微分方程的最优控制
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摘要
分布参数系统的最优控制理论主要包括：庞特里雅金最大值原理；可控性；哈密顿-雅可比方程(即动态规划方程)；时间最优控制等等。
在这篇博士论文中，我们建立了非线性微分方程(包括抛物型微分方程，椭圆型微分方程以及3维Navier-Stokes方程)最优控制问题的庞特里雅金最大值原理，特别地，这些方程可能只有局部解或存在多解(我们称这样的系统为非适定系统，相应的最优控制问题为非适定最优控制问题)，以及适定的非线性发展方程最优控制问题的庞特里雅金最大值原理；我们研究了phase-field系统的时间最优控制问题以及Boussinesq系统的局部内可控性。
这篇博士论文共四章．第二章是引言部分，其中我们给出了在第三章和第四章要用到的定义与主要结果．在第三章，我们建立了非线性微分方程最优控制问题的庞特里雅金最大值原理。在第四章，我们讨论了Carleman不等式及其在最优控制问题中的应用。
Optimal control theory of distributed parameter systems mainly includes: Pontryagin's maximum principle; controllability; Hamilton-Jacobi equation (i.e., dynamic programming equation); time optimal control, etc.
In this dissertation, we establish Pontryagin's maximum principle of optimal control problems governed by some nonlinear differential equations (parabolic differential equations, elliptic differential equations and 3-dimensional Navier-Stokes equations), which in particular could have local solution only or could admit more than one solution (we shall call such systems as non-well-posed systems and the corresponding optimal control problems as non-well-posed optimal control problems), and some well-posed nonlinear evolution systems; we study time optimal control of phase-field system and we obtain local internal controllability of Boussinesq system.
This dissertation consists of four chapters. Chapter 2 is preliminary in which we give all definitions and main results used in Chapter 3 and Chapter 4. Chapter 3 presents Pontryagin's maximum principle of optimal control problems governed by nonlinear differential equations. Chapter 4 is concerned with Carleman inequality and its applications to optimal control problems.
在这篇博士论文中，我们建立了非线性微分方程(包括抛物型微分方程，椭圆型微分方程以及3维Navier-Stokes方程)最优控制问题的庞特里雅金最大值原理，特别地，这些方程可能只有局部解或存在多解(我们称这样的系统为非适定系统，相应的最优控制问题为非适定最优控制问题)，以及适定的非线性发展方程最优控制问题的庞特里雅金最大值原理；我们研究了phase-field系统的时间最优控制问题以及Boussinesq系统的局部内可控性。
这篇博士论文共四章．第二章是引言部分，其中我们给出了在第三章和第四章要用到的定义与主要结果．在第三章，我们建立了非线性微分方程最优控制问题的庞特里雅金最大值原理。在第四章，我们讨论了Carleman不等式及其在最优控制问题中的应用。
Optimal control theory of distributed parameter systems mainly includes: Pontryagin's maximum principle; controllability; Hamilton-Jacobi equation (i.e., dynamic programming equation); time optimal control, etc.
In this dissertation, we establish Pontryagin's maximum principle of optimal control problems governed by some nonlinear differential equations (parabolic differential equations, elliptic differential equations and 3-dimensional Navier-Stokes equations), which in particular could have local solution only or could admit more than one solution (we shall call such systems as non-well-posed systems and the corresponding optimal control problems as non-well-posed optimal control problems), and some well-posed nonlinear evolution systems; we study time optimal control of phase-field system and we obtain local internal controllability of Boussinesq system.
This dissertation consists of four chapters. Chapter 2 is preliminary in which we give all definitions and main results used in Chapter 3 and Chapter 4. Chapter 3 presents Pontryagin's maximum principle of optimal control problems governed by nonlinear differential equations. Chapter 4 is concerned with Carleman inequality and its applications to optimal control problems.
引文
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[9] V.Barbu, Exact controllability of the superlinear heat equation, Appl. Math. Optim., 42(2000), pp. 73-89.
