积微分方程的时间最优控制和无限维空间中时间最优控制的Meyer逼近
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摘要
本文主要讨论了Banach空间中受控系统是一类半线性积微分方程的时间最优控制存在性以及受控系统分别是半线性和非线性方程的时间最优控制的Meyer逼近。
首先，作者研究了如下的受控系统其中A是Banach空间X中C_0-半群{T(t)，t≥0}的无穷小生成元，积分算子如下
(Sx)(t)=integral from n=0 to 1 k(t，τ)g(τ，x(τ))dτ。在得到了方程(1)的温和解的存在唯一性之后，作者着重对目标是固定点和活动靶两种情形的时间最优控制的存在性问题分别进行了详细的讨论，并得到了两个新的存在性结果。
其次，在Banach空间中，作者分别讨论了下面的半线性方程(2)以及非线性方程(3)的时间最优控制的Meyer逼近。
借助紧半群的一致算子拓扑收敛性和变换思想，作者构造了一串Meyer问题序列去逼近半线性发展方程(2)的时间最优控制闯题，从而揭示了时间最优控制问题与Meyer问题之间本质联系。同时，给出了时间最优控制的存在性证明的新方法。这里，算子B满足较弱的条件如B本征有界。
运用C_0-半群强算子拓扑收敛性和变换思想，作者重新构造了一串Meyer问题序列去逼近半线性发展方程(3)的时间最优控制问题，再次揭示了时间最优控制问题与Meyer问题之间的本质联系。作为结果，也得到了(3)的时间最优控制的存在性。这里，虽然半群的条件将低了，但是算子B要求强连续。关键词：C_0-半群；紧半群；存在性；温和解；时间最优控制；变换；Meyer逼近
In this thesis, we pay great attention to discuss the existence of time optimal control of system governed by a class of semi-linear integro-differential equations and Meyer Approximation of time optimal control of system governed by semi-linear equations and nonlinear equations respectively on infinite dimensional Banach spaces.
First, the following control system governed by semi-linear integro-differential equations (7) is considered
where A is the infinitesimal generator of a Co-semigroup, I = [0,1] and l is fixed and integro operator is given by
After getting the extisence of mild soultion of equations (7), we pay great attention to study the existence of time optimal controls about the target which are corresponding to a fixed point and a set varying with time respectively. As conclusion, we obtain two new existence results.
Second, another remarkable problem that Meyer approximation of time optimal control of system governed by semi-linear equations
and nonlinear equations
are also investigated respectively on infinite dimensional Banach spaces.
Using the uniformly operator topology of compact semigroup theory and transformation idea, a sequence of solution of Meyer problems are constructed to approach to a class of time optimal control problems of system governed by semi-linear equations (8) on infinite dimensional Banach spaces. A deep relation between Meyer problem and time optimal control is also presented. Consequently, the existence of time optimal control is obtained. Here, operator B satisfies relative weak conditions such as B satisfies essence bounded.
By virtue of the strongly operator topology of Co-semigroup theory and transformation idea, a sequence of solution of Meyer problems are constructed to approach to a class of time optimal control problems of system governed by nonlinear equations (9) on infinite dimensional Banach spaces. A deep relation between Meyer problem and time optimal control is also presented. Consequently, the existence of time optimal control is obtained. Though Co-semigroup is not as strong as compact semigroup, B satisfies relative strong condition such as B satisfies essence bounded and it is a strong continuous operator.
