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Qi Lü

Publications and source records attributed to Qi Lü.

At least 37 records · Page 2Linked to original sources
arXiv2024

Inverse problems for stochastic partial differential equations

This book aims to provide a brief overview of recent advancements in the theory of inverse problems for stochastic partial differential equations. In order to keep the content concise, we will only discuss the inverse problems of two typical classes of stochastic partial differential equations: second-order stochastic parabolic equations and secondorder stochastic hyperbolic equations. The main tool for studying these inverse problem is Carleman estimate. We do not intend to pursue any general treatment of the Carleman estimates themselves and choose direct arguments based on basic stochastic calculus, rather than more general sophisticated methods. As this field is still developing and there are many challenging issues to be addressed, the purpose of this book is not to serve as a comprehensive summary, but rather to spark interest and encourage further exploration in this area among readers. We prefer to present results that, from our perspective, include fresh and promising ideas. In cases where a complete mathematical theory is lacking, we only provide the available results. We do not intend for the current book to be encyclopedic in any sense, and the references are limited.

math.PR↗
arXiv2023

Time-inconsistent Linear Quadratic Optimal Control Problem for Forward-Backward Stochastic Differential Equations

We study the time-inconsistent linear quadratic optimal control problem for forward-backward stochastic differential equations with potentially indefinite cost weighting matrices for both the state and the control variables. Our research makes two contributions. Firstly, we introduce a novel type of Riccati equation system with parameters and constraint conditions, known as the generalized equilibrium Riccati equation. This equation system offers a comprehensive solution for the closed-loop equilibrium strategy of the problem at hand. Secondly, we establish the well-posedness of the generalized equilibrium Riccati equation for the one-dimensional case, provided certain conditions are met.

math.OC↗
arXiv2023

Carleman Estimates for Second Order Elliptic Operators with Limiting Weights, an Elementary Approach

By using some deep tools from microlocal analysis, the authors of the papers (Ann. of Math., 165 (2007), 567--591, J. Amer. Math. Soc., 23 (2010), 655--691; Invent. Math., 178 (2009), 119--171; Duke Math. J., 158(2011), 83--120) have successfully established various Carleman estimates for elliptic operators that possess limiting Carleman weight. In this study, we revisit these problems and present a unified and fundamental approach for deriving these estimates. The main tool we employ is an elementary pointwise estimate for second-order elliptic operators.

math.AP↗
arXiv2023

Exact controllability for a refined Stochastic Wave Equation

In this paper, we obtain the exact controllability for a refined stochastic wave equation with three controls by establishing a novel Carleman estimate for a backward hyperbolic-like operator. Compared with the known result, the novelty of this paper is twofold: (1) Our model contains the effects in the drift terms when we put controls directly in the diffusion terms, which is more sensible for practical applications; (2) We provide an explicit description of the waiting time which is sharp in the case of dimension one and is independent of the coefficients of lower terms.

math.OC↗
arXiv2023

Stability Estimate for an Inverse Stochastic Parabolic Problem of Determining Unknown Time-varying Boundary

Stochastic parabolic equations are widely used to model many random phenomena in natural sciences, such as the temperature distribution in a noisy medium, the dynamics of a chemical reaction in a noisy environment, or the evolution of the density of bacteria population. In many cases, the equation may involve an unknown moving boundary which could represent a change of phase, a reaction front, or an unknown population. In this paper, we focus on an inverse problem where the goal is to determine an unknown moving boundary based on data observed in a specific interior subdomain for the stochastic parabolic equation and prove that the unknown boundary depends logarithmically on the interior measurement. This allows us, theoretically, to track and to monitor the behavior of unknown boundary from observation in an arbitrary interior domain. The stability estimate is based on a new Carleman estimate for stochastic parabolic equations. As a byproduct, we obtain a quantitative unique continuation property for stochastic parabolic equations.

math.AP↗
arXiv2022

Exact Controllability for a Refined Stochastic Plate Equation

A widely used stochastic plate equation is the classical plate equation perturbed by a term of Itô's integral. However, it is known that this equation is not exactly controllable even if the controls are effective everywhere in both the drift and the diffusion terms and also on the boundary. In some sense, this means that some key feature has been ignored in this model. Then, a one-dimensional refined stochastic plate equation is proposed and its exact controllability is established in [28]. In this paper, by means of a new global Carleman estimate, we establish the exact controllability of the multidimensional refined stochastic plate equation with two interior controls and two boundary controls. Moreover, we give a result about the lack of exact controllability, which shows that the action of two interior controls and at least one boundary control is necessary.

math.OC↗
arXiv2022

Strong Unique Continuation Property for Stochastic Parabolic Equations

We establish a strong unique continuation property for stochastic parabolic equations. Our method is based on a suitable stochastic version of Carleman estimate. As far as we know, this is the first result for strong unique continuation property of stochastic partial differential equations.

math.AP↗
arXiv2022

Stochastic Verification Theorem for Infinite Dimensional Stochastic Control Systems

The verification theorem serving as an optimality condition for the optimal control problem, has been expected and studied for a long time. The purpose of this paper is to establish this theorem for control systems governed by stochastic evolution equations in infinite dimensions, in which both the drift and the diffusion terms depend on the controls.

math.OC↗
arXiv2022

Optimal Feedback Controls of Stochastic Linear Quadratic Control Problems in Infinite Dimensions with Random Coefficients