[10] V.Barbu, Controllability of Evolution Equations of Parabolic Type, to appear.
[11] V.Barbu, Local controllability of the phase-field system, Nonlinear Analysis. Theory, Methods Applications, to appear.
[12] V.Barbu, Local internal controllability of Navier-Stokes equations, Differential and Integral Equation, to appear.
[13] V.Barbu and N.Pavel, Optimal control problems with two point boundary conditions, J. Optim. Theory Appl., 77(1993), pp. 51-78.
[14] V.Barbu and N.Pavel, Periodic optimal control in Hilbert space, Appl. Math. Optim., 33(1996), pp. 169-188.
[15] V.Barbu and N.Pavel, Flow-invarianec closed set with respect to nonlinear semigroup flows, to appear.
[16] V.Barbu and S.Srithanaran, Flow-invariance preserving feedback controllers for Navier-Stokes equations, J. Math. Ann. Appl., 255(2001), pp. 281-307.
[17] V.Barbu and G.S.Wang, State-constrained optimal control problems governed by semilinear equations, Numerical Functional Analysis and Optimization, 21(2000), pp. 639-667.
[18] M.Bergounioux and F.Troltzsch, Optimal conditions and generalized bang-bang principle for a state constrained semilinear parabolic problem, Numer. Funct. Anal. Optim., 15(1996), pp. 517-537.
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for the semilinear heat equation with nonlinearities involving gradient term, Journal of Optimization Theory and Applications, 110(2001), pp. 245-265.
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[27] E.Casas, L.A.Fernandez, Distributed control of system governed by a general class of quasiIinear equations, J. Differential Equations, 104(1993), pp. 20-47.
[28] E.Casas and J.Yong, Maximum principle for state-constrained control problems governed by quasilinear equations, Differential Integral Equations, 8(1995), pp. 1-18.
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[36] H.O.Fattorini and S.S.Sritharan, Necessary and sufficient conditions for optimal controls in viscous flow problems, Proc. Royal Soc. Edinburgh, 1994, 124A, pp. 211-251.
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Island, 2000.
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[43] A.V.Fursikov, O.Yu.Imanuvilov, Local exact boundary controllability of the Boussinesq equation, SIAM J. Control Optim., 36(1998), pp. 391-421.
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[45] H.Gao, Optimality condition of systems governed by semilinear parabolic equations, Acta Mathematica Sinica, 42(1999), pp. 705-714. (Chinese)
[46] H.Gao and X.Li, Necessary conditions for optimal control of elliptic systems, J. Austral. Math. Soc. Ser. B41, 2000, pp. 542-567.
[47] Z.X.He, State constrained optimal control problems governed by variational inequalities, SIAM J. Control Optim., 25(1987), pp. 1071-1085.
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[53] V.M.Isakov, On the uniqueness of the solution of the Cauchy problem, Dokl. Akad. Nauk SSSR, 255(1980), pp. 18-21; English Transl., Soviet Math. Dokl, 22(1980), pp. 639-642.
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[57] G.E.James, Monotone methods for semilinear elliptic equations in unbounded domains, J. Mathematical Analysis and Applications, 137(1989), pp. 122-131.
[58] D.Jerison, Carleman inequalities for the Dirac and Laplace operators and unique continuation, Adv. Math., 62(1986), pp. 118-134.
[59] Y.M.Kim, Carleman inequalities for the Dirac operator and strong unique continuation, Proc. Amer. Math. Soc., 123(1995), pp. 2103-2112.
[60] X.Li and J.Yong, Necessary conditions for optimal control of distributed parameter systems, SIAM J. Control Optim., 29(1991), pp. 895-908.
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[78] G.S.Wang, Optimal controls of 3-dimensional Navier-Stokes equations with state constraints, SIAM J. Control Optim., to appear.
[79] G.S.Wang, Pontryagin maximum principle of optimal control governed by dynamic systems with two point boundary state constraint, Nonlinear Analysis, to appear.
