Hilbert空间中具无界控制的无限时区线性二次最优控制问题

本文研究了Hilbert空间中一类由解析半群支配的具无界控制的无限时区线性二次最优控制问题,其中指标中的控制项加权算子要求强制而状态项加权算子可允许为不定号.在指数能稳条件下,证明了任意的最优控制及其最优轨线必定连续,建立了正实引理作为此问题唯一可解的充要条件,并用代数Riccati方程的解给出了最优控制的闭环综合。     

Backward Stochastic Riccati Equations and Infinite Horizon L-Q Optimal Control with Infinite Dimensional State Space and Random Coefficients

We study the Riccati equation arising in a class of quadratic optimal control problems with infinite dimensional stochastic differential state equation and infinite horizon cost functional. We allow the coefficients, both in the state equation and in the cost, to be random. In such a context backward stochastic Riccati equations are backward stochastic differential equations in the whole positive real axis that involve quadratic non-linearities and take values in a non-Hilbertian space. We prove existence of a minimal non-negative solution and, under additional assumptions, its uniqueness. We show that such a solution allows to perform the synthesis of the optimal control and investigate its attractivity properties. Finally the case where the coefficients are stationary is addressed and an example concerning a controlled wave equation in random media is proposed.     

Indefinite Stochastic Linear Quadratic Control with Markovian Jumps in Infinite Time Horizon

This paper studies a stochastic linear quadratic (LQ) control problem in the infinite time horizon with Markovian jumps in parameter values. In contrast to the deterministic case, the cost weighting matrices of the state and control are allowed to be indinifite here. When the generator matrix of the jump process – which is assumed to be a Markov chain – is known and time-invariant, the well-posedness of the indefinite stochastic LQ problem is shown to be equivalent to the solvability of a system of coupled generalized algebraic Riccati equations (CGAREs) that involves equality and inequality constraints. To analyze the CGAREs, linear matrix inequalities (LMIs) are utilized, and the equivalence between the feasibility of the LMIs and the solvability of the CGAREs is established. Finally, an LMI-based algorithm is devised to slove the CGAREs via a semidefinite programming, and numerical results are presented to illustrate the proposed algorithm.     

随机线性二次最优控制:从离散到连续时间模型

在一般情形下,分析了离散时间LQ问题与连续时间情形两者之间的自然联系.首先回顾了连续时间和离散时间随机LQ问题及对应Riccati微分/差分方程的相关结论.接下来在假设Riccati微分方程有解的前提下,证明了离散化步长足够小时,Riccati差分方程有解.然后针对连续和离散时间模型,采用配对问题最优控制的反馈形式,分别构造了一个辅助反馈控制,并证明该控制可驱使对应模型的性能指标逼近于配对问题的值函数,以此得到了关于两个模型之间联系的初步结论.最后藉由前述结论以及控制问题的特性,揭晓了连续时间和离散时间模型之间的自然联系,并给出了Riccati差分方程和微分方程的解之间的误差估计.由此联系,可构造相应离散系统和LQ问题,以适当的阶估计连续时间LQ问题的解,抑或为离散时间模型构造一个近似最优控制.无论哪种思路,都旨在降低直接求解原问题的难度和复杂性.     

On the Bellman Equation for Infinite Horizon Problems with Unbounded Cost Functional

We study a class of infinite horizon control problems for nonlinear systems, which includes the Linear Quadratic (LQ) problem, using the Dynamic Programming approach. Sufficient conditions for the regularity of the value function are given. The value function is compared with sub- and supersolutions of the Bellman equation and a uniqueness theorem is proved for this equation among locally Lipschitz functions bounded below. As an application it is shown that an optimal control for the LQ problem is nearly optimal for a large class of small unbounded nonlinear and nonquadratic pertubations of the same problem. Accepted 8 October 1998     

一类非线性二阶微分方程无穷边值问题的多重正无界解

本文通过构造—个特殊的锥,利用锥拉压不动点定理,证明了一类非线性二阶微分方程无穷边值问题的两个正无界解的存在性。     

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??We study the linear quadratic optimal stochastic control problem which is jointly driven by Brownian motion and L\'{e}vy processes. We prove that the new affine stochastic differential adjoint equation exists an inverse process by applying the profound section theorem. Applying for the Bellman's principle of quasilinearization and a monotone iterative convergence method, we prove the existence and uniqueness of the solution of the backward Riccati differential equation. Finally, we prove that the optimal feedback control exists, and the value function is composed of the initial value of the solution of the related backward Riccati differential equation and the related adjoint equation.     

A Finite Horizon Linear Quadratic Optimal Stochastic Control Problem Driven by Both Brownian Motion and L\'{e}vy Processes

We study the linear quadratic optimal stochastic control problem which is jointly driven by Brownian motion and L\'{e}vy processes. We prove that the new affine stochastic differential adjoint equation exists an inverse process by applying the profound section theorem. Applying for the Bellman's principle of quasilinearization and a monotone iterative convergence method, we prove the existence and uniqueness of the solution of the backward Riccati differential equation. Finally, we prove that the optimal feedback control exists, and the value function is composed of the initial value of the solution of the related backward Riccati differential equation and the related adjoint equation.     

