of this dissertation,we consider an optimalcontrol problem for a class of multi-solution ordinary differential equations with two-point boundary value condition.Since the control domain and the objective functionalmay be non-convex,the corresponding penalizing problem may have no solution.Inthis case,we introduce a new tool,i.e.relaxed control theory.By constructing relaxedpenalizing problem and then passing to the limit,we obtain the Pontryagin's maximumprinciple for the optimal pairs of the original problem.
The third part consists of Chapter 5 and this chapter consists of two sections.Inthe first section,under the non-convex case we are devoted to investigate an optimalcontrol problem for a class of semilinear elliptic type equations which admit exactly twosolutions.By analyzing the property of a soltuion of the state equation and establishing the proper energy estimate,we prove the relaxed control method is fit for our problemand gain the Pontryagin's maximum principle.In the second section,by using relaxedcontrols we prove the existence and nonexistence results of a class of optimal controlproblems.
引文
[1] Pontryagin L S, Boltyanskii V G, Gamkrelidze R V, et al. The Mathematical Theory of Optimal Processes[M]. New York: Interscience Publishers, 1962.
[2] Lions J L. Contr(?)le Optimal de Syst(?)mes Gouvern(?)s Par Des Equations aux D(?)riv(?)es Partielles[M]. Paris: Dunod, 1968.
[3] Lions J L. Optimal Control of Systems Governed by Partial Differential Equations[M]. New York: Springer-Verlag, 1971.
[4] Li X J. Maximum principle of optimal periodic control for functiomal differential systems[J]. J Optim Theory Appl, 1986, 50: 421-429.
[5] Li X J, Chow S N. Maximum principle of optimal control for functiomal differential systems[J]. J Optim Theory Appl, 1987, 54: 335-360.
[6] Li X J, Yao Y. Time optimal control of distributed parameter sysyems[J]. Scientia Sinica, 1981, 24:455-465.
[7] Li X J, Yao Y. Maximum principle of distributed parameter sysyems with time lags[J]. Lecture Notes in Control and Information Sci, Springer-Verlag, 1985, 410-427.
[8] Fattorini H O. The time optimal control problems in Banach spaces[J]. Appl Math Optim, 1974, 1: 163-188.
[9] Fattorini H O. The maximum principle for nonlinear nonconvex systems in infinite dimensional spaces[J]. Lecture Notes in Control and Information Sci, Springer-Verlag, 1985,75: 162-178.
[10] Fattorini H O. A unified theory of necessary conditions for nonlinear nonconvex systems[J]. Appl Math Optim, 1987, 15: 141-185.
[11] Fattorini H O. Optimal control problems for distributed parameter systems in Banach spaces[J]. Appl Math Optim, 1993, 28: 225-257.
[12] Fattorini H O. The maximum principle for linear infinite dimentional control systems with state constraints[J]. Discrete Continuous Dynam Systems, 1995, 1:77-101.
[13] Fattorini H O. Infinite Dimensional Optimization Theory and Optimal Control[M]. UK:Cambridge Univ Press, 1999.
[14] Fattorini H O. Optimal control problems with state contraints for semilinear distributed parameter systems[J]. J Optim Theory Appli, 1996, 88: 25-59.
[15] Fattorini H O. The maximum principle for control systems described by linear parabolic equations[J]. J Math Anal Appl, 2001, 259: 630-651.
[16] Li X, Yong J. Optimal Control Theory for Infinite Dimensional Systems[M]. Boston: Birkh(?)user, 1995.
[17] Barbu V. Analysis and Control of Nonlinear Infinite Dimensional Systems[M]. New York: Academic Press, 1993.
[18] Lasiecka I, Triggiani R. Control Theory for Partial Differential Equations: Continuous and Approximation Theories, vol. I and II[M]. UK: Cambridge Univ Press,2000.
[19] Clarke F H. Optimization and Nonsmooth Analysis[M]. New York: John Wiley, 1983.
[20] Neittaanmaki P, Tiba D, Sprekels J. Optimization of Elliptic Systems. Theory and Applications[M]. New York: Springer, 2006.
[21] Casas E, Troltzsch F. Second-order necessary optimality conditions for some state-constrained control problems of semilinear elliptic equation[J]. Appli Math Optim, 1999, 39: 211-227.
[22] Fattorini H O, Frankowska H. Necessary conditions for infinite dimensional control problems[J]. Math Control Signals Systems, 1991, 4: 225-257.
[23] Mackenroth U. On some elliptic optimal control problems with state constraints[J]. Optimization, 1986, 17: 595-607.