[80] G.S.Wang, Optimal control of parablic differential equations with two point boundary state constraints, SIAM J. Control Optim., 38(2000), pp. 1639-1654.
[81] G.S.Wang, Optimal control of parabolic variational inequality involving state constraint, Nonlinear Anal., 42(2000), pp. 789-801.
[82] G.S.Wang and S.R.Chen, Maximum principle for optimal control of some parabolic systems
with two point boundary conditions, Numer. Funct. Anal. Optim., 20(1999), pp. 163-174.
[83] G.S.Wang and L.J.Wang, State-Constrained Optimal Control Governed by Non-Well-Posed Parabolic Differential Equations, SIAM J. Control Optim., 40(2002), pp. 1517-1539.
[84] G.S.Wang and L.J.Wang, Maximum Principle for Optimal Control of Non-Well-Posed Elliptic Differential Equations, Nonlinear Analysis, to appear.
[85] G.S.Wang and L.J.Wang, Maximum Principle of State Constrained Optimal Control Governed by Fluid Dynamical Systems, Nonlinear Analysis, to appear.
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[87] G.S.Wang and L.J.Wang, State-Constrained Optimal Control in Hilbert Space, Numerical Functional Analysis and Optimization., 22(2001), pp. 255-276.
[88] G.S.Wang and L.J.Wang, The Carleman Inequality and Its Application to Periodic Optimal Control Governed by Semilinear Parabolic Differential Equations, Journal of Optimization Theory and Applications, to appear.
[89] G.S.Wang, Y.C.Zhao and W.D.Li, Some optimal control problems governed by elliptic variational inequalities with control and state constraint on boundary, Journal of Optimization Theory and Applications, 106(2000), pp. 639-667.
[90] L.J.Wang and G.S.Wang, Local internal controllability of the Boussinesq system, Nonlinear Analysis, 53(2003), pp. 637-652.
[91] L.J.Wang and G.S.Wang, The Time Optimal Control of the Phase-Field System, SIAM J. Control Optim., to appear.
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[93] J.Yong, Optimal controls for distributed parameter systems with mixed constraints, Colloquim Math., 60/61(1990), pp. 35-48.
[94] J.Yong, Pontryagin maximum principle for semilinear second order elliptic partial differential equations and variational inequalities with state constraints, Differential Integral Equations,5(1992), pp. 1307-1334.
[2] V.I.Arnold and B.A.Khesin, Topological Methods in Hydrodynamics, Springer-Verlag, 1998.
[3] J.P.Aubin and I.Ekeland, Applied Nonlinear Analysis, John Wiley and Sons, New York, 1984.
[4] V.Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Nordhoff, Leyden, the Netherlands, 1976.
[5] V.Barbu, Optimal Control of Variational Inequalities, Res. Notes Math., 100, Pitman, 1984.
[6] V.Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, Academic Press, New York, 1993.
[7] V.Barbu, The time optimal control of Navier-Stokes equations, Systems and Control Letters, 30(1997), pp. 93-100.
[8] V.Barbu, Optimal control of Navier-Stokes equations with periodic inputs, Nonlinear Analysis TMA, 31(1998), pp. 15-31.
[9] V.Barbu, Exact controllability of the superlinear heat equation, Appl. Math. Optim., 42(2000), pp. 73-89.
[10] V.Barbu, Controllability of Evolution Equations of Parabolic Type, to appear.
[11] V.Barbu, Local controllability of the phase-field system, Nonlinear Analysis. Theory, Methods Applications, to appear.
[12] V.Barbu, Local internal controllability of Navier-Stokes equations, Differential and Integral Equation, to appear.
[13] V.Barbu and N.Pavel, Optimal control problems with two point boundary conditions, J. Optim. Theory Appl., 77(1993), pp. 51-78.
[14] V.Barbu and N.Pavel, Periodic optimal control in Hilbert space, Appl. Math. Optim., 33(1996), pp. 169-188.
[15] V.Barbu and N.Pavel, Flow-invarianec closed set with respect to nonlinear semigroup flows, to appear.