首先，作者研究了如下的受控系统其中A是Banach空间X中C_0-半群{T(t)，t≥0}的无穷小生成元，积分算子如下
(Sx)(t)=integral from n=0 to 1 k(t，τ)g(τ，x(τ))dτ。在得到了方程(1)的温和解的存在唯一性之后，作者着重对目标是固定点和活动靶两种情形的时间最优控制的存在性问题分别进行了详细的讨论，并得到了两个新的存在性结果。
其次，在Banach空间中，作者分别讨论了下面的半线性方程(2)以及非线性方程(3)的时间最优控制的Meyer逼近。
借助紧半群的一致算子拓扑收敛性和变换思想，作者构造了一串Meyer问题序列去逼近半线性发展方程(2)的时间最优控制闯题，从而揭示了时间最优控制问题与Meyer问题之间本质联系。同时，给出了时间最优控制的存在性证明的新方法。这里，算子B满足较弱的条件如B本征有界。
运用C_0-半群强算子拓扑收敛性和变换思想，作者重新构造了一串Meyer问题序列去逼近半线性发展方程(3)的时间最优控制问题，再次揭示了时间最优控制问题与Meyer问题之间的本质联系。作为结果，也得到了(3)的时间最优控制的存在性。这里，虽然半群的条件将低了，但是算子B要求强连续。关键词：C_0-半群；紧半群；存在性；温和解；时间最优控制；变换；Meyer逼近
In this thesis, we pay great attention to discuss the existence of time optimal control of system governed by a class of semi-linear integro-differential equations and Meyer Approximation of time optimal control of system governed by semi-linear equations and nonlinear equations respectively on infinite dimensional Banach spaces.
First, the following control system governed by semi-linear integro-differential equations (7) is considered
where A is the infinitesimal generator of a Co-semigroup, I = [0,1] and l is fixed and integro operator is given by
After getting the extisence of mild soultion of equations (7), we pay great attention to study the existence of time optimal controls about the target which are corresponding to a fixed point and a set varying with time respectively. As conclusion, we obtain two new existence results.
Second, another remarkable problem that Meyer approximation of time optimal control of system governed by semi-linear equations
and nonlinear equations
are also investigated respectively on infinite dimensional Banach spaces.
Using the uniformly operator topology of compact semigroup theory and transformation idea, a sequence of solution of Meyer problems are constructed to approach to a class of time optimal control problems of system governed by semi-linear equations (8) on infinite dimensional Banach spaces. A deep relation between Meyer problem and time optimal control is also presented. Consequently, the existence of time optimal control is obtained. Here, operator B satisfies relative weak conditions such as B satisfies essence bounded.
By virtue of the strongly operator topology of Co-semigroup theory and transformation idea, a sequence of solution of Meyer problems are constructed to approach to a class of time optimal control problems of system governed by nonlinear equations (9) on infinite dimensional Banach spaces. A deep relation between Meyer problem and time optimal control is also presented. Consequently, the existence of time optimal control is obtained. Though Co-semigroup is not as strong as compact semigroup, B satisfies relative strong condition such as B satisfies essence bounded and it is a strong continuous operator.
引文
[1] N. U. Ahmed, Elements of Finite Dimensional System and Control Theory [M], Longman Scientific Technical, 1988.
[2] N. U. Ahmed and K. L. Teo, Optimal Control of Distribute Parameter Systems [M], Elsevier North Holland, Inc, New York, 1981.
[3] N. U. Ahmed, Semigroup Theory with Applications to System and Control [M], Pitman Research Notes in Maths series 246, Longman Scientific Technical, New York, 1991.
[4] H. Amann, Periodic Solutions of Semilinear Parabolic Equations [M], Nonlinear Analysis: "A collection of papers in Honor of Erich Rothe", Academic Press, New York, 1978, 1-29.
[5] H. Amann, Invariant Sets and Existence for Semilinear Parabolic and Elliptic Systems, J. Mathematical Analysis and Applications, 65, 1978, 432-469.
[6] J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis [M]. Wiley-Interscience, 1984.
[7] J. P. Aubin and H. Frankowska, Set-value Analysis [M]. Boston Birkhauser, 1990.
[8] R. Aubin, C. Coucoubetis, P.H. Ho, X. Nicollin, A. Olivero, J. Sifakis, S. Yovine, The Algorithmic Analysis of Hybrid Systems, Theorem Computation Science, 138, 1995, 3-34.
