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Fengjiao Liu

Publications and source records attributed to Fengjiao Liu.

14 recordsLinked to original sources
arXiv2026

Continuous-Time Covariance Steering with Common Free-Final Time: Finite-Horizon Solutions and Infinite-Horizon Limits

This article studies the optimal common free-final time problem for steering the state covariance of a continuous-time stochastic linear system between prescribed initial and terminal covariance matrices. We first establish a deterministic reformulation of the SDE-constrained free-final time stochastic optimal control problem (SOCP). For the ensuing SOCP, we provide necessary conditions for optimality and establish sufficient conditions for the optimal common final time to be finite. Subsequently, we characterize the asymptotic behavior of the finite-horizon optimal solutions as the final time tends to infinity, the invariant sets and trajectories associated with this limiting regime, and derive the sensitivity of the Hamiltonian with respect to the common final time. Finally, leveraging these sensitivities, we develop a trust-region line-search algorithm together with an infinite-horizon case detection method, and demonstrate its performance on three different problems: (a) an illustrative covariance-steering example, (b) a demonstration of spacecraft maneuver using covariance control, and (c) Gaussian mixture to Gaussian mixture steering.

math.OC↗
arXiv2026

Reachability Analysis of the State Transition and State Covariance Matrices for an LTV System

In this paper, we study the reachability of two closely related matrices appearing in the analysis of linear time-varying (LTV) systems over a finite time interval, namely, its closed-loop state transition matrix via a state feedback control and its state covariance matrix starting from some given initial state covariance matrix. Under a mild assumption, we first characterize the set of closed-loop terminal state transition matrices reachable from the identity matrix using controls of the state feedback form. Then, we provide the set of terminal state covariance matrices reachable from any given positive definite initial state covariance matrix when the LTV system is not necessarily controllable. Both results are based on the solutions of corresponding matrix Riccati differential equations (RDE).

eess.SY↗
arXiv2026

Schrodinger Bridges and Density Steering Problems for Gaussian Mixtures Models in Discrete-Time

In this work, we revisit the discrete-time Schr\"{o}dinger Bridge (SB) and Density Steering (DS) problems for Gaussian mixture model (GMM) boundary distributions. Building on the existing literature, we construct a set of feasible Markovian policies that transport the initial distribution to the final distribution, and are expressed as mixtures of elementary component-to-component optimal policies. We then study the policy optimization within this feasible set in the context of discrete-time SBs and density-steering problems, respectively. We show that for minimum-effort density-steering problems, the proposed policy achieves the same control cost as existing approaches in the literature. For discrete-time SB problems, the proposed policy yields a cost smaller than or equal to that in the literature, resulting in a less conservative approximation. Finally, we study the continuous-time limit of our proposed discrete-time approach and show that it agrees with recently proposed approximations to the continuous-time SB for GMM boundary distributions. We illustrate this new result through two numerical examples.

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arXiv2025

Go With the Flow: Fast Diffusion for Gaussian Mixture Models

Schrodinger Bridges (SBs) are diffusion processes that steer, in finite time, a given initial distribution to another final one while minimizing a suitable cost functional. Although various methods for computing SBs have recently been proposed in the literature, most of these approaches require computationally expensive training schemes, even for solving low-dimensional problems. In this work, we propose an analytic parametrization of a set of feasible policies for steering the distribution of a dynamical system from one Gaussian Mixture Model (GMM) to another. Instead of relying on standard non-convex optimization techniques, the optimal policy within the set can be approximated as the solution of a low-dimensional linear program whose dimension scales linearly with the number of components in each mixture. The proposed method generalizes naturally to more general classes of dynamical systems, such as controllable linear time-varying systems, enabling efficient solutions to multi-marginal momentum SBs between GMMs, a challenging distribution interpolation problem. We showcase the potential of this approach in low-to-moderate dimensional problems such as image-to-image translation in the latent space of an autoencoder, learning of cellular dynamics using multi-marginal momentum SBs, and various other examples. The implementation is publicly available at https://github.com/georgeRapa/GMMflow.

cs.LG↗
arXiv2025

Reachability and Controllability Analysis of the State Covariance for Linear Stochastic Systems

This paper studies the set of terminal state covariances that are reachable over a finite time horizon from a given initial state covariance for a linear stochastic system with additive noise. For discrete-time systems, a complete characterization of the set of reachable state covariances is given. For continuous-time systems, we present an upper bound on the set of reachable state covariances. Moreover, for both linear discrete-time and continuous-time systems, necessary and sufficient conditions are provided for the controllability of the state covariance over a finite horizon.

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arXiv2024

Optimal Covariance Steering for Discrete-Time Linear Stochastic Systems

In this paper, we study the optimal control problem for steering the state covariance of a discrete-time linear stochastic system over a finite time horizon. First, we establish the existence and uniqueness of the optimal control law for a quadratic cost function. Then, we show the separation of the optimal mean and the covariance steering problems. We also develop efficient computational methods to solve for the optimal control law, which is identified as the solution to a semi-definite program. The effectiveness of the proposed approach is demonstrated through numerical examples. In the process, we also obtain some novel theoretical results for a matrix Riccati difference equation, which may be of independent interest.