It is a longstanding unsolved problem to characterize the optimal feedback controls for general linear quadratic optimal control problem of stochastic evolution equation with random coefficients. A solution to this problem is given in [21] under some assumptions which can be verified for interesting concrete models, such as controlled stochastic wave equations, controlled stochastic Schrödinger equations, etc. More precisely, the authors establish the equivalence between the existence of optimal feedback operator and the solvability of the corresponding operator-valued, backward stochastic Riccati equations. However, their result cannot cover some important stochastic partial differential equations, such as stochastic heat equations, stochastic stokes equations, etc. A key contribution of the current work is to relax the $C_0$-group assumption of unbounded linear operator $A$ in [21] and using contraction semigroup assumption instead. Therefore, our result can be well applicable in the linear quadratic problem of stochastic parabolic partial differential equations. To this end, we introduce a suitable notion to the aforementioned Riccati equation, and some delicate techniques which are even new in the finite dimensional case.

math.OC↗
arXiv2022

Null controllability for stochastic fourth order parabolic equations

We establish the null controllability for linear stochastic fourth order parabolic equations. Utilizing the duality argument, the null controllability is reduced to the observability for backward fourth order stochastic parabolic equations, and the desired observability estimate is obtained by a new global Carleman estimate. Our Carleman estimate is based on a new fundamental identity for a stochastic fourth order parabolic operator.

math.OC↗
arXiv2021

Relationships Between the Maximum Principle and Dynamic Programming for Infinite Dimensional Stochastic Control Systems

Pontryagin type maximum principle and Bellman's dynamic programming principle serve as two of the most important tools in solving optimal control problems. There is a huge literature on the study of relationship between them. The main purpose of this paper is to investigate the relationships between Pontryagin type maximum principle and dynamic programming principle for control systems governed by stochastic evolution equations in infinite dimensional space, with the control variables appearing into both the drift and the diffusion terms. To do so, we first establish dynamic programming principle for those systems without employing the martingale solutions. Then we establish the desired relationships in both cases that value function associated is smooth and nonsmooth. For the nonsmooth case, in particular, by employing the relaxed transposition solution, we discover the connection between the superdifferentials and subdifferentials of value function and the first-order and second-order adjoint equations.

math.OC↗
arXiv2021

Numerics for Stochastic Distributed Parameter Control Systems: a Finite Transposition Method

In this chapter, we present some recent progresses on the numerics for stochastic distributed parameter control systems, based on the \emph{finite transposition method} introduced in our previous works. We first explain how to reduce the numerics of some stochastic control problems in this respect to the numerics of backward stochastic evolution equations. Then we present a method to find finite transposition solutions to such equations. At last, we give an illuminating example.

math.OC↗
arXiv2021

A Concise Introduction to Control Theory for Stochastic Partial Differential Equations

The aim of this notes is to give a concise introduction to control theory for systems governed by stochastic partial differential equations. We shall mainly focus on controllability and optimal control problems for these systems. For the first one, we present results for the exact controllability of stochastic transport equations, null and approximate controllability of stochastic parabolic equations and lack of exact controllability of stochastic hyperbolic equations. For the second one, we first introduce the stochastic linear quadratic optimal control problems and then the Pontryagin type maximum principle for general optimal control problems. It deserves mentioning that, in order to solve some difficult problems in this field, one has to develop new tools, say, the stochastic transposition method introduced in our previous works.

math.OC↗
arXiv2019

Carleman Estimates for Parabolic Operators with Discontinuous and Anisotropic Diffusion Coefficients, an Elementary Approach

By using some deep tools from microlocal analysis, J. Le Rousseau and L. Robbiano (Invent. Math., 183 (2011), 245--336) established several Carleman estimates for parabolic operators with isotropic diffusion coefficients which have jumps at interfaces. In this paper, we revisit the same problem but for the general case of anisotropic diffusion coefficients. Our main tools are a pointwise estimate for parabolic operators and a suitable chosen weight function.

math.OC↗
arXiv2019

First and Second Order Necessary Optimality Conditions for Controlled Stochastic Evolution Equations with Control and State Constraints

The purpose of this paper is to establish first and second order necessary optimality conditions for optimal control problems of stochastic evolution equations with control and state constraints. The control acts both in the drift and diffusion terms and the control region is a nonempty closed subset of a separable Hilbert space. We employ some classical set-valued analysis tools and theories of the transposition solution of vector-valued backward stochastic evolution equations and the relaxed-transposition solution of operator-valued backward stochastic evolution equations to derive these optimality conditions. The correction part of the second order adjoint equation, which does not appear in the first order optimality condition, plays a fundamental role in the second order optimality condition.

math.OC↗
arXiv2019

Well-posedness of Stochastic Riccati Equations and Closed-Loop Solvability for Stochastic Linear Quadratic Optimal Control Problems

We study the closed-loop solvability of a stochastic linear quadratic optimal control problem for systems governed by stochastic evolution equations. This solvability is established by means of solvability of the corresponding Riccati equation, which is implied by the uniform convexity of the quadratic cost functional. At last, conditions ensuring the uniform convexity of the cost functional are discussed.

math.OC↗
arXiv2019

Exact Controllability for a Refined Stochastic Wave Equation

A widely used stochastic wave equation is the classical wave equation perturbed by a term of Itô's integral. We show that this equation is not exactly controllable even if the controls are effective everywhere in both the drift and the diffusion terms and also on the boundary. In some sense this means that some key feature has been ignored in this model. Then, based on a stochastic Newton's law, we propose a refined stochastic wave equation. By means of a new global Carleman estimate, we establish the exact controllability of our stochastic wave equation with three controls. Moreover, we give a result about the lack of exact controllability, which shows that the action of three controls is necessary. Our analysis indicates that, at least from the point of view of control theory, the new stochastic wave equation introduced in this paper is a more reasonable model than that in the existing literatures.

math.OC↗