Banach空间非线性脉冲微分方程无穷边值问题

采用上了解方法和单调迭代技术研究Ｂａｎａｃｈ空间一阶非线性微分方程无穷边值问题,并建立了其最大解和最小解的存性定理。     

Clifford分析中无界域上正则函数的边值问题

在引入修正Cauchy核的基础上,讨论了无界域上正则函数的带共轭值的边值问题：a(t)Φ (t) b(t)Φ (t) c(t)Φ-(t) d(t)Φ=g(t)．首先给出了无界域上正则函数的Plemelj公式,然后利用积分方程方法和压缩不动点原理证明了问题解的存在唯一性．     

Banach空间中含有无穷多个跳跃点的一阶脉冲积分-微分方程的无穷边值问题

通过建立一个新的比较引理,应用上下解方法和单调迭代技术,研究了Banach空间中含有无穷多个跳跃点的一阶脉冲积分-微分方程无穷边值问题在任意闭区间上最小解和最大解的存在性.     

四元数分析中无界域上正则函数的带位移非线性边值问题

讨论了四元数分析中无界域上正则函数的一类带位移非线性边值问题.利用积分方程方法和Schauder不动点理论证明了问题解的存在性,并给出了解的积分表达式.     

带裂缝的不同材料拼接平面混合问题

本文讨论了带若干条任意形状裂缝的不同材料拼接平面的混合问题，即已知裂缝一侧的位移和另一侧的外应力求弹性平衡，给出了混合问题的正确提法，问题是用复就方法求解的，并归结为求解某种正则型奇异积分方程组，证明了适当且唯一地选择某些待定常数的值，该方程组有唯一解。     

Asymptotic Behavior of the Eigenvalues of the Schrodinger Operator with Transversal Potential in a Weakly Curved Infinite Cylinder

In this paper, we derive sufficient conditions for the existence of an eigenvalue for the Laplace and the Schrodinger operators with transversal potential for homogeneous Dirichlet boundary conditions in a tube, i.e., in a curved and twisted infinite cylinder. For tubes with small curvature and small internal torsion, we derive an asymptotic formula for the eigenvalue of the problem. We show that, under certain relations between the curvature and the internal torsion of the tube, the above operators possess no discrete spectrum.__________Translated from Matematicheskie Zametki, vol. 77, no. 5, 2005, pp. 656–664.Original Russian Text Copyright ©2005 by V. V. Grushin.     

Banach空间一类奇异脉冲微分方程边值问题的正解

讨论了Banach空间中一类具有奇异性脉冲微分方程的边值问题,利用M(o|¨)nch不动点定理,在与相应的线性算子谱半径有关的条件下,获得了该边值问题正解的存在性.     

General Linear Quadratic Optimal Stochastic Control Problem Driven by a Brownian Motion and a Poisson Random Martingale Measure with Random Coefficients

In this article, we consider a linear-quadratic optimal control problem (LQ problem) for a controlled linear stochastic differential equation driven by a multidimensional Browinan motion and a Poisson random martingale measure in the general case, where the coefficients are allowed to be predictable processes or random matrices. By the duality technique, the dual characterization of the optimal control is derived by the optimality system (so-called stochastic Hamilton system), which turns out to be a linear fully coupled forward-backward stochastic differential equation with jumps. Using a decoupling technique, the connection between the stochastic Hamilton system and the associated Riccati equation is established. As a result, the state feedback representation is obtained for the optimal control. As the coefficients for the LQ problem are random, here, the associated Riccati equation is a highly nonlinear backward stochastic differential equation (BSDE) with jumps, where the generator depends on the unknown variables K, L, and H in a quadratic way (see (5.9) herein). For the case where the generator is bounded and is linearly dependent on the unknown martingale terms L and H, the existence and uniqueness of the solution for the associated Riccati equation are established by Bellman's principle of quasi-linearization.     

Clifford 分析中无界域上正则函数带Haseman位移的边值问题

该文在引入修正的Cauchy核的基础上,讨论了Clifford 分析中无界域上正则函数带 Haseman 位移的边值问题. 首先给出了无界域上Cauchy 型积分的Plemelj公式,再利用积分方程方法和压缩不动点定理证明了问题解的存在唯一性.     

Existence of Three Weak Solutions for a Class of Quasi-Linear Elliptic Operators with a Mixed Boundary Value Problem Containing $p(\cdot )$-Laplacian in a Variable Exponent Sobolev Space

In this paper, we consider a mixed boundary value problem to a class of nonlinear operators containing $p(\cdot )$-Laplacian. More precisely, we are concerned with the problem with the Dirichlet condition on a part of the boundary and the Steklov boundary condition on an another part of the boundary. We show the existence of at least three weak solutions under some hypotheses on given functions and the values of parameters.     

实Clifford分析中,广义双正则函数向量的带位移带共轭的非线性边值问题

首先得到了广义正则函数向量的 Plemelj公式 ,然后利用积分方程的方法和 Arzela- Ascoli定理 ,讨论了实 Clifford分析中广义双正则函数向量的带位移带共轭的非线性边值问题解的存在性 .     

双正则函数向量的带Haseman位移带共轭值的线性及非线性边值问题

首先利用积分方程的方法和Arzela-Ascoli定理讨论了实Clifford分析中双正则函数向量的带Haseman位移带共轭值的非线性边值问题解的存在性及其积分表达式,其次利用压缩映射原理解决了其线性边值问题解的存在唯一性及其积分表达式.