一类离散HJB方程的数值解法
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摘要
Hamilton-Jacobi-Bellman方程(简称HJB方程)最早出现于用动态规划解最优控制问题,之后在科学、工程、经济领域中得到广泛应用.因此HJB方程数值解的研究是一个非常热门的话题;它是偏微分方程数值解领域中重要课题之一.本文主要是研究离散HJB方程数值解法,我们在文中构造了若干新算法并证明了相应算法的收敛性,然后通过数值试验,证明了算法的有效性.
离散的HJB方程在一定的条件下可用拟变分不等式组来逼近.对此拟变分不等式组,我们构造了松弛迭代格式,当ω= 1时即Gauss-Seidel型迭代算法.然后我们考虑基于此算法的区域分解方法,并给出了上述算法的收敛性分析.数值试验显示松弛算法中适当选取松弛因子,能显著提高算法的有效性.
Lions和Mercier[1]对离散的HJB方程的数值解提出了两种迭代格式,其中的格式I是在迭代的每一步中对一个变分不等式进行求解.我们对此格式引进一个松弛因子ω,我们称它为Lions-Mercier型的松弛算法.我们给出了此算法的收敛性证明.数值例子表明,合理地选取松弛因子,能大大提高算法的运算速度.
我们还提出了求解HJB方程的一种新的松弛迭代格式,称为Gauss-Seidel型迭代.它在每一步迭代只需进行简单的算术运算,而不需求解线性方程组或线性互补问题,且每一步迭代都用到了上一步的最新结果.此算法的收敛性比传统算法快,我们用数值试验表明了这一点.算法的单调收敛性也得到了证明.
最后,我们对离散的HJB方程的提出了新的多重网格法.在磨光算子的选取上我们选择了一个非线性的光滑算子,即上段所述的松弛型迭代算法.数值试验显示修改磨光算子的新的多重网格法是有效的,并且算法的运算速度明显高于已有求解HJB方程的多重网格法.
The Hamilton-Jacobi-Bellman equation(in short HJB equation) are encoun-tered first solving optimal control problem using dynamic programming. Af-terward, the HJB equation have been widely used in sciences、engineering andeconomics. So, the numerical solutions for the HJB equation have attracted muchattention. The HJB equations are important problems in the numerical solutionsof partial di?erential equations, too. In the thesis, we discuss mainly the numer-ical solutions of the discrete problems of a kind of HJB equation. We constructmany new iterative algorithms and prove their convergence. The numerical re-sults show the e?ectiveness of algorithms.
The discrete HJB equation may be approximated by a quasivariational in-equality system on the condition. We consider a relaxation algorithm for theapproximate quasivariational inequality system of HJB equation, the relaxationalgorithm is Gauss-Seidel type algorithm onω= 1. Then we construct a domaindecomposition method for the quasivariational inequality system based on therelaxation algorithm. We give the corresponding monotone convergence theoriesabout their algorithms. Numerical experiments show that the e?ectiveness of thealgorithms are improved using properω.
Lions and Mercier have proposed two iterative schemes for the numericalsolution of the discrete HJB equation. At each iteration, a variational inequalityis solved in Scheme I. We propose, based on Scheme I, a relaxation scheme witha parameterω. We call it Lions-Mercier relaxation algorithm. The monotoneconvergence of the algorithm has been proved. In our numerical examples, thealgorithm is faster than Scheme I choosing a proper parameterω.
A new successive relaxation iterative algorithm for discrete HJB equation isproposed. It does not solve a linear equation system or a linear complementarityproblem but carry out simple arithmetic operations at each iteration, and weuse the newest results at each iteration. Numerical tests show it is faster thantraditional algorithms. Monotone convergence has been proved for the algorithm.Finally, we construct a new multigrid method for discrete HJB equation.
We choose a nonlinear smoother, that is a relation iterative algorithm in the lastparagraph, as a smoothing operator. Numerical experiments show it is faster than mulitgrid methods that they used to solve the discrete HJB equation.
离散的HJB方程在一定的条件下可用拟变分不等式组来逼近.对此拟变分不等式组,我们构造了松弛迭代格式,当ω= 1时即Gauss-Seidel型迭代算法.然后我们考虑基于此算法的区域分解方法,并给出了上述算法的收敛性分析.数值试验显示松弛算法中适当选取松弛因子,能显著提高算法的有效性.
Lions和Mercier[1]对离散的HJB方程的数值解提出了两种迭代格式,其中的格式I是在迭代的每一步中对一个变分不等式进行求解.我们对此格式引进一个松弛因子ω,我们称它为Lions-Mercier型的松弛算法.我们给出了此算法的收敛性证明.数值例子表明,合理地选取松弛因子,能大大提高算法的运算速度.