[24] Yong J M. Pontryagin maximum principle for semilinear second order elliptic partial differential equations and variational inequalities with state constraints[J]. Differential Integral Equations, 1992, 5: 1307-1334.
[25] Bergounioux M, Troltzsch F. Optimal conditions and generalized bang-bang principle for a state constrained semilinear parabolic problems[J]. Numer Funct Anal Optim, 1996, 15: 517-537.
[26] Bors D. Superlinear elliptic systems with distributed abd boundary controls[J]. Control Cybernet, 2005, 34(4): 987-1004.
[27] Bors D, Walczak S. Optimal control elliptic systems with distributed and boundary controls[J]. Nonlinear Anal, 2005, 63: el367-el376.
[28] Mackenroth U. On some elliptic optimal control problems with state constraints[J]. Optimization, 1986, 17: 595-607.
[29] Mordukhovich B S, Zhang K. Minimax control of parabolic systems with Dirichlet boundary conditions and state constraints[J]. Appli Math Optim, 1997, 36: 323-360.
[30] Mordukhovich B S, Zhang K. Dirichlet boundary condition control of parabolic systems with pointwise state constraints[J]. Internat Ser Numer Math, 1998, 126: 223-235.
[31] Raymond J P, Zidani H. Pontryagin's principle for state-constrained control problems governed by parabolic equations with unbounded controls[J]. SIAM J Control Optim, 1998, 36(6): 1853-1879.
[32] Alibert J J, Raymond J P. Boundary control of semilinear elliptic equations with discontinuous leading coefficient and unbounded controls[J]. Numer Funct Anal Optim, 1997, 18: 235-250.
[33] Bonnans J F, Tiba D. Pontryagin's principle in the control of semilinear elliptic variational inequalities[J]. Appli Math Optim, 1991, 23: 273-289.
[34] Casas E. Boundary control of semiliner elliptic equations with pointwise state constraints[J]. SIAM J Control Optim, 1993. 31: 993-1006.
[35] Arada N, Raymond J P. State-constrained relaxed problems for semilinear elliptic equations[J]. J Math Anal Appl, 1998, 223: 248-271.
[36] Bonnans J F, Casas E. An extension of Pontryagin's principle for state-constrained optimal control of semilinear elliptic equations and variational inequalities[J]. SIAM J Control Optim, 1995, 33(1): 274-298.
[37] Casas E. Pontryagin's principle for state-constrained boundary control problems of semilinear parabolic equations[J]. SIAM J COntrol Optim, 1997, 35(4): 1297-1327.
[38] Casas E, Mateos M. Second order optimality conditions for semilinear elliptic control problems with finitely many state constraints[J]. SIAM J COntrol Optim, 2002, 40(5):1431-1454.
[39] Casas E, Yong J. Maximum principle for state-constrained optimal control problems governed by quasilinear elliptic equations, Differential Integral Equations[J]. 1995, 8(1):1-18.
[40] Crandall M G, Rabinowitz P. Bifurcation peturbation of simple eigenvalues, and linearized stability[J]. Arch Rational Mech Anal, 1973, 53: 161-180.
[41] Lions J L. Contr(?)le de Syst(?)mes Distribu(?)s Singuliers[M]. Paris: Dunod, 1983.
[42] Lions J L. Some Methods in the Mathematical Analysis of System and Their Control[M]. Beijing: Science Press, 1981.
[43] Bonnans J F, Casas E. Optimal control of semilinear multistate systems with state constraints[J]. SIAM J Control Optim, 1989, 27(2):446-455.
[44] Casas E, Kavian O, Puel J P. Optimal control of an ill-posed elliptic semilinear equation with an exponential non linearity[J]. ESAIM: COCV, 1998,3: 361-380.
[45] Abergel F, Casas E. Some optimal control problems of multistate equations appearing in fluid mechanichs[J]. RAIRO Mod(?)l Math Anal Num(?)r, 1993, 27(2): 223-247.
[46] Gao H. Optimality condition for a class of semi-linear elliptic equations[J]. Acta Math Sinica, 2001, 44(2): 319-332.
[47] Gao H, Pavel N H. Optimal control problems for a class of semilinear multisolution elliptic equations[J]. J Optim Theory Appl, 2003, 1: 353-380.
[48] Wang G S. Optimal control problems governed by non well posed semilinear elliptic equation[J]. Nonlinear Anal, 2000, 42(5): 789-801.
[49] Wang G S. Optimal control problems governed by non-well-posed semilinear elliptic equation[J]. Nonlinear Anal, 2002, 49: 315-333.