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arXiv2023

Optimal Covariance Steering for Continuous-Time Linear Stochastic Systems With Additive Noise

In this paper, we study the problem of how to optimally steer the state covariance of a general continuous-time linear stochastic system over a finite time interval subject to additive noise. Optimality here means reaching a target state covariance with minimal control energy. The additive noise may include a combination of white Gaussian noise and abrupt "jump noise" that is discontinuous in time. We first establish the controllability of the state covariance for linear time-varying stochastic systems. We then turn to the derivation of the optimal control, which entails solving two dynamically coupled matrix ordinary differential equations (ODEs) with split boundary conditions. We show the existence and uniqueness of the solution to these coupled matrix ODEs, and thus those of the optimal control.

eess.SY↗
arXiv2023

Optimal Covariance Steering for Continuous-Time Linear Stochastic Systems With Multiplicative Noise

In this paper we study the finite-horizon optimal covariance steering problem for a continuous-time linear stochastic system subject to both additive and multiplicative noise. The noise can be continuous or it may contain jumps. Additive noise does not depend on the state or the control, whereas multiplicative noise has a magnitude proportional to the current state. The cost is assumed to be quadratic in both the state and the control. First, the controllability of the state covariance is established under mild assumptions. Then, the optimal control for steering the covariance is provided. Lastly, the existence and uniqueness of the optimal control is shown. In the process, we provide a result of independent interest regarding the maximal interval of existence of the solution to a matrix Riccati differential equation.

math.OC↗
arXiv2022

Distributed Feedback Control of Multi-Channel Linear Systems

In this paper it is established that any jointly controllable, jointly observable, multi-channel, discrete or continuous time linear system with a strongly connected neighbor (communication) graph can be exponentially stabilized with any pre-specified convergence rate using a time-invariant distributed linear control. As an illustration of how this finding can be used to deal with certain distributed tracking problems, a solution is given to a distributed set-point control problem for a continuous-time, multi-channel linear system in which each and every agent with access to the system is able to independently adjust its scalar-valued, controlled output to any desired set-point value. To better understand the constraints on controller design, the distributed control problem is recast as a classical decentralized control problem. Armed with the tools of decentralized control, including the notion of a "fixed spectrum", it is possible to show quite surprisingly that if the only information each agent is allowed to share with its neighbors is its measured output, then distributed stabilization in some cases is impossible. Using well-known decentralized control concepts, lower bounds are derived on the dimensions of the shared sub-states of local controllers which, if satisfied, guarantee that there will be no fixed closed-loop system eigenvalues to contend with. The decentralized control perspective also enables one to assert that without imposing a partitioning constraint, the closed-loop spectrum of any jointly controllable, jointly observable multi-channel linear system with a strongly connected neighbor graph, can be freely assigned with distributed feedback control. It is also explained why in the face of finite delays, exponential stabilization at any prescribed convergence rate can still be achieved with distributed control, at least for discrete-time multi-channel linear systems.

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arXiv2019

Structural Completeness of a Multi-channel Linear System with Dependent Parameters

It is well known that the "fixed spectrum" {i.e., the set of fixed modes} of a multi-channel linear system plays a central role in the stabilization of such a system with decentralized control. A parameterized multi-channel linear system is said to be "structurally complete" if it has no fixed spectrum for almost all parameter values. Necessary and sufficient algebraic conditions are presented for a multi-channel linear system with dependent parameters to be structurally complete. An equivalent graphical condition is also given for a certain type of parameterization.

eess.SP↗
arXiv2018

The Power Allocation Game on A Network: Computation Issue

In this paper two algorithms with the goal of generating the equilibrium set of the power allocation game first developed in \cite{allocation} are proposed. Based on the first algorithm, the geometric property of the pure strategy Nash equilibrium set will be proven to be a collection of convex polytopes. The second, simulation-based, algorithm is developed to overcome the shortcoming of the first algorithm in terms of generating the equilibrium set efficiently and then making policy-relevant predictions based on the set. The second algorithm will be usefully applied to a real-world case study, which draws on the current crisis between North Korea and certain key players including the US and China.

cs.GT↗
arXiv2017

A Distributed, Dynamical System View of Finite, Static Games

This paper contains a reformulation of any $n$-player finite, static game into a framework of distributed, dynamical system based on agents' payoff-based deviations. The reformulation generalizes the method employed in the second part of the study of countries' relation formation problem in Li and Morse (2017) to the case of any finite, static game. In the paper two deviation rules are provided and possible applications of this framework are discussed.

cs.GT↗
arXiv2017

A Graphical Characterization of Structurally Controllable Linear Systems with Dependent Parameters

One version of the concept of structural controllability defined for single-input systems by Lin and subsequently generalized to multi-input systems by others, states that a parameterized matrix pair $(A, B)$ whose nonzero entries are distinct parameters, is structurally controllable if values can be assigned to the parameters which cause the resulting matrix pair to be controllable. In this paper the concept of structural controllability is broadened to allow for the possibility that a parameter may appear in more than one location in the pair $(A, B)$. Subject to a certain condition on the parameterization called the "binary assumption", an explicit graph-theoretic characterization of such matrix pairs is derived.

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