一般情形下的平均场随机最大值原理

发布时间：2017-12-27 19:26

本文关键词：一般情形下的平均场随机最大值原理　出处：《山东大学》2017年硕士论文　论文类型：学位论文

【摘要】：本文中,考虑了两种平均场类型的随机控制问题,状态方程的系数依赖于解以及解的分布,且代价泛函也是平均场类型的。我们首先来看如下的状态过程,平均场SDE的控制问题:代价泛函为有如下假设:(H3.1)(2)b,σ关于(x,μ,v)的导数满足Lipschitz条件并有界。(3)h,Φ关于(X,μ,v)和(x,μ)的导数是Lipschitz连续的且被C(1 + |x| + |v|)和C(1+|x|)界住。(4)b,σ关于(x,μ)是Lipschitz连续和线性增长的,关于控制v是一致的。(H3.2)H(x,μ,p,q,v)关于v是凸的。在控制域为凸的情况下,我们考虑使得代价泛函达到最小值随机最优控制所满足的条件。通过凸扰动和对偶的技巧得到最优化控制的必要条件,也得到了控制最优的充分条件。我们接下来看如下的状态过程,解耦的控制问题:其中b,σ,f,Φ是给定的映射且初值ζ是一个F0可测的随机变量。有如下假设:(2)b,σ,f 关于(x,μ,u),(x,y,z,v,u)的导数是 Lipschitz 连续的且有界。(3)关于(x,μ)的导数是Lipschitz连续的且被C(1+|x|)界住。(4)对任意的控制 u,f(·,0,0,δ0,u)∈HF2(0,T;Rm)。(5)b,σ关于(x,μ)是 Lipschitz 连续和线性增长的,f 关于(x,y,z,v)是 Lipschitz连续的,关于控制u是一致的。代价泛函为同理,我们利用凸扰动和对偶的技巧,便可以得到满足最优控制的必要条件。
[Abstract]:In this paper, two stochastic control problems of mean field type are considered. The coefficients of state equations depend on the solution and the distribution of solutions, and the cost functional is also the mean field type. We first look at the following state process, the control problem of mean field SDE: the cost function assumes the following assumption: (H3.1) (2) B, and the derivative of (x, V, Lipschitz) satisfies Lipschitz condition and bounded. (3) h, the derivative of (X, mu, V) and (x, x) is Lipschitz continuous and is bounded by C (1 + |x| + |v|) and C (1+|x|). (4) B, sigma (x, mu) is a continuous and linear growth of Lipschitz, and is consistent with the control of v. (H3.2) H (x, mu, P, Q, V) about V is convex. When the control domain is convex, we consider the conditions that the cost functional satisfies the minimum random optimal control. The necessary conditions for optimal control are obtained by the technique of convex and duality, and the optimal sufficient conditions are also obtained. We then look at the state of the process is as follows: the decoupling control problem, including B, F, Phi sigma, and zeta mapping is given the initial value is a random variable F0 can be measured. There are the following assumptions: (2) B, sigma, f about (x, u, U), (x, y, Z, V, U) are Lipschitz continuous and bounded. (3) the derivative of (x, mu) is Lipschitz continuous and is bounded by the C (1+|x|) boundary. (4) to control for any u, f (-, 0,0, 8 0, U) and HF2 (0, T; Rm). (5) B, sigma (x, mu) is Lipschitz continuous and linear growth, f (x, y, Z, V) is Lipschitz continuous, and the control u is consistent. The cost functional is the same, and we can get the necessary condition to satisfy the optimal control by using the technique of convex and dual.
【学位授予单位】：山东大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O231

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