我们还提出了求解HJB方程的一种新的松弛迭代格式,称为Gauss-Seidel型迭代.它在每一步迭代只需进行简单的算术运算,而不需求解线性方程组或线性互补问题,且每一步迭代都用到了上一步的最新结果.此算法的收敛性比传统算法快,我们用数值试验表明了这一点.算法的单调收敛性也得到了证明.
最后,我们对离散的HJB方程的提出了新的多重网格法.在磨光算子的选取上我们选择了一个非线性的光滑算子,即上段所述的松弛型迭代算法.数值试验显示修改磨光算子的新的多重网格法是有效的,并且算法的运算速度明显高于已有求解HJB方程的多重网格法.
The Hamilton-Jacobi-Bellman equation(in short HJB equation) are encoun-tered first solving optimal control problem using dynamic programming. Af-terward, the HJB equation have been widely used in sciences、engineering andeconomics. So, the numerical solutions for the HJB equation have attracted muchattention. The HJB equations are important problems in the numerical solutionsof partial di?erential equations, too. In the thesis, we discuss mainly the numer-ical solutions of the discrete problems of a kind of HJB equation. We constructmany new iterative algorithms and prove their convergence. The numerical re-sults show the e?ectiveness of algorithms.
The discrete HJB equation may be approximated by a quasivariational in-equality system on the condition. We consider a relaxation algorithm for theapproximate quasivariational inequality system of HJB equation, the relaxationalgorithm is Gauss-Seidel type algorithm onω= 1. Then we construct a domaindecomposition method for the quasivariational inequality system based on therelaxation algorithm. We give the corresponding monotone convergence theoriesabout their algorithms. Numerical experiments show that the e?ectiveness of thealgorithms are improved using properω.
Lions and Mercier have proposed two iterative schemes for the numericalsolution of the discrete HJB equation. At each iteration, a variational inequalityis solved in Scheme I. We propose, based on Scheme I, a relaxation scheme witha parameterω. We call it Lions-Mercier relaxation algorithm. The monotoneconvergence of the algorithm has been proved. In our numerical examples, thealgorithm is faster than Scheme I choosing a proper parameterω.
A new successive relaxation iterative algorithm for discrete HJB equation isproposed. It does not solve a linear equation system or a linear complementarityproblem but carry out simple arithmetic operations at each iteration, and weuse the newest results at each iteration. Numerical tests show it is faster thantraditional algorithms. Monotone convergence has been proved for the algorithm.Finally, we construct a new multigrid method for discrete HJB equation.
We choose a nonlinear smoother, that is a relation iterative algorithm in the lastparagraph, as a smoothing operator. Numerical experiments show it is faster than mulitgrid methods that they used to solve the discrete HJB equation.
引文
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[2] Hoppe R H W. Multigrid methods for Hamilton-Jacobi-Belman equations. Nu-merische Mathematik. 1996, 49: 239-254
[3] Bensoussan A, Lions J L.Applications of Variational Inequalities in StochasticControl. North-Holland, Amsterdam, 1982
[4] Boulbrachene M, Haiour M. The finite elementapproximation of Hamilton-Jacobi-Bellman equations. Computers & Mathematics with Applications. 2001, 41: 993-1007
[5] Sun M. Domain decomposition method for solving HJB equations, Numer. Funct.Anal. Optim. 1993, 14: 145-166
[6] Sun M. Alternating direction algorithms for solving HJB equations. Applied Math-ematics and Optimization. 1996, 34: 267-277
[7] Bellman R. Dynamic Programming. New Jersey: Princeton Univ Press, 1957.
[8] Stanislaw S. Hamilton-Jacobi-Bellman equations and dynamic programming forpower-maximizing relaxation of radiation. International Journal of Heat and MassTransfer, 2007, 50: 2714-2732
[9] Bellman R. Adaptive Control Processes: A Guided Tour.New Jersey: PrincetonUniversity Press, 1961, 1-42
[10] Bensoussan A, Lions J L.Impulse Control and QuasiVariational Inequalities. Paris:Gauthier Villars, 1984, 1-31
[11] Fleming W H, Rishel R. Deterministic and Stochastic Optimal Control.Berlin:Springer, 1975, 1-27
[12] Yong J M, Zhou X Y. Stochastic Controls: Hamilton Systems and HJB equations.New York: Springer, 1998, 1-33
[13] Zhou X Y. A unified treatment of maximum principle and dynamic Programminginstochastic controls. Stochastics and Stochastics Reports, 1991, 36: 137-161.
