The Maximum Principle of Forward Backward Transformation Stochastic Control System
Li Zhou^{1}, Hong Zhang^{1}, Jie Zhu^{1}, Shucong Ming^{2}
^{1}School of Information, Beijing Wuzi University, Beijing, China
^{2}Chinese Academy of Finance and Development, Central University of Finance and Economics, Beijing, China
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To cite this article:
Li Zhou, Hong Zhang, Jie Zhu, Shucong Ming. The Maximum Principle of Forward Backward Transformation Stochastic Control System. Pure and Applied Mathematics Journal. Vol. 4, No. 3, 2015, pp. 109-114. doi: 10.11648/j.pamj.20150403.18
Abstract: In the paper, we discuss the maximum principle for the forward backward stochastic system. Assume the system follows a coupled forward backward stochastic differential equation modulated by a Marlcov chain and the control domain is convex. By convex variable method, we give the necessary and sufficient conditions for the existence of optimal control.
Keywords: Maximum Principle, Stochastic Control System, Forward Backward Transformation
1. Introduction
By the convex variation method, we give both the necessary and sufficient condition for the optimal control.
In this chapter, we study the maximum principle system forward backward conversion system. The control system is described by forward backward stochastic differential equations with continuous time Markov chain with finite state.
This chapter is structured as follows: in Section 1, we give the preliminary knowledge and problems; in section second, we obtain the necessary and sufficient conditions.
2. The Introduction of Optimal Control Problem
be a probability space, T>0 is a fixed time. is a D dimension Brown, is a finite state Markov chain, The state space is the transfer density function is for where is bounded nonnegative function. . is flow field generated by and all P zero sets. is the counting measure of . Where is the number of times that Markova chain jump to state j. within 0 to t. is a saddle measure. At this point, the markov chain can be expressed as
We defined the following space:
Consider the following forward and backward stochastic system conversion system:
(2.1)
Where are generators of BSDEs with a Markov chain.
In this chapter, for the convenience of marking, we only consider the situation that the X and Y are one dimension and change the backward equation into:
are measurable functions with certainty. D is a nonempty convex subset of. satisfy Lipschitz Condition as to x. satisfy the following Lipschitz Condition:
(2.2)
(2.3)
In this paper, we make the following assumptions:
(2.4)
3. Maximum Principle
In this part, we will get the maximum principle of stochastic variational method for optimal control problems in Section 1 of the value by classical convex optimal control. We will give the necessary and sufficient condition for the existence.
3.1. Necessary Condition
Let be an optimal control of the optimal control system (1.1),whose corresponding trajectory is denoted as . Let be another adaptation process (not necessarily valued in U) ,which satisfies .Since the control domain D is convex,. For the convenience of marking,
Introduction of variational equations as follows:
(3.1)
Obviously, the above linear FBSDE has a unique solution
Note is the rail line corresponding to .,
Note
We have a convergence results are as follows:
Lemma 1.
Proof.
We show that the other three equations. Firstly, we have
(3.2)
Noted
Similarly
This (2.2) could also be written as follows.
Attention to the measurable variation for
(3.3)
And , we apply division integral formula to
And we can get the following from (2.3) and (1.3)
where
We know that , By Gronwall inequality, we can prove the three convergence results.
Because. is an optimal control, we have
(3.4)
we prove the following variational inequalities,
Lemma .2. We suppose that theorem are tenable , The following variational inequality was established
(3.5)
Proof.
The convergence results of lemma 1
Similarly, we have
Thus (3.5) can be obtained by (3.4)
we introduce the dual equations as follows,
(3.6)
The define of is
(3.7)
(3.8)
Now we can prove stochastic maximum principle
Theorem 1 (maximum principle). Set as an optimal control, Is the rail line correspondingly. is the only solution of the dual equations(3.6).thus for any , the following can be obtain,
(3.9)
Proof.
We apply division integral formula to
According the Lemma .2.
We suppose that is the following form,
Where , Thus we can obtain that
We suppose that , Thus we can obtain that
For and , We suppose that ,Thus we can obtain that
Thus
3.2. Sufficient Conditions
In this section, under certain conditions, we obtain sufficient conditions for the existence of the optimal control.
Theorem 2. We suppose that theorem are tenable, h, r, H are concave to and can be written as , where I is the measurable function with certainty. is the solution of the Dual equations as to the control , which will be an optimal control if and on if theorem (3.9) is satisfied.
Proof
We suppose that is arbitrary feasible control, and noted is rail line correspondingly. In order to mark the simple, we note:
Thus
We could know from the concavity of the.
We apply the division integral formula to and
By the condition (3.9) can be obtained
Acknowledgements
This paper is funded by the project of National Natural Science Fund, Logistics distribution of artificial order picking random process model analysis and research(Project number: 71371033); and funded by intelligent logistics system Beijing Key Laboratory (No.BZ0211); and funded by scientific-research bases--- Science & Technology Innovation Platform---Modern logistics information and control technology research (Project number: PXM2015_014214_000001); and funded by 2014-2015 school year, Beijing Wuzi University, College students' scientific research and entrepreneurial action plan project (No.68); and funded by Beijing Wuzi University, Yunhe scholars program (00610303/007); and funded by Beijing Wuzi University, Management science and engineering Professional group of construction projects. (No. PXM2015_014214_000039). University Cultivation Fund Project of 2014-Research on Congestion Model and algorithm of picking system in distribution center (0541502703)
References