[16] V.Barbu and S.Srithanaran, Flow-invariance preserving feedback controllers for Navier-Stokes equations, J. Math. Ann. Appl., 255(2001), pp. 281-307.
[17] V.Barbu and G.S.Wang, State-constrained optimal control problems governed by semilinear equations, Numerical Functional Analysis and Optimization, 21(2000), pp. 639-667.
[18] M.Bergounioux and F.Troltzsch, Optimal conditions and generalized bang-bang principle for a state constrained semilinear parabolic problem, Numer. Funct. Anal. Optim., 15(1996), pp. 517-537.
[19] J.F.Bonnans and E.Casas, An extension of Pontryagin's principle for state-constrained optimal control of semilinear equation and variational inequalities, SIAM J. Control Optim., 33(1995), pp. 274-298.
[20] J.F.Bonnans and D.Tiba, Pontryagin's principle in the control of semilinear elliptic variational inequalities, Applied Math. Optim., 23(1991), pp. 273-289.
[21] V R,Cabanillas; S B, De Menezes and E.Zuazua, Null controllability in unbounded domains
for the semilinear heat equation with nonlinearities involving gradient term, Journal of Optimization Theory and Applications, 110(2001), pp. 245-265.
[22] G.Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92(1986), pp. 205-245.
[23] Z.Cai, N.Pavel and S.L.Wen, Optimal control of some partial differential equations with two-point boundary conditions, Optimal Control of Differential Equations (Athens, OH, 1993), Lecture Notes in Pure and Applied Mathematics, Dekker, New York, 160(1994), pp. 49-68.
[24] D.Cao and H.S.Zhou, Multiple positive solutions of nonhomogeneous semilinear elliptic equations, Proceedings of the Royal Society of Edinburgh, (1996), 126A, pp. 443-463.
[25] E.Casas, Boundary control problems for quasi-linear equations: a Pontryagin's principle, Applied Math. Optim., 33(1996), pp. 265-291.
[26] E.Casas, Pontryagin's principle for state-constrained boundary control problems of semilinear parabolic equation, SIAM J. Control Optim., 35(1997), pp. 1297-1327.
[27] E.Casas, L.A.Fernandez, Distributed control of system governed by a general class of quasiIinear equations, J. Differential Equations, 104(1993), pp. 20-47.
[28] E.Casas and J.Yong, Maximum principle for state-constrained control problems governed by quasilinear equations, Differential Integral Equations, 8(1995), pp. 1-18.
[29] E.Casas and J.Yong, Maximal principle for state-constrained optimal control problems governed by quasilinear equations, Differential Integral Equations, 130(1996), pp. 179-200.
[30] D.Chae, O.Yu.Imanuvilov and M.S.Kim, Exact controllability for semilinear parabolic equations with Neumann boundary conditions, J. Dynamical and Control Systems, 2(1996), 449-483.
[31] F.H.Clarke, Optimization and Non Smooth Analysis, John Wiley and Sons, New York, 1983.
[32] P.Constantin, C.Foias, Navier-Stokes Equations, The University Chicago Press, 1998.
[33] J.M.Coron, On the controllability of the 2-D incompressible Navier-Stokes equations with Navier-slip boundary conditions, ESIAM Control Optimz. Calculus of Variations, 1(1996), pp. 35-75.
[34] H.O.Fattorini and H.Frankowska, Necessary conditions for infinite dimensional control problems, Math. Control Signals Systems, 4(1991), pp. 225-257.
[35] H.O.Fattorini and T.Murphy, Optimal problems for nonlinear parabolic boundary control systems, SIAM J. Control Optim., 32(1994), pp. 1577-1596.
[36] H.O.Fattorini and S.S.Sritharan, Necessary and sufficient conditions for optimal controls in viscous flow problems, Proc. Royal Soc. Edinburgh, 1994, 124A, pp. 211-251.
[37] E.Fernandez-Cara, Null controllability of the semilinear heat equation, ESAIM Control Optim: Calc. Var. 2(1997), pp. 87-103.