[14] Li S P, Liu K H. HJB Equations Subject to Stochastic Control Theory and Se-curities Investment Models, Journal of Northern Jiaotong University, 2003, 27:63-67
[15] Tangs J, Yong J M. Finite horizon stochastic optimal switching and impulse con-trols with a viscosity solution approach, Stochastics and Stochastic Reports, 1993,45: 145-176
[16] Zhang L. Optimal financing and divided control of a corporation with transactioncosts. Control Theory & Applications, 2004, 21: 895-900
[17] Lions P L. Optimal control of di?usion processes and HJB equations, Part I,Comm PDE, 1983, 8: 1101-1174
[18] Lions P L. Optimal control of di?usion processes and HJB equations, Part II,Comm PDE, 1983, 8: 1129-127
[19] Lions P L, Formule de trotter et equations de Hamilton-Jacobi-Bellman. Calcolo,1980, 17: 321-331
[20] Zhou S Z, Zhan W P. A new domain decomposition method for an HJB equation.Journal of Computational and Applied Mathematics, 2003, 159: 195-204
[21] Mete S H. On the Hamilton-Jacobi-Bellman equations in Banach spaces. Journalof Optimization Theory and Applications, 1988, 57: 429-437
[22] Carj O. Lower semicontinuous solutions for a class of Hamilton-Jacobi-Bellmanequations. Journal of Optimization Theory and Applications, 1996, 3: 637-657
[23] Beard R W, Saridis G N, Wen J T. Approximate Solutions to the Time-InvariantHamilton-Jacobi-Bellman Equation. Journal of Optimization Theory and Appli-cations, 1998, 96: 589-626
[24] Evans L C. Classical solutions of the Hamilton-Jacobi-Bellman equation for uni-formly elliptic operators, Transactions of the AMC, 1983, 275: 245-255
[25] Rainer B, Naoyuki I. Limit Theorem for Controlled Backward SDEs and Ho-mogenization of Hamilton-Jacobi-Bellman Equations. Applied Mathematics andOptimization, 2005, 51: 1-33
[26] Dimentberg M, Iourtchenko D, Bratus A. Bounded Control of Random Vibration:Hybrid Solution to HJB Equations.Meccanica, 2007, 37: 129-141
[27] Bai L h, Guo J Y. Utility maximization with partial information: Hamilton-Jacobi-Bellman equation approach. Frontiers of Mathematics in China, 2007, 2:527-537
[28] Bratus A S, Ivanova A P, Iourtchenko D V, Menaldi J L. Local solutions of theHamilton-Jacobi-Bellman equation for some stochastic problems. Automation andRemote Control, 2007, 74: 1023-1038
[29] Ito K. Existence of solutions to the Hamilton-Jacobi-Bellman equation underquadratic growth conditions. Journal of Di?erential Equations, 2001, 176: 1-28
[30] Bratus A S, Volosov K A. Exact solutions to the Hamilton-Jacobi-Bellman equa-tion for optimal correction with an integral constraint on the total resource ofcontrol. Doklady Mathematics. 2002, 66: 148-151
[31] Bratus A S, Volosov K A. Exact solutions of the Hamilton-Jacobi-Bellman equa-tion for problems of optimal correction with a constrained overall control resource.Journal of Applied Mathematics and Mechanics, 2004, 68: 731-742
[32] Bratus A S, Ivanova AP. Local solutions to the Hamilton-Jacobi-Bellman equationand their application to the problem of optimal control of vibrations of elasticdistributed systems. Journal of Computer and Systems Sciences Intenational. 2004,43: 189-197
[33] Alexandre M B, Christian Claude, Patrick Saint-Pierre. Viability-Based Compu-tations of Solutions to the Hamilton-Jacobi-Bellman Equation. Lecture Notes inComputer Science, 2007, 645-649
[34] Evans L C, Friedman A. Optimal stochastic switching and Dirichlet problem forthe Bellman equations. Transactions of the American Mathematical Society, 1979,253: 365-389
[35] Lions P L, Menaldi J L. Optimal control of stochastic integrals and Hamilton-Jacobi-Bellman Equations. SIAM Control and Optimization, 1982, 1: 58-81
[36] Lions P L. Optimal control of di?usion processes and Hamilton-Jacobi-Bellmanequations. Part 2: Viscosity solutions and uniqueness, Comm. Partial Di?erentialEquations, 1983, 8: 1103-1280.
[37] Lions P L. On the Hamilton-Jacobi-Bellman equations. Acta Applicandae Math-ematicae: An International Survey Journal on Applying Mathematics and Math-ematical Applications, 1983, 1: 17-41
[38] Barron E N. Application of viscosity solutions of infinite-dimensional Hamilton-Jacobi-Bellman equations to some problems in distributed optimal control. Journalof Optimization Theory and Applications, 1990, 64: 245-268
[39] Wang X, Shimizu K. Approximate solution of Hamilton-Jacobi-Bellman equationby using neural networks and matrix calculus techniques. IEICE Transactions onFundamentals of Electronics Communications and Computer Sciences, 2001, 84:1549-1556
[40] Nosovskij G V. Nonlinear potentials for Hamilton-Jacobi-Bellman equations, ActaApplicandae Mathematicae. 1993,30: 101-123.
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