arXiv subjects

Qi Lü

Publications and source records attributed to Qi Lü.

At least 19 recordsLinked to original sources

arXiv2026

Autonomous Flows: Exact Finite Interpolation and Uniform Approximation Obstructions

In dimension at least two, a class of locally Lipschitz vector fields realizes every finite correspondence between distinct inputs and distinct targets at any prescribed positive time, provided that it is linear, its members generate global flows, and it approximates every smooth vector field of bounded support uniformly on bounded sets. The input and target sets may overlap. The proof constructs a smooth autonomous reference flow and finitely many localized correction fields, then uses uniform stability and Brouwer degree to obtain exact endpoints within the approximating class. The argument uses only uniform approximation of the vector fields; it does not require control of their derivatives. Applied to shallow ReLU vector fields, the result gives exact finite interpolation without time-dependent coefficients or additional state variables, together with bounds on width and normalized coefficient strength. In contrast, no continuous autonomous semiflow can exchange and compress two disjoint balls. Its time maps therefore fail to approximate all continuous maps uniformly on compact sets. The results distinguish exact interpolation at finitely many points from uniform control of neighborhoods.

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arXiv2026

Stability Estimates for the Inverse Recovery of Drift and Diffusion Point Sources in Stochastic Parabolic Equations

This paper studies the simultaneous recovery of point-source locations and temporal strengths in both the drift and diffusion terms of a stochastic parabolic equation with Neumann boundary conditions, using only boundary observations on a nonempty portion of the boundary after the sources have become inactive. By separating the stochastic equation into its expectation and centered parts, the drift channel reduces to a deterministic source problem with scalar Laplace moments, while the diffusion channel is controlled by Itô's isometry, which preserves the full $L^2$-energy of the integrand. Under suitable nondegeneracy and separation assumptions, we establish Lipschitz stability for the location of a single source in either channel, logarithmic stability for the drift strength, and Lipschitz stability for the complete diffusion strength including its sign. For multiple sources, we obtain Lipschitz stability up to permutation for drift locations and weighted temporal moments in dimensions two and three, and for diffusion locations and strengths in dimensions one through three. The proof employs explicit adjoint probes, including complex-isotropic polynomial annihilators in dimensions two and three and spectral synthesis in one dimension, combined with boundary-to-source estimates derived from Green's identities. These results extend and improve upon existing deterministic stability theories by revealing the fundamentally different temporal information carried by stochastic integrals, a distinction that is further illustrated by numerical experiments.

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arXiv2026

A Geometric Inverse Source Problem for Stochastic Parabolic Equations

This paper addresses the geometric inverse problem of simultaneously recovering two unknown deterministic source supports in a stochastic parabolic equation, where one source appears in the drift term and the other in the diffusion term. We establish that partial boundary flux measurements alone uniquely determine both supports. Moreover, we prove the existence of minimizers for a perimeter-regularized objective functional. For smooth interfaces, we derive a Hadamard-type boundary representation of the shape derivative. To the best of our knowledge, this is the first such formula for the simultaneous recovery of a drift-source support and a diffusion-source support in an SPDE setting. The derivative exhibits a genuinely stochastic two-channel structure: the adjoint state governs the sensitivity of the drift-source interface, while the martingale component controls the sensitivity of the diffusion-source interface. Based on this formula, we develop a shape-gradient reconstruction method. Numerical experiments demonstrate its effectiveness and its capacity to distinguish between the two source channels under both full and partial boundary observations.

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arXiv2026

Stackelberg Stochastic Linear-Quadratic Differential Games: A Closed-Loop Equilibrium Approach

This paper addresses a Stackelberg stochastic linear-quadratic (LQ) differential game under closed-loop information, a problem inherently time-inconsistent. Existing approaches rely on solving two coupled Hamilton-Jacobi-Bellman (HJB) equations derived via time discretization and a limiting argument, whose convergence remains an open problem. We propose an alternative framework based on closed-loop equilibrium strategies. We reformulate the leader's problem as a forward-backward optimal control problem involving a coupled system of forward SDEs and backward Riccati equations. Due to the presence of controlled Riccati equations, the leader's problem becomes essentially nonlinear. Using a variational method, we characterize the leader's closed-loop equilibrium strategy and derive the associated equilibrium Riccati equation (ERE). A key conceptual distinction is that the follower adopts a globally optimal strategy against any admissible control of the leader, whereas in previous literature the follower's strategy was only locally optimal along the leader's specific equilibrium path. This makes the follower's strategy more robust and the leader's commitment more credible. In our LQ setting, the resulting ERE coincides exactly with the coupled HJB system from the literature, showing the leader's strategy is equivalent to the feedback Stackelberg solution. Thus, our framework provides not only an alternative derivation but also a rigorous justification of the limiting argument. We establish a priori estimates for the ERE, covering 1D and high-dimensional cases, ensuring global well-posedness for any finite horizon. This significantly extends existing results which require a sufficiently short time horizon or control-independent diffusion. An application to an asset management problem with numerical simulations illustrates the theoretical results.