[9] R. Aubin, D. L. Dill, A Theory of Timed Automata, Theoretical Computer Science, 126, 1994, 183-235.
[10] V. N. Afanasev, V. B. Kolmanovskii and V. R. Nosov, Mathematical Theory of Control Systems Design [M], Kluwer Academic Publishers, 1996.
[11] Balder E., Necessary and Sufficient Conditions for L1—strong-weak Lower Semicon-tinuity of Integral Functional, Nonlinear Analysis TMA, 11, 1987, 1399-1404.
[12] E. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems [M], Academic Press Boston, 1993.
[13] D. P. Bertsekas, Dynamic Programming and Optimal Control [M], Athena Scientific, 1995.
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[15] Carja, On the Minimal Time Function for Distributed Control Systems in Banach spaces, Journal of Optimization Theory, Appl. 44, 1984, 397-406.
[16] Carja, On the Continuity of The Minimal Time Function for Distributed Control Systems, Boll. Un. Math. Ital., 6, 1985, 293-302.
[17] Qian Chen, JinRong Wang and ZiJiang Luo, n~+n~-n~+-GAS Monte Carlo Simulation, J. Guizhou University, 22, 2005, 239-243.
[18] Yu. V. Egorov, Optimal Control in Banach Spaces, Dokl. Acad. Nank SSSR, Russian, 150, 1963, 241-244.
[19] H. O. Fattorini, Time-optlmal Control of Solutions of Operational Differential Equations, SIAM Journal on Control, 2, 1964, 54-59.
[20] H. O. Fattorini, Some Remarks on Complete Controllabitility, SIAM Journal on Control 4, 1966, 686-694.
[21] H. O. Fattorini, Control in Finite Time of Differential Equations in Banach Space, Communication on Pure and Applied Mathematics 19, 1966, 17-34.
[22] H. O. Fattorini, On Complete Controllability of Linear System, Journal of Diffrential Equations 3, 1967, 391-402.
[23] H. O. Fattorini, The Time Optimal Control Problems in Banach Space, Applied Mathematics Optimization 1, 1974, 163-188.
[24] H. O. Fattorini, The Time-optimal Control for Boundary Control of the Heat Equation, Calculus of Variations and Control Theory, Academic Press, New York, San Francisco, London, 1976.
[25] H. O. Fattorini, The Time Optimal Problem for Distributed Control of Systems Described by the Wave Equation, Proceedings of Conference on Control Theory of Systems Governed by Partial Differential Equations, Naval Surface Weqpons Centre, Baltimore, Academic Press, New York, 1976, 151-176.
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[27] H. O. Fattorini, The Maximum Principle in Infinite Dimensional, Discrete and Continuous Dynanical Systems, 6(3), 2000, 557-574.
[28] R. V. Gamlcrelidze, Theory of Time-optimal Control Progrcesses for Linear System, Izvestia Akad. Nank, SSSR, 22, 1958, 449-474.
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[39] J. P. LaSalle, The Time Optimal Control Problem, Contributions to the Theory of Nonlinear Oscillations 5, Princeton University Press, Princeton, 1960, 1-24.
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[41] E. B. Lee and L. Markus, Fundations of Optimal Control Theory[M], Wiley, New York, 1967.
[42] J. L. Lions, Optimal Control of Systems Goverened by Paratial Differential Equations[M], Springer-Verlag, New York, 1971.
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[45] V. Lakshmikantham, S. Leela, Nonlinear differential equations in abstract space[M], Pergamon Press, New York, 1981.
[46] V. Lakshmikantham, Some problems in integro-differential equations of volterra type, J. Integral Equations 10, 1985, 137-146.
[47] Jiang Qiyuan, Mathematical Model[M], published by high education, 1998.
[48] Li Xunjing and Yong Jiongmin, Optimal Control Theory for Infinite Dimensional Systems[M], Birkhauser Boston, 1995.