[50] Deng Y B, Wang G S. Optimal control of some semilinear elliptic equations with critical exponent[J]. Nonlinear Anal, 1999, 36: 915-922.
[51] Wang G S, Wang L J. Maximum principle for optimal control of non-well-posed elliptic differential equations[J]. Nonlinear Anal, 2003, 52: 41-67.
[52] Young L C. On approximation by polygon's in the calculus of variations[J]. Proceeding of the Royal Society, (A), 1933, 14: 325-341.
[53] Young L C. Generalized curves and the existence of an attained absolute minimum in the calculus of variations[J]. C R Sci Letters Varsovie, C Ⅲ, 1937, 30: 212-234.
[54] Young L C. Necessary conditions in the calculus of variations[J]. Acta Math, 1938, 69: 239-258.
[55] McShane E J. Relaxed controls and variational problems[J]. SIAM J Control, 1967, 5:438-485.
[56] Filippov A F. On certain questions in the theory of optimal control[J]. SIAM J Control Ser A, Control, 1962, 1: 76-84.
[57]Gamkrelidze R V．On sliding optimal states[J]．Dokl Akad Nauk SSSR，1962，143： 1243-1245．
[58]Warge J．Relaxed variational problems[J]J Math Anal Appl，1962，4： 111-128．
[59]Warga J．Optimal Control of Differential and Functional Equations[M]．New York： Academic Press，1972．
[60]Lou H．Existence and nonexistence results of an optimal control problem by using relaxed control[J]．SIAM J Control Optim，2007，46(6)： 1923-1941．
[61]Adams R A．Sobolev Spaces[M]．New York： Academic Press，1975．
[62]Amann H．Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces[J]．SIAM Rev，1976，18： 620-709．
[63]Evans L C．Partial Differential Equations[M]．Providence： AMS，1998．
[64]雍炯敏，楼红卫．最优控制理论简明教程[M]．北京：高等教育出版社，2006．
[65]Hsu T S．Existence of multiple positive solutions of subcritical semilinear elliptic problems in exterior strip domains[J]．Nonlinear Anal，2006，64： 1203-1228．
[66]Courant R，Hilbert D．Methods of Mathematical Physics-Volume I[M]．New York： Interscience，1953．
[67]魏光祖，袁忠信．索伯列夫空间与偏微分方程[M]．开封：河南大学出版社，1994．
[68]Lou H．Maximum principle of optimal control for degenerate quasi-linear elliptic equations[J]．SIAM J Control Optim，2003，42(1)： 1-23．
[69]Amann H，Crandall M G．On some existence theorems for semi-linear elliptic equations[J]．Indiana Univ Math J，1978，27(5)： 779-790．
[70]Lou H．Existence of optimal controls for semilinear elliptic equations without Cesari type conditions[J]．ANZIAM J，2003，45： 115-131．
[2] Lions J L. Contr(?)le Optimal de Syst(?)mes Gouvern(?)s Par Des Equations aux D(?)riv(?)es Partielles[M]. Paris: Dunod, 1968.
[3] Lions J L. Optimal Control of Systems Governed by Partial Differential Equations[M]. New York: Springer-Verlag, 1971.
[4] Li X J. Maximum principle of optimal periodic control for functiomal differential systems[J]. J Optim Theory Appl, 1986, 50: 421-429.
[5] Li X J, Chow S N. Maximum principle of optimal control for functiomal differential systems[J]. J Optim Theory Appl, 1987, 54: 335-360.
[6] Li X J, Yao Y. Time optimal control of distributed parameter sysyems[J]. Scientia Sinica, 1981, 24:455-465.
[7] Li X J, Yao Y. Maximum principle of distributed parameter sysyems with time lags[J]. Lecture Notes in Control and Information Sci, Springer-Verlag, 1985, 410-427.
[8] Fattorini H O. The time optimal control problems in Banach spaces[J]. Appl Math Optim, 1974, 1: 163-188.
[9] Fattorini H O. The maximum principle for nonlinear nonconvex systems in infinite dimensional spaces[J]. Lecture Notes in Control and Information Sci, Springer-Verlag, 1985,75: 162-178.
[10] Fattorini H O. A unified theory of necessary conditions for nonlinear nonconvex systems[J]. Appl Math Optim, 1987, 15: 141-185.
[11] Fattorini H O. Optimal control problems for distributed parameter systems in Banach spaces[J]. Appl Math Optim, 1993, 28: 225-257.