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arXiv2026

Unique Continuation Property for Stochastic Wave Equations

This paper establishes a fundamental and surprising phenomenon in the theory of stochastic wave equations: the restoration of the unique continuation property (UCP) across characteristic hypersurfaces, a property that is known to fail generically in the deterministic setting. We prove that if a solution to a linear stochastic wave equation vanishes on one side of a characteristic surface $Γ$, then it must vanish in a full neighborhood of any point on $Γ$, provided the stochastic diffusion coefficient is non-degenerate. This result stands in sharp contrast to the classical Hörmander-type counterexamples for deterministic waves. Furthermore, we extend the UCP to equations with non-homogeneous stochastic sources and establish a global unique continuation result from the interior of an arbitrarily narrow characteristic cone. Our proofs rely on a novel stochastic Carleman estimate, where the Itô diffusion term introduces a crucial positive energy contribution that is absent in deterministic models. These findings demonstrate a qualitative difference between deterministic and stochastic hyperbolic dynamics and open new avenues for control theory and inverse problems in stochastic setting.

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arXiv2026

Exact Controllability for Stochastic First-Order Multi-Dimensional Hyperbolic Systems

This paper investigates the exact controllability problem for multi-dimensional stochastic first-order symmetric hyperbolic systems with control inputs acting in two distinct ways: an internal control applied to the diffusion term and a boundary control applied to the drift term. By means of a classical duality argument, the controllability problem is reduced to an observability estimate for the corresponding backward stochastic system. The main technical contribution is the establishment of a new global Carleman estimate for such backward systems, combined with a weighted energy identity. This enables us to prove the desired observability inequality under a geometric structural condition (Condition \ref{cond1}), which ensures that all characteristic rays propagate toward the boundary within a finite time. As a result, we obtain exact controllability provided the control time $T$ exceeds a sharp threshold $T_0$ given explicitly in terms of the system geometry. Furthermore, we complement the positive result with several negative controllability theorems, which demonstrate that both controls are necessary and must act in a distributed manner. Our analysis not only extends controllability theory from deterministic to stochastic multi-dimensional hyperbolic systems but also provides, as a byproduct, new results for deterministic systems under a structural hypothesis. Applications to stochastic traffic flow, epidemiological models, and shallow-water equations are discussed.

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arXiv2025

Mean-Square Stability of Continuous-Time Stochastic Model Predictive Control

We propose a stochastic model predictive control (SMPC) framework for a broad class of unconstrained controlled stochastic differential equations (SDEs) and establish its mean-square exponential stability in the infinite-horizon limit. At each prediction step of the MPC iteration, the nonlinear controlled SDE is approximated by its linearization at the origin, with the sampled state of the nonlinear system as initial condition, yielding a finite-horizon stochastic linear-quadratic (SLQ) optimal control problem. The resulting optimal control is then applied to the original nonlinear stochastic dynamics until the next sampling instant. This construction leads to a delayed SMPC scheme whose closed-loop behavior is governed by a coupled time-delay SDE system, a setting that has not been analyzed before. We prove global mean-square exponential stability for linear and mildly nonlinear SDEs by exploiting the exponential convergence of the Riccati equation to the algebraic Riccati equation (ARE). For strongly nonlinear SDEs, we establish local mean-square exponential stability by combining exponential Riccati convergence with stopping-time techniques and Grönwall-type estimates. It is observed that, to ensure the desired local stability properties, the nonlinearities of the SDE are allowed to have polynomial growth but not exponential growth, distinguishing SMPC from its deterministic counterpart. These results provide the first rigorous mean-square stability guarantees for SMPC of SDE systems with delayed state information, thereby advancing the theoretical foundations of stochastic predictive control.

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arXiv2025

Null control of heat equations with analytic memory kernels

We analyze the control properties of heat equations with memory terms. We recall previous results showing that if the moving support of the control covers the whole domain where heat diffuses, the system is null controllable when the memory kernel is polynomial. We formulate the problem of extending this result to the case of some more general memory kernels, in particular analytic ones.