[49] Li Xunjing and Yong Jiongmin, Basic Theory of Control[M], published by high education, perking, 2002.
[50] Li Shugang and Wang Gengsheng, The Time Optimal Control of the Boussinesq Equations, Numerical Functional Analysis and Optimization, 24, 2003, 163-180.
[51] C. C. H. Ma. and T. W. T. Qian, Rapid Tracking with Automatic Trajectory Optimization for Speed, ASME. Dyn. Sys. Meas. and Control, 121, 1999, 697-702.
[52] A. Pazy, Semigroup of Linear Operators and Applications to Partial Differential Equations[M], Springer-Verlag, New York, 1983.
[53] R. Pallu dela Barriere, Optimal Control Theory[M], W. B. Saunders Company, Philadaphia, 1967.
[54] E. Polak, Computation Methods in Optimization A Unified Approach[M], Academic Press, New York and London, 1971.
[55] B. Piccolic, Time-optimal Control Problems for the Swing and the Ski, International Journal of Control, 62(6), 1995, 1409-1429.
[56] D. L. Pepyne and C. G. Cassandras, Optimal Control of Linear Systems in Manufacturing, Proceeding IEEE, 88(7), 1108-1123.[57] S. Pettersson and B. Lennartson, Time Optimal Control and Disturbanc Compensation for A class of Hybrid Systems, In Proceeding 13th IFAC World Congress, 1996, 281-286.
[58] Z. Shiller, On singular Time Optimal Control Along Specified Paths, IEEE Transactions on Robotics and Automation, 10(4), 1994, 561-571.
[59] Z. Shiller, Time Energy Optimal Control of Articulated Systems with Geometric Path Constraints, In IEEE International Conference on Robotics and Automation, San Diego, California, 1994, 2680-2685.
[60] I. Sergam, J. Werschnik, and E. K. U. Gross, Optimal Control of Time-dependent Targets, Physical Review A 71, 053810, 2005.
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[69] JinRong Wang and Xiang Xiaoling, Time Optimal Controls of System Governed by Integro-Differential Equations on Banach Spaces, J. Guizhou University, 22, 2005, 119-125.
[70] JingRong Wang, Peng Yunfei and X. Xiang, Meyer Approxrimation of Time Optimal Control of System Governed by Semi-linear Equation on Infinite Dimensional Banach Spaces, to appear.
[71] JingRong Wang and W. Wei Meyer Approximation of Time Optimal Control System Governed by Nonlinear Equation on Infinite Dimensional Space, to appear.
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[78] Tao Yang, Impulsive Control Theory[M], Springer-Verlag BerlinHeidelberg, 2001.
[79] Jiongmin Yong, Time Optimal Control for Semilinear Distributed Parameter Systems-Existence Theory and Necessary Conditions, J. Kodai Math., 14, 1991, 239-253.
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[2] N. U. Ahmed and K. L. Teo, Optimal Control of Distribute Parameter Systems [M], Elsevier North Holland, Inc, New York, 1981.
[3] N. U. Ahmed, Semigroup Theory with Applications to System and Control [M], Pitman Research Notes in Maths series 246, Longman Scientific Technical, New York, 1991.
[4] H. Amann, Periodic Solutions of Semilinear Parabolic Equations [M], Nonlinear Analysis: "A collection of papers in Honor of Erich Rothe", Academic Press, New York, 1978, 1-29.
[5] H. Amann, Invariant Sets and Existence for Semilinear Parabolic and Elliptic Systems, J. Mathematical Analysis and Applications, 65, 1978, 432-469.
[6] J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis [M]. Wiley-Interscience, 1984.
[7] J. P. Aubin and H. Frankowska, Set-value Analysis [M]. Boston Birkhauser, 1990.
[8] R. Aubin, C. Coucoubetis, P.H. Ho, X. Nicollin, A. Olivero, J. Sifakis, S. Yovine, The Algorithmic Analysis of Hybrid Systems, Theorem Computation Science, 138, 1995, 3-34.
