Heisenberg群上与Schr\"odinger算子相关的Poisson半群的分数阶导数估计

令$\mathcal{L}=-{\Delta}_{\mathbb{H}^{n}}+V$为Heisenberg群$\mathbb{H}^{n}$上的Schr\"odinger算子, 其中${\Delta}_{\mathbb{H}^{n}}$为次Laplace算子, 非负位势$V$属于逆H\"{o}lder类. 本文中, 利用从属性公式, 我们给出与$\mathcal{L}$相关的Poisson半群的分数阶导数的正则性估计, 作为应用, 我们得到了与$\mathcal{L}$相关的Campanato型空间的一个刻画.     

带磁场的一维Schr\"{o}dinger方程的逆散射问题

本文研究带磁场的一维Schr\"{o}dinger方程的逆散射问题. 利用其散射矩阵的有关性质以及函数变换的方法, 获得位势$p$可以被散射系数决定的结果, 并推广了迹公式的成立范围.     

积分方程方法在层状散射问题中的应用研究

本文研究声波在分层均匀介质中碰到不可穿透障碍物产生的混合边值散射问题. 应用边界积分方程法将原问题转化为与之等价的边界积分方程组, 通过分析积分算子的Fredholm性质, 得到正问题解的适定性. 应用Nystr\"om方法将积分算子离散, 给出远场模式的计算方法, 并利用具体的数值实验验证方法的有效性, 为进一步展开反问题的研究奠定理论基础.     

Rota-Baxter簇系统与Gr\"obner-Shirshov基

从Yang-Baxter簇方程和Volterra积分方程得到的Rota-Baxter簇代数的概念出发,我们引入Rota-Baxter簇系统的概念,推广了Brzezinski提出的Rota-Baxter系统.我们证明这个概念也与结合Yang-Baxter簇对和pre-Lie簇代数有关.此外,作为Rota-Baxter簇系统的一个类比,我们引入平均簇系统的概念,并证明平均簇系统会得到dialgebra簇结构.我们还研究dendriform代数上的Rota-Baxter簇系统,并展示它们如何诱导quadri簇代数结构.最后,我们用Gr\"obner-Shirshov基的方法给出Rota-Baxter簇系统的一个线性基.     

Orlicz空间内的M\"{u}ntz有理函数的逼近

本文研究了Orlicz空间内M\"{u}ntz有理函数逼近问题, 相比于前人对同类问题的研究, 本文改进了系指数$\{\lambda_{n}\}^\infty_{n=1}$所满足的条件, 利用H\"{o}lder不等式、Hardy-Littlewood极大函数、$K$-泛函、连续模、$N$函数的凸性等技巧, 得到逼近的Jackson型定理. 由于Orlicz空间的拓扑结构比连续函数空间和Lp空间复杂, 所以本文的结果具有一定的拓展意义.     

非线性分数阶积分微分发展方程适度解的存在唯一性（英文）

本文研究一类在Banach空间中分数阶积分微分发展方程的问题,利用分数阶幂算子和解析半群理论来证明所给方程适度解的存在唯一性.并进一步给出适度解的H?lder连续性.     

$l^p$-值Wiener过程增量在H\"older范数下的局部Strassen重对数律

应用$l^p$-值Wiener过程在H\"older范数下的大偏差, 研究了$l^p$-值Wiener过程增量在H\"older范数下的局部Strassen重对数律.     

由拟从属定义的单叶函数的逆在$k$次根变换下的Fekete-Szego问题

本文在逆函数$n$次根变换的条件下,利用拟从属定义了一个单叶解析函数类,估计了它的Fekete-Szeg\"o泛函.     

分数阶Schr\"{o}dinger-Poisson系统规范化解的存在性

本文研究分数阶Schr\"{o}dinger-Poisson系统规范化解的存在性, 首先在变分框架下将其规范化解转化为约束极小化问题的极小元, 然后利用集中紧性原理证明了极小元的存在性与不存在性.     

$\alpha$-L\"uroth展式的若干度量性质的研究[英文]

本文研究$\alpha$-L\"uroth展式中的若干度量性质. 获得了该展式数字的“0-1”率, 基于该结果, 得到了相应的重对数率, 进一步完善了该展式的度量性质, 作为交错L\"uroth展式的推广, 该论文的结论包含了交错L\"uroth展式相应的结果.     

一类具漂移的“跳-停”奇异型随机控制

以随机分析和最优控制理论为基础,讨论了一类带停时的奇异型随机控制问题.在原模型状态过程的基础上添加了漂移因子,并将原模型中的控制费用函数推广为一般的费用函数.在某些条件下,得到"跳一停"策略是其最优控制策略,并给出了"跳一停"策略存在的条件以及控制方法,所得的结论在实际中有较深的应用背景.     