[12] Fattorini H O. The maximum principle for linear infinite dimentional control systems with state constraints[J]. Discrete Continuous Dynam Systems, 1995, 1:77-101.
[13] Fattorini H O. Infinite Dimensional Optimization Theory and Optimal Control[M]. UK:Cambridge Univ Press, 1999.
[14] Fattorini H O. Optimal control problems with state contraints for semilinear distributed parameter systems[J]. J Optim Theory Appli, 1996, 88: 25-59.
[15] Fattorini H O. The maximum principle for control systems described by linear parabolic equations[J]. J Math Anal Appl, 2001, 259: 630-651.
[16] Li X, Yong J. Optimal Control Theory for Infinite Dimensional Systems[M]. Boston: Birkh(?)user, 1995.
[17] Barbu V. Analysis and Control of Nonlinear Infinite Dimensional Systems[M]. New York: Academic Press, 1993.
[18] Lasiecka I, Triggiani R. Control Theory for Partial Differential Equations: Continuous and Approximation Theories, vol. I and II[M]. UK: Cambridge Univ Press,2000.
[19] Clarke F H. Optimization and Nonsmooth Analysis[M]. New York: John Wiley, 1983.
[20] Neittaanmaki P, Tiba D, Sprekels J. Optimization of Elliptic Systems. Theory and Applications[M]. New York: Springer, 2006.
[21] Casas E, Troltzsch F. Second-order necessary optimality conditions for some state-constrained control problems of semilinear elliptic equation[J]. Appli Math Optim, 1999, 39: 211-227.
[22] Fattorini H O, Frankowska H. Necessary conditions for infinite dimensional control problems[J]. Math Control Signals Systems, 1991, 4: 225-257.
[23] Mackenroth U. On some elliptic optimal control problems with state constraints[J]. Optimization, 1986, 17: 595-607.
[24] Yong J M. Pontryagin maximum principle for semilinear second order elliptic partial differential equations and variational inequalities with state constraints[J]. Differential Integral Equations, 1992, 5: 1307-1334.
[25] Bergounioux M, Troltzsch F. Optimal conditions and generalized bang-bang principle for a state constrained semilinear parabolic problems[J]. Numer Funct Anal Optim, 1996, 15: 517-537.
[26] Bors D. Superlinear elliptic systems with distributed abd boundary controls[J]. Control Cybernet, 2005, 34(4): 987-1004.
[27] Bors D, Walczak S. Optimal control elliptic systems with distributed and boundary controls[J]. Nonlinear Anal, 2005, 63: el367-el376.
[28] Mackenroth U. On some elliptic optimal control problems with state constraints[J]. Optimization, 1986, 17: 595-607.
[29] Mordukhovich B S, Zhang K. Minimax control of parabolic systems with Dirichlet boundary conditions and state constraints[J]. Appli Math Optim, 1997, 36: 323-360.
[30] Mordukhovich B S, Zhang K. Dirichlet boundary condition control of parabolic systems with pointwise state constraints[J]. Internat Ser Numer Math, 1998, 126: 223-235.
[31] Raymond J P, Zidani H. Pontryagin's principle for state-constrained control problems governed by parabolic equations with unbounded controls[J]. SIAM J Control Optim, 1998, 36(6): 1853-1879.
[32] Alibert J J, Raymond J P. Boundary control of semilinear elliptic equations with discontinuous leading coefficient and unbounded controls[J]. Numer Funct Anal Optim, 1997, 18: 235-250.
[33] Bonnans J F, Tiba D. Pontryagin's principle in the control of semilinear elliptic variational inequalities[J]. Appli Math Optim, 1991, 23: 273-289.
[34] Casas E. Boundary control of semiliner elliptic equations with pointwise state constraints[J]. SIAM J Control Optim, 1993. 31: 993-1006.
[35] Arada N, Raymond J P. State-constrained relaxed problems for semilinear elliptic equations[J]. J Math Anal Appl, 1998, 223: 248-271.
[36] Bonnans J F, Casas E. An extension of Pontryagin's principle for state-constrained optimal control of semilinear elliptic equations and variational inequalities[J]. SIAM J Control Optim, 1995, 33(1): 274-298.
[37] Casas E. Pontryagin's principle for state-constrained boundary control problems of semilinear parabolic equations[J]. SIAM J COntrol Optim, 1997, 35(4): 1297-1327.
[38] Casas E, Mateos M. Second order optimality conditions for semilinear elliptic control problems with finitely many state constraints[J]. SIAM J COntrol Optim, 2002, 40(5):1431-1454.