[9] R. Aubin, D. L. Dill, A Theory of Timed Automata, Theoretical Computer Science, 126, 1994, 183-235.
[10] V. N. Afanasev, V. B. Kolmanovskii and V. R. Nosov, Mathematical Theory of Control Systems Design [M], Kluwer Academic Publishers, 1996.
[11] Balder E., Necessary and Sufficient Conditions for L1—strong-weak Lower Semicon-tinuity of Integral Functional, Nonlinear Analysis TMA, 11, 1987, 1399-1404.
[12] E. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems [M], Academic Press Boston, 1993.
[13] D. P. Bertsekas, Dynamic Programming and Optimal Control [M], Athena Scientific, 1995.
[14] J. E. Bobrow, S. Dubowsky, and J. S. Gibson, Time Optimal Control of Robotic Manipulators Along Speified Paths, International Journal of Robotics Research, 4, 1985, 3-17.
[15] Carja, On the Minimal Time Function for Distributed Control Systems in Banach spaces, Journal of Optimization Theory, Appl. 44, 1984, 397-406.
[16] Carja, On the Continuity of The Minimal Time Function for Distributed Control Systems, Boll. Un. Math. Ital., 6, 1985, 293-302.
[17] Qian Chen, JinRong Wang and ZiJiang Luo, n~+n~-n~+-GAS Monte Carlo Simulation, J. Guizhou University, 22, 2005, 239-243.
[18] Yu. V. Egorov, Optimal Control in Banach Spaces, Dokl. Acad. Nank SSSR, Russian, 150, 1963, 241-244.
[19] H. O. Fattorini, Time-optlmal Control of Solutions of Operational Differential Equations, SIAM Journal on Control, 2, 1964, 54-59.
[20] H. O. Fattorini, Some Remarks on Complete Controllabitility, SIAM Journal on Control 4, 1966, 686-694.
[21] H. O. Fattorini, Control in Finite Time of Differential Equations in Banach Space, Communication on Pure and Applied Mathematics 19, 1966, 17-34.
[22] H. O. Fattorini, On Complete Controllability of Linear System, Journal of Diffrential Equations 3, 1967, 391-402.
[23] H. O. Fattorini, The Time Optimal Control Problems in Banach Space, Applied Mathematics Optimization 1, 1974, 163-188.
[24] H. O. Fattorini, The Time-optimal Control for Boundary Control of the Heat Equation, Calculus of Variations and Control Theory, Academic Press, New York, San Francisco, London, 1976.
[25] H. O. Fattorini, The Time Optimal Problem for Distributed Control of Systems Described by the Wave Equation, Proceedings of Conference on Control Theory of Systems Governed by Partial Differential Equations, Naval Surface Weqpons Centre, Baltimore, Academic Press, New York, 1976, 151-176.
[26] H. O. Fattorini, Infinite Dimensional Optimization and Control Theory[M], Cambridge University Press, 1999.
[27] H. O. Fattorini, The Maximum Principle in Infinite Dimensional, Discrete and Continuous Dynanical Systems, 6(3), 2000, 557-574.
[28] R. V. Gamlcrelidze, Theory of Time-optimal Control Progrcesses for Linear System, Izvestia Akad. Nank, SSSR, 22, 1958, 449-474.
[29] D. J. Guo, V. Lakshmikantham, X. Z. Liu, Nonlinear integral equations in abstract space, Kluwer Academic Publishers, Dorderecht, 1996.
[30] A. Giua, C. Seatzu and C. Van Der Mee, Optimal Control of Switched Autonomous Linear Systems, In Proceeding of the 40th IEEE CDC, 2001, 2472-2477.
[31] S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis (Theory)[M], Kiuwer Academic Publishers, Dordrecht Boston, London, 1997.