一类非对称的“跳-停”奇异型随机控制

研究了一类带停时的非对称的奇异型随机控制的折扣问题,不论是从受控状态过程还是从费用函数均推广为较一般的情形,得到"跳-停"策略是其最优控制策略,并给出了"跳-停"策略存在的条件、最优费用函数以及控制方法,所得的结论在实际中有较深的应用背景。     

Solution of a class of stochastic linear-convex control problems using deterministic equivalents

In this paper, we identify a new class of stochastic linearconvex optimal control problems, whose solution can be obtained by solving appropriate equivalent deterministic optimal control problems. The term linear-convex is meant to imply that the dynamics is linear and the cost function is convex in the state variables, linear in the control variables, and separable. Moreover, some of the coefficients in the dynamics are allowed to be random and the expectations of the control variables are allowed to be constrained. For any stochastic linear-convex problem, the equivalent deterministic problem is obtained. Furthermore, it is shown that the optimal feedback policy of the stochastic problem is affine in its current state, where the affine transformation depends explicitly on the optimal solution of the equivalent deterministic problem in a simple way. The result is illustrated by its application to a simple stochastic inventory control problem.This research was supported in part by NSERC Grant A4617, by SSHRC Grant 410-83-0888, and by an INRIA Post-Doctoral Fellowship.     

A NARMAX model-based state-space self-tuning control for nonlinear stochastic hybrid systems

A novel state-space self-tuning control methodology for a nonlinear stochastic hybrid system with stochastic noise/disturbances is proposed in this paper. via the optimal linearization approach, an adjustable NARMAX-based noise model with estimated states can be constructed for the state-space self-tuning control in nonlinear continuous-time stochastic systems. Then, a corresponding adaptive digital control scheme is proposed for continuous-time multivariable nonlinear stochastic systems, which have unknown system parameters, measurement noise/external disturbances, and inaccessible system states. The proposed method enables the development of a digitally implementable advanced control algorithm for nonlinear stochastic hybrid systems.     

An inverse optimal problem in discrete-time stochastic control

In this paper, we study an inverse optimal problem in discrete-time stochastic control. We give necessary and sufficient conditions for a solution to a system of stochastic difference equations to be the solution of a certain optimal control problem. Our results extend to the stochastic case the work of Dechert. In particular, we present a stochastic version of an important principle in welfare economics.     

带停时的奇异型随机控制问题在金融投资模型中的应用

以随机分析的知识和最优控制理论为基础,讨论了一类带停时的奇异型随机控制的折扣费用问题在金融投资模型中的应用,将该带停时的奇异型随机控制模型的受控状态过程和费用函数结构都推广到了最一般的形式,使该模型的应用范围更加广泛.通过讨论一组相应的变分不等式的解,分别对退化和非退化两种情况给出了此随机控制问题的最优策略,相应得出了投资模型中的最佳决策,并且证明了变分不等式的解即为最优费用函数.与以往不同的是,所得的相关结论应用到了金融投资模型中,从而解决了一类金融投资问题.     

随机递归最优控制和混合最优控制问题

对随机递归最优控制问题即代价函数由特定倒向随机微分方程解来描述和递归混合最优控制问题即控制者还需 决定最优停止时刻, 得到了最优控制的存在性结果. 在一类等价概率测度集中,还给出了递归最优值函数的最小和最大数学期望.     

An optimal control problem with a random stopping time

This paper deals with a stochastic optimal control problem where the randomness is essentially concentrated in the stopping time terminating the process. If the stopping time is characterized by an intensity depending on the state and control variables, one can reformulate the problem equivalently as an infinite-horizon optimal control problem. Applying dynamic programming and minimum principle techniques to this associated deterministic control problem yields specific optimality conditions for the original stochastic control problem. It is also possible to characterize extremal steady states. The model is illustrated by an example related to the economics of technological innovation.This research has been supported by NSERC-Canada, Grants 36444 and A4952; by FCAR-Québec, Grant 88EQ3528, Actions Structurantes; and by MESS-Québec, Grant 6.1/7.4(28).     

Optimal control of random evolutions

We consider a stochastic control problem for a random evolution. We study the Bellman equation of the problem and we prove the existence of an optimal stochastic control which is Markovian. This problem enables us to approximate the general problem of the optimal control of solutions of stochastic differential equations.     

Sto chastic Maximum Principle for Optimal Control of Forward-backward Sto chastic Pantograph Systems with Regime Switching

In this paper, we derive the stochastic maximum principle for optimal control problems of the forward-backward Markovian regime-switching system. The control system is described by an anticipated forward-backward stochastic pantograph equation and modulated by a continuous-time finite-state Markov chain. By virtue of classical variational approach, duality method, and convex analysis, we obtain a stochastic maximum principle for the optimal control.