[38] E.Fernandez-Cara, Null controllability for semilinear parabolic equations with critical growth of the nonlinearity, C. R. Acad. Sci. Paris ser. Ⅰ Math..324(1997), pp. 1371-1376.
[39] A.V.Fursikov, Control problems and results on the unique solution of mixed problems to three dimensional Navier-Stokes and Euler equations, Mat. Sbornik, 115(1981), pp. 281-306.
[40] A.V.Fursikov, Optimal Control of Distributed Systems, Theory and Application, Translations of Mathematical Monography, Vol. 187, American Mathematical Society Providence, Rhode
Island, 2000.
[41] A.V.Fursikov, O.Yu. Imanuvilov, Local exact controllability of two-dimensional Navier-Stokes system with control on the part of the boundary, Math. Sbornik, 187(1996), pp. 1355-1390.
[42] A.V.Fursikov, O.Yu.Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series, No. 34, Seoul National University, Korea, 1996.
[43] A.V.Fursikov, O.Yu.Imanuvilov, Local exact boundary controllability of the Boussinesq equation, SIAM J. Control Optim., 36(1998), pp. 391-421.
[44] A.V.Fursikov, O.Yu.Imanuvilov, Exact controllability of the Navier-Stokes and Boussinesq equations, Russian Math. Surveys, 54(1999), pp. 565-618.
[45] H.Gao, Optimality condition of systems governed by semilinear parabolic equations, Acta Mathematica Sinica, 42(1999), pp. 705-714. (Chinese)
[46] H.Gao and X.Li, Necessary conditions for optimal control of elliptic systems, J. Austral. Math. Soc. Ser. B41, 2000, pp. 542-567.
[47] Z.X.He, State constrained optimal control problems governed by variational inequalities, SIAM J. Control Optim., 25(1987), pp. 1071-1085.
[48] L.H(?)rmander, Linear partial differential operators, Springer-Verlag, Berlin-New York 1963; Russian transl., Mir, Moscow 1965.
[49] L.H(?)rmander, The analysis of linear partial differential operators, I. Distribution Theory and Fourier Analysis, Springer-Verlag, Berlin-New York 1983; Russian transl., Mir, Moscow 1986.
[50] O.Yu.Imanuvilov, Local exact controllability for the 2-D Navier-Stokes equations with the Navier slip conditions, Lecture Notes in Physics, 491(1997), pp. 148-168.
[51] O.Yu.Imanuvilov, On exact controllability of Navier-Stokes equations, ESIAM Control Optimz. Calculus of Variations, 3(1998), pp. 97-131.
[52] O.Yu.Imanuvilov, Remarks on controllability of Navier-Stokes equations, ESIAM Control Optimz. Calculus of Variations, 6(2001), pp. 47-97.
[53] V.M.Isakov, On the uniqueness of the solution of the Cauchy problem, Dokl. Akad. Nauk SSSR, 255(1980), pp. 18-21; English Transl., Soviet Math. Dokl, 22(1980), pp. 639-642.
[54] V.Isakov, Inverse parabolic problems with the final overdetermination, Comm. Pure Appl. Math, 64(1991), pp. 185-209.
[55] V.Isakov, Carleman type estimates in an anisotropic case and applications, J. Differential Equations, 105(1993), pp. 217-239.
[56] V.Isakov, On uniqueness in a lateral Cauchy problem with multiple characteristics, J. Differential Equations, 134(1997), pp. 134-147.
[57] G.E.James, Monotone methods for semilinear elliptic equations in unbounded domains, J. Mathematical Analysis and Applications, 137(1989), pp. 122-131.
[58] D.Jerison, Carleman inequalities for the Dirac and Laplace operators and unique continuation, Adv. Math., 62(1986), pp. 118-134.
[59] Y.M.Kim, Carleman inequalities for the Dirac operator and strong unique continuation, Proc. Amer. Math. Soc., 123(1995), pp. 2103-2112.
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