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A Study on Optimal Theory and Its Application for Uncertain Systems

Author: DingXiaoDong
Tutor: ShaoShiHuangï¼›WuRangQuan
School: Donghua University
Course: Control Theory and Control Engineering
Keywords: Uncertain programming Simplex method Genetic algorithm Fuzzy and stochastic simulation Stochastic optimal control BSDE Stochastic dynamic programming Hamilton-Jacobi-Bellman equation
CLC: O224
Type: PhD thesis
Year: 2002
Downloads: 1098
Quote: 11
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Abstract


Uncertain programming is an important content of the optimization theory for uncertain systems. Traditional uncertain programming mainly contains stochastic programming and fuzzy programming, which have many applications in manufacture, economy and management etc. Especially, the theory of stochastic or fuzzy linear programming is more complete, so it has more applications than stochastic or fuzzy nonlinear programming. Stochastic or fuzzy chance constrained programming refer to the objective functions and the constraint conditions contain stochastic or fuzzy parameters, the meaning of chance is the probability or possibility that the constraint conditions are satisfied. The usual methods of dealing with stochastic or fuzzy chance constrained linear programming are converting the chance constraint conditions to respective definitive or clear mathematic programming problems to compute them, according to given belief level. However, these methods are indirect algorithm, which have some limitations. In the case of complex situation, the obtained definitive or clear mathematic programming problems are usually nonlinear programming which bring complexity to computing, and even some stochastic or fuzzy problems are difficult to convert to definitive or clear problems. Therefore, It is important problem how to give and design direct computing methods for stochastic or fuzzy chance constrained linear programming according to which characteristic.The existence of uncertain factors really cause that decision systems contain stochastic or fuzzy parameters, but the uncertainty is not just stochastic or fuzzy, which might be hybrid of two factors of stochastic and fuzzy. It is need for not only theory but also applications to establish the hybrid programming models that contains stochastic and fuzzy parameters.Stochastic optimal control is an important content of the optimization theory for uncertain systems too. The objective function for the stochastic optimal control can be classified by the discounted cost problem and average expectation cost problem etc. The expression of specific objective function often depends its actual application problems, thus there are many types of theory study under the several objective functions in the usual stochastic optimal control, but the study methods are very similar. So it is the aim of many authors to give a uniform objective function for studying stochastic optimal control problems. For the appearance of the backward stochastic differential equations (BSDE), the studies of the stochastic optimal control problems are one of the main factors, and along with studies of BSDE. a uniform objective function for the stochastic optimal control can be defined using the solution of BSDE by the coupled forward-backward stochastic differential equations. It is not trivial generalization for the usual theory of the stochastic optimal control to study the stochastic optimal control problems.The above problems motivated the author to: (1) conquer the lack of the indirectcomputing methods for the uncertain linear programming to seek the direct computing method; (2) conquer the Singularity of stochastic or fuzzy factor in the usual uncertain programming models to give the hybrid programming models which contains stochastic and fuzzy parameters; (3) further strengthen the applications of BSDE in the stochastic optimal control to extend the related theories of the usual stochastic optimal control, and to enlarge the applied field.Upon to date, there is no existing review on uncertain programming theory and its applications, and there is no existing review on the applications of BSDE in the stochastic optimal control problems. In the dissertation, recent studies on uncertain programming theories and their applications and the optimal control for continuous stochastic systems are first systematically overviewed.In the dissertation, the simplex method basing on stochastic simulations is first presented, which provides a direct approach for computing stochastic chance constrained line

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CLC: > Mathematical sciences and chemical > Mathematics > Operations Research > Optimization of the mathematical theory
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