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arXiv2025

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

This paper investigates the relationship between Pontryagin's maximum principle and dynamic programming principle in the context of stochastic optimal control systems governed by stochastic evolution equations with random coefficients in separable Hilbert spaces. Our investigation proceeds through three contributions: (1). We first establish the formulation of the dynamic programming principle for this class of infinite-dimensional stochastic systems, subsequently deriving the associated stochastic Hamilton-Jacobi-Bellman equations that characterize the value function's evolution. (2). For systems with smooth value functions, we develop explicit correspondence relationships between Pontryagin's maximum principle and dynamic programming principle, elucidating their fundamental connections through precise mathematical characterizations. (3). In the more challenging non-smooth case, we employ tools in nonsmooth analysis and relaxed transposition solution techniques to uncover previously unknown sample-wise relationships between the two principles.

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arXiv2025

Dynamic Programming Principle for Stochastic Control Problems on Riemannian Manifolds

In this paper, we first establish the dynamic programming principle for stochastic optimal control problems defined on compact Riemannian manifolds without boundary. Subsequently, we derive the associated Hamilton-Jacobi-Bellman (HJB) equation for the value function. We then prove the existence, uniqueness of viscosity solutions to the HJB equation, along with their continuous dependence on initial data and model parameters. Finally, under appropriate regularity conditions on the value function, we establish a verification theorem that characterizes optimal controls.

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arXiv2025

An Inverse Source Problem for Semilinear Stochastic Hyperbolic Equations

This paper investigates an inverse source problem for general semilinear stochastic hyperbolic equations. Motivated by the challenges arising from both randomness and nonlinearity, we develop a globally convergent iterative regularization method that combines Carleman estimate with fixed-point iteration. Our approach enables the reconstruction of the unknown source function from partial lateral Cauchy data, without requiring a good initial guess. We establish a new Carleman estimate for stochastic hyperbolic equations and prove the convergence of the proposed method in weighted spaces. Furthermore, we design an efficient numerical algorithm that avoids solving backward stochastic partial differential equations and is robust to randomness in both the model and the data. Numerical experiments are provided to demonstrate the effectiveness of the method.

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arXiv2025

Forward-Backward Stochastic Linear-Quadratic Optimal Controls: Equilibrium Strategies and Non-Symmetric Riccati Equations

Linear-quadratic optimal control problem for systems governed by forward-backward stochastic differential equations has been extensively studied over the past three decades. Recent research has revealed that for forward-backward control systems, the corresponding optimal control problem is inherently time-inconsistent. Consequently, the optimal controls derived in existing literature represent pre-committed solutions rather than dynamically consistent strategies. In this paper, we shift focus from pre-committed solutions to addressing the time-inconsistency issue directly, adopting a dynamic game-theoretic approach to derive equilibrium strategies. Owing to the forward-backward structure, the associated equilibrium Riccati equation (ERE) constitutes a coupled system of matrix-valued, non-local ordinary differential equations with a non-symmetric structure. This non-symmetry introduces fundamental challenges in establishing the solvability of the EREs. We overcome the difficulty by establishing a priori estimates for a combination of the solutions to EREs, which, interestingly, is a representation of the equilibrium value function.

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arXiv2025

Exact Controllability for a Refined Stochastic Hyperbolic Equation with Internal Controls

We establish the internal exact controllability of a refined stochastic hyperbolic equation by deriving a suitable observability inequality via Carleman estimates for the associated backward stochastic hyperbolic equation. In contrast to existing results on boundary exact controllability--which require longer waiting times, we demonstrate that the required waiting time for internal exact controllability in stochastic hyperbolic equations coincides exactly with that of their deterministic counterparts.

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arXiv2024

On Inverse Problems for Mean Field Games with Common Noise via Carleman Estimate

In this paper, we study two kinds of inverse problems for Mean Field Games (MFGs) with common noise. Our focus is on MFGs described by a coupled system of stochastic Hamilton-Jacobi-Bellman and Fokker-Planck equations. Firstly, we establish the Lipschitz and Hölder stability for determining the solutions of a coupled system of stochastic Hamilton-Jacobi-Bellman and Fokker-Planck equations based on terminal observation of the density function. Secondly, we derive a uniqueness theorem for an inverse source problem related to the system under consideration. The main tools to establish those results are two new Carleman estimates.

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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.

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arXiv2024

Finite Codimensionality Method in Infinite-dimensional Optimization Problems

This paper is devoted to establishing an enhanced Fritz John type first-order necessary condition for a general constrained nonlinear infinite-dimensional optimization problem. Unlike traditional constraint qualifications in optimization theory, a condition of finite codimensionality is employed to ensure the existence of nontrivial Lagrange multipliers. As applications, first-order necessary conditions for optimal control problems of some deterministic/stochastic control systems are derived in a unified manner. Compared with the existing constraint qualifications, the finite codimensionality condition, which is equivalent to some suitable {\it a priori} estimates, can offer a more straightforward verification process in these applications.

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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.

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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.

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