[32] H. Hermes, and J. P. LaSalle, Functional Analysis and Time Optimal Control[M], Academic Press, New York, 1969.
[33] L. M. Hocking, Optimal Control-An Introduction to the Theory with Applications, Clarendon Press Oxford, 1991.
[34] W. John and Nicklow, Discrete-Time Optimal Control for Water Resources Engineering and Mangement, Water International, 25(1), 2000, 89-95.
[35] N. N. Krasovskii, Concerning the Theory of Optimal Control, Avtomat. i Telemehm., 18, 1957, 960-970.[36] R. E. Kalmau, On the General Theory of Control Systems, Proc. 1st IFAC Congress, Moscow, 1960, Butterworth, London, 1961, 481-492.
[37] Kowalewski and Adam, Time-optimal Control of a Parabolic System with Time Lags Given in the Integral Form, IMA Journal of Mathematical Control and Information, 17(3), 2000, 209-225.
[38] J. P. LaSalle, Time Optimal Control Systemds, Mathmatics, 45, 1959, 573-577.
[39] J. P. LaSalle, The Time Optimal Control Problem, Contributions to the Theory of Nonlinear Oscillations 5, Princeton University Press, Princeton, 1960, 1-24.
[40] M. Leslie, Optimal Control[M], Clarendon Press Oxford, New York, 1991.
[41] E. B. Lee and L. Markus, Fundations of Optimal Control Theory[M], Wiley, New York, 1967.
[42] J. L. Lions, Optimal Control of Systems Goverened by Paratial Differential Equations[M], Springer-Verlag, New York, 1971.
[43] Tatsien Li, Tiehu Qin, Physical and Partial Differential Equation[M], Higher Education Press, Beijing, 2000.
[44] V. Lakshmikantham, D. D. Bainor and P. S. Simeonov, Theory of impulsive differential equations, World Scientific, Singapore-London, 1989.
[45] V. Lakshmikantham, S. Leela, Nonlinear differential equations in abstract space[M], Pergamon Press, New York, 1981.
[46] V. Lakshmikantham, Some problems in integro-differential equations of volterra type, J. Integral Equations 10, 1985, 137-146.
[47] Jiang Qiyuan, Mathematical Model[M], published by high education, 1998.
[48] Li Xunjing and Yong Jiongmin, Optimal Control Theory for Infinite Dimensional Systems[M], Birkhauser Boston, 1995.
[49] Li Xunjing and Yong Jiongmin, Basic Theory of Control[M], published by high education, perking, 2002.
[50] Li Shugang and Wang Gengsheng, The Time Optimal Control of the Boussinesq Equations, Numerical Functional Analysis and Optimization, 24, 2003, 163-180.
[51] C. C. H. Ma. and T. W. T. Qian, Rapid Tracking with Automatic Trajectory Optimization for Speed, ASME. Dyn. Sys. Meas. and Control, 121, 1999, 697-702.
[52] A. Pazy, Semigroup of Linear Operators and Applications to Partial Differential Equations[M], Springer-Verlag, New York, 1983.
[53] R. Pallu dela Barriere, Optimal Control Theory[M], W. B. Saunders Company, Philadaphia, 1967.
[54] E. Polak, Computation Methods in Optimization A Unified Approach[M], Academic Press, New York and London, 1971.
[55] B. Piccolic, Time-optimal Control Problems for the Swing and the Ski, International Journal of Control, 62(6), 1995, 1409-1429.
[56] D. L. Pepyne and C. G. Cassandras, Optimal Control of Linear Systems in Manufacturing, Proceeding IEEE, 88(7), 1108-1123.[57] S. Pettersson and B. Lennartson, Time Optimal Control and Disturbanc Compensation for A class of Hybrid Systems, In Proceeding 13th IFAC World Congress, 1996, 281-286.
[58] Z. Shiller, On singular Time Optimal Control Along Specified Paths, IEEE Transactions on Robotics and Automation, 10(4), 1994, 561-571.
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