arXiv subjects

Zhexian Li

Publications and source records attributed to Zhexian Li.

6 recordsLinked to original sources

arXiv2026

FNO$^{\angle \theta}$: Extended Fourier neural operator for learning state and optimal control of distributed parameter systems

We propose an extended Fourier neural operator (FNO) architecture for learning state and linear quadratic additive optimal control of systems governed by partial differential equations. Using the Ehrenpreis-Palamodov fundamental principle, we show that any state and optimal control of linear PDEs with constant coefficients can be represented as an integral in the complex domain. The integrand of this representation involves the same exponential term as in the inverse Fourier transform, where the latter is used to represent the convolution operator in FNO layer. Motivated by this observation, we modify the FNO layer by extending the frequency variable in the inverse Fourier transform from the real to complex domain to capture the integral representation from the fundamental principle. We illustrate the performance of FNO in learning state and optimal control for the nonlinear Burgers' equation, showing order of magnitude improvements in training errors and more accurate predictions of non-periodic boundary values over FNO.

cs.LG↗

arXiv2025

Constrained bilinear optimal control of reactive evolution equations

We consider constrained bilinear optimal control of second-order linear evolution partial differential equations (PDEs) with a reaction term on the half line, where control arises as a time-dependent reaction coefficient and constraints are imposed on the state and control variables. These PDEs represent a wide range of physical phenomena in fluid flow, heat, and mass transfer. Existing computational methods for this type of control problems only consider constraints on the control variable and lack global convergence guarantee. In this paper, we propose a novel optimize-then-discretize framework for computing constrained bilinear optimal control with both state and control constraints. Unlike existing methods that derive optimality conditions directly from the PDE constraint, this framework first replaces the PDE constraint with an equivalent integral representation of the PDE solution and then derives optimality conditions for the reformulated problem. The integral representation, derived from the unified transform method, does not involve differential operators, and thus explicit expressions for necessary conditions of optimality can be derived using the Karush-Kuhn-Tucker conditions for infinite-dimensional optimization. Discretizing the optimality conditions results in a system of finite-dimensional smooth nonlinear equations, which can be efficiently solved using existing algorithms with guaranteed global convergence at a quadratic rate. This is in contrast with discretize-then-optimize methods that discretize the PDE first and then solve the optimality conditions of the approximated finite-dimensional problem. Computational results for two applications, namely nuclear reactivity control and water quality treatment in a reactor, are presented to illustrate the effectiveness of the proposed framework.

physics.comp-ph↗

arXiv2025

Adaptive model predictive control for traffic signal timing with unknown demand and parameters

This paper designs traffic signal control policies for a network of signalized intersections without knowing the demand and parameters. Within a model predictive control (MPC) framework, control policies consist of an algorithm that estimates parameters and a one-step MPC that computes control inputs using estimated parameters. The algorithm switches between different terminal sets of the MPC to explore different regions of the state space, where different parameters are identifiable. The one-step MPC minimizes a cost that approximates the sum of squares of all the queue lengths within a constant and does not require demand information. We show that the algorithm can estimate parameters exactly in finite time, and the one-step MPC renders maximum throughput in terms of input-to-state practical stability. Simulations indicate better transient performance regarding queue lengths under our proposed policies than existing ones.

eess.SY↗

arXiv2024

A complex spatial frequency approach to optimal control of finite-extent linear evolution systems

We consider the linear quadratic regulator (LQR) for one-dimensional linear evolution partial differential equations (PDEs) on a finite interval in space. The control is applied as an additive forcing term to PDEs. Existing methods for closed-form optimal control only apply to homogeneous (zero) boundary conditions, often resulting in series representations. In this paper, we consider general smooth boundary conditions. We use the unified transform, namely the Fourier transform restricted to the bounded spatial domain, to decouple PDEs into a family of ordinary differential equations (ODEs) parameterized by complex spatial frequency variables. Then, optimal control in the frequency domain is derived using LQR theory for ODEs. The inverse Fourier transform leads to non-causal terms in optimal control corresponding to integrals, over the real line, of future values of unspecified boundary conditions. To eliminate this non-causality, we deform the integrals to well-constructed contours in the complex plane along which the contribution of unknowns vanishes. For the reaction-diffusion equation, we show that the integral representation can be reformulated as a series representation, which leads to a state-feedback convolution form for optimal control, with the boundary conditions appearing as an additive term. In numerical experiments, we illustrate the computational advantages of the integral representation in comparison to the series representation and structural properties of the convolution kernel.

eess.SY↗

arXiv2023

Exploring the sensing power of mixed vehicle fleets

Vehicle-based mobile sensing, also known as drive-by sensing, efficiently surveys urban environments at low costs by leveraging the mobility of urban vehicles. While recent studies have focused on drive-by sensing for fleets of a single type, our work explores the sensing power and cost-effectiveness of a mixed fleet that consists of vehicles with distinct and complementary mobility patterns. We formulate the drive-by sensing coverage (DSC) problem, proposing a method to quantify sensing utility and an optimization procedure that determines fleet composition, sensor allocation, and vehicle routing for a given budget. Our air quality sensing case study in Longquanyi District (Chengdu, China) demonstrates that using a mixed fleet enhances sensing utilities and achieves close approximations to the target sensing distribution at a lower cost. Generalizing these insights to two additional real-world networks, our regression analysis uncovers key factors influencing the sensing power of mixed fleets. This research provides quantitative and managerial insights into drive-by sensing, showcasing a positive externality of urban transport activities.

math.OC↗

arXiv2023

Output-feedback adaptive model predictive control for ramp metering: a set-membership approach

Ramp metering, which regulates the flow entering the freeway, is one of the most effective freeway traffic control methods. This paper introduces an output-feedback adaptive approach to ramp metering that combines model predictive control (MPC) with set-membership parameter and state estimation. The set-membership estimator is based on a mixed-monotone embedding of underlying traffic dynamics. The embedding is also used as the modeling basis for MPC optimization. For a freeway stretch with unknown parameters and partial measurement on the freeway mainline, we provide sufficient conditions on the control horizon, cost functions, terminal sets of MPC, and inflow demand at the ramps such that the queue lengths in the closed-loop system remain bounded. The sufficient condition on the demand matches the necessary condition, thereby proving maximal throughput under the proposed controller. The result is strengthened to input-to-state stability when model parameters and demand are known. The stability analysis is conducted for the case of constant demand and unbounded on-ramps. The closed-loop trajectory data generated by the proposed controller is shown to facilitate finite time estimation of free-flow model parameters, i.e., free-flow speed and turning ratios. Simulation results illustrate stability of the closed-loop system under the proposed controller with time-varying demand and few mainline measurements, for which the system becomes unstable under a well-known approach from the literature. This indicates that the proposed controller renders higher throughput than the well-known approach, possibly using more computing resources.

eess.SY↗