[39] Casas E, Yong J. Maximum principle for state-constrained optimal control problems governed by quasilinear elliptic equations, Differential Integral Equations[J]. 1995, 8(1):1-18.
[40] Crandall M G, Rabinowitz P. Bifurcation peturbation of simple eigenvalues, and linearized stability[J]. Arch Rational Mech Anal, 1973, 53: 161-180.
[41] Lions J L. Contr(?)le de Syst(?)mes Distribu(?)s Singuliers[M]. Paris: Dunod, 1983.
[42] Lions J L. Some Methods in the Mathematical Analysis of System and Their Control[M]. Beijing: Science Press, 1981.
[43] Bonnans J F, Casas E. Optimal control of semilinear multistate systems with state constraints[J]. SIAM J Control Optim, 1989, 27(2):446-455.
[44] Casas E, Kavian O, Puel J P. Optimal control of an ill-posed elliptic semilinear equation with an exponential non linearity[J]. ESAIM: COCV, 1998,3: 361-380.
[45] Abergel F, Casas E. Some optimal control problems of multistate equations appearing in fluid mechanichs[J]. RAIRO Mod(?)l Math Anal Num(?)r, 1993, 27(2): 223-247.
[46] Gao H. Optimality condition for a class of semi-linear elliptic equations[J]. Acta Math Sinica, 2001, 44(2): 319-332.
[47] Gao H, Pavel N H. Optimal control problems for a class of semilinear multisolution elliptic equations[J]. J Optim Theory Appl, 2003, 1: 353-380.
[48] Wang G S. Optimal control problems governed by non well posed semilinear elliptic equation[J]. Nonlinear Anal, 2000, 42(5): 789-801.
[49] Wang G S. Optimal control problems governed by non-well-posed semilinear elliptic equation[J]. Nonlinear Anal, 2002, 49: 315-333.
[50] Deng Y B, Wang G S. Optimal control of some semilinear elliptic equations with critical exponent[J]. Nonlinear Anal, 1999, 36: 915-922.
[51] Wang G S, Wang L J. Maximum principle for optimal control of non-well-posed elliptic differential equations[J]. Nonlinear Anal, 2003, 52: 41-67.
[52] Young L C. On approximation by polygon's in the calculus of variations[J]. Proceeding of the Royal Society, (A), 1933, 14: 325-341.
[53] Young L C. Generalized curves and the existence of an attained absolute minimum in the calculus of variations[J]. C R Sci Letters Varsovie, C Ⅲ, 1937, 30: 212-234.
[54] Young L C. Necessary conditions in the calculus of variations[J]. Acta Math, 1938, 69: 239-258.
[55] McShane E J. Relaxed controls and variational problems[J]. SIAM J Control, 1967, 5:438-485.
[56] Filippov A F. On certain questions in the theory of optimal control[J]. SIAM J Control Ser A, Control, 1962, 1: 76-84.
[57]Gamkrelidze R V．On sliding optimal states[J]．Dokl Akad Nauk SSSR，1962，143： 1243-1245．
[58]Warge J．Relaxed variational problems[J]J Math Anal Appl，1962，4： 111-128．
[59]Warga J．Optimal Control of Differential and Functional Equations[M]．New York： Academic Press，1972．
[60]Lou H．Existence and nonexistence results of an optimal control problem by using relaxed control[J]．SIAM J Control Optim，2007，46(6)： 1923-1941．
[61]Adams R A．Sobolev Spaces[M]．New York： Academic Press，1975．
[62]Amann H．Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces[J]．SIAM Rev，1976，18： 620-709．
[63]Evans L C．Partial Differential Equations[M]．Providence： AMS，1998．
[64]雍炯敏，楼红卫．最优控制理论简明教程[M]．北京：高等教育出版社，2006．
[65]Hsu T S．Existence of multiple positive solutions of subcritical semilinear elliptic problems in exterior strip domains[J]．Nonlinear Anal，2006，64： 1203-1228．
[66]Courant R，Hilbert D．Methods of Mathematical Physics-Volume I[M]．New York： Interscience，1953．
[67]魏光祖，袁忠信．索伯列夫空间与偏微分方程[M]．开封：河南大学出版社，1994．
[68]Lou H．Maximum principle of optimal control for degenerate quasi-linear elliptic equations[J]．SIAM J Control Optim，2003，42(1)： 1-23．
[69]Amann H，Crandall M G．On some existence theorems for semi-linear elliptic equations[J]．Indiana Univ Math J，1978，27(5)： 779-790．
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