Optimal control of membrane filtration systems

Optimal stopping, randomized stopping and singular control with partial information flow

The purpose of this paper is two-fold: We extend the well-known relation between optimal stopping and randomized stopping of a given stochastic process to a situation where the available information flow is a filtration with no a priori assumed relation to the filtration of the process. We call these problems optimal stopping and randomized stopping with general information. Following an idea of Krylov [K] we introduce a special singular stochastic control problem with general information and show that this is also equivalent to the partial information optimal stopping and randomized stopping …

两相膜问题的最优控制

Optimal Control of Two-Phase Membrane Problem

本文研究状态由两相膜自由边界问题支配、控制作用于解正负部特征函数系数的最优控制问题，通过正则化处理不可微的控制-状态映射，建立了最优对的存在性、唯一性与刻画。

Stochastic Linear-Quadratic Optimal Control with Partial Observation

The paper studies a class of quadratic optimal control problems for partially observable linear dynamical systems. In contrast to the full information case, the control is required to be adapted to the filtration generated by the observation system, which in turn is influenced by the control. The variation method fails in this case due to the fact that the filtration is not fixed. To overcome the difficulty, we use the orthogonal decomposition of the state process to write the cost functional as the sum of two parts: one is a functional of the control and the filtering process and the other pa…

Optimal configuration and symmetry breaking phenomena in the composite membrane problem with fractional Laplacian

We consider the following eigenvalue optimization in the composite membrane problem with fractional Laplacian: given a bounded domain $Ω\subset \mathbb{R}^n$, $α>0$ and $0<A<|Ω|$, find a subset $D\subset Ω$ of area $A$ such that the first Dirichlet eigenvalue of the operator $(-Δ)^s+αχ_D$ is as small as possible. The solution $D$ is called as an optimal configuration for the data $(Ω,α,A)$. Looking at the well-known extension definition for the fractional Laplacian, in the case $s=1/2$ this is essentially the composite membrane problem for which the mass is concentrated at the boundary as one …

McKean-Vlasov optimal control: limit theory and equivalence between different formulations

We study a McKean-Vlasov optimal control problem with common noise, in order to establish the corresponding limit theory, as well as the equivalence between different formulations, including the strong, weak and relaxed formulation. In contrast to the strong formulation, where the problem is formulated on a fixed probability space equipped with two Brownian filtrations, the weak formulation is obtained by considering a more general probability space with two filtrations satisfying an $(H)$-hypothesis type condition from the theory of enlargement of filtrations. When the common noise is uncontr…

Stochastic linear-quadratic control with a jump and regime switching on a random horizon

In this paper, we study a stochastic linear-quadratic control problem with random coefficients and regime switching on a horizon $[0,T\wedgeτ]$, where $τ$ is a given random jump time for the underlying state process and $T$ is a constant. We obtain an explicit optimal state feedback control and explicit optimal cost value by solving a system of stochastic Riccati equations (SREs) with jumps on $[0,T\wedgeτ]$. By the decomposition approach stemming from filtration enlargement theory, we express the solution of the system of SREs with jumps in terms of another system of SREs involving only Brown…

Backward stochastic differential equations with conditional reflection and related recursive optimal control problems

We introduce a new type of reflected backward stochastic differential equations (BSDEs) for which the reflection constraint is imposed on its main solution component, denoted as $Y$ by convention, but in terms of its conditional expectation $\mathbb{E}[Y_t|\mathcal{G}_{t}]$ on a general sub-filtration $\{\mathcal{G}_{t}\}.$ We thus term such equation as conditionally reflected BSDE (for short, conditional RBSDE). Conditional RBSDE subsumes classical RBSDE with a pointwise reflection barrier, and the recent developed BSDE with a mean reflection constraint, as its two special and extreme cases: …

有限时间内实现膜分离的最小能量消耗的测地线界

Geodesic bound of the minimum energy expense to achieve membrane separation within finite time

该研究推导出有限时间内膜分离最小额外能耗的基本极限，证明其与初终态几何距离平方成正比、与操作时间成反比，指出优化分离方案等价于找几何空间测地线，且最优控制由对称破缺方案实现。

具有可分离漂移不确定性的最优自适应控制

Optimal adaptive control with separable drift uncertainty

研究可分离漂移不确定性下的随机最优控制问题，将非马尔可夫问题转化为高维马尔可夫全信息控制问题，用随机Perron方法刻画值函数并构造ε-最优控制，验证强弱公式值一致，数值表明自适应控制优于确定性等价控制。

McKean-Vlasov optimal control: the dynamic programming principle

We study the McKean-Vlasov optimal control problem with common noise in various formulations, namely the strong and weak formulation, as well as the Markovian and non-Markovian formulations, and allowing for the law of the control process to appear in the state dynamics. By interpreting the controls as probability measures on an appropriate canonical space with two filtrations, we then develop the classical measurable selection, conditioning and concatenation arguments in this new context, and establish the dynamic programming principle under general conditions.

Time-symmetric optimal stochastic control problems in space-time domains

We present a pair of adjoint optimal control problems characterizing a class of time-symmetric stochastic processes defined on random time intervals. The associated PDEs are of free-boundary type. The particularity of our approach is that it involves two adjoint optimal stopping times adapted to a pair of filtrations, the traditional increasing one and another, decreasing. They are the keys of the time symmetry of the construction, which can be regarded as a generalization of "Schrödinger's problem" (1931-32) to space-time domains. The relation with the notion of "Hidden diffusions" is also de…