arXiv subjects
Publications and source records attributed to Huan Yu.
In this paper, we present a new and elementary proof of the local existence and uniqueness of the classical solution to the Cauchy problem of the two-dimensional generalized surface quasi-geostrophic (SQG) equation via the method of the contraction mapping principle. Also, same result holds true for a kind of transport equation with nonlocal velocity via the method of the contraction mapping principle.
We develop reinforcement learning (RL) boundary controllers to mitigate stop-and-go traffic congestion on a freeway segment. The traffic dynamics of the freeway segment are governed by a macroscopic Aw-Rascle-Zhang (ARZ) model, consisting of $2\times 2$ quasi-linear partial differential equations (PDEs) for traffic density and velocity. Boundary stabilization of the linearized ARZ PDE model has been solved by PDE backstepping, guaranteeing spatial $L^2$ norm regulation of the traffic state to uniform density and velocity and ensuring that traffic oscillations are suppressed. Collocated Proportional (P) and Proportional-Integral (PI) controllers also provide stability guarantees under certain restricted conditions, and are always applicable as model-free control options through gain tuning by trail and error, or by model-free optimization. Although these approaches are mathematically elegant, the stabilization result only holds locally and is usually affected by the change of model parameters. Therefore, we reformulate the PDE boundary control problem as a RL problem that pursues stabilization without knowing the system dynamics, simply by observing the state values. The proximal policy optimization, a neural network-based policy gradient algorithm, is employed to obtain RL controllers by interacting with a numerical simulator of the ARZ PDE. Being stabilization-inspired, the RL state-feedback boundary controllers are compared and evaluated against the rigorously stabilizing controllers in two cases: (i) in a system with perfect knowledge of the traffic flow dynamics, and then (ii) in one with only partial knowledge. We obtain RL controllers that nearly recover the performance of the backstepping, P, and PI controllers with perfect knowledge and outperform them in some cases with partial knowledge.
Underwater robots in shallow waters usually suffer from strong wave forces, which may frequently exceed robot's control constraints. Learning-based controllers are suitable for disturbance rejection control, but the excessive disturbances heavily affect the state transition in Markov Decision Process (MDP) or Partially Observable Markov Decision Process (POMDP). Also, pure learning procedures on targeted system may encounter damaging exploratory actions or unpredictable system variations, and training exclusively on a prior model usually cannot address model mismatch from the targeted system. In this paper, we propose a transfer learning framework that adapts a control policy for excessive disturbance rejection of an underwater robot under dynamics model mismatch. A modular network of learning policies is applied, composed of a Generalized Control Policy (GCP) and an Online Disturbance Identification Model (ODI). GCP is first trained over a wide array of disturbance waveforms. ODI then learns to use past states and actions of the system to predict the disturbance waveforms which are provided as input to GCP (along with the system state). A transfer reinforcement learning algorithm using Transition Mismatch Compensation (TMC) is developed based on the modular architecture, that learns an additional compensatory policy through minimizing mismatch of transitions predicted by the two dynamics models of the source and target tasks. We demonstrated on a pose regulation task in simulation that TMC is able to successfully reject the disturbances and stabilize the robot under an empirical model of the robot system, meanwhile improve sample efficiency.
eHWC J2019+368 is one of the sources emitting $γ$-rays with energies higher than 100 TeV based on the recent measurement with the High Altitude Water Cherenkov Observatory (HAWC), and the origin is still in debate. The pulsar PSR J2021$+$3651 is spatially coincident with the TeV source. We investigate theoretically whether the multiband nonthermal emission of eHWC J2019+368 can originate from the pulsar wind nebula (PWN) G75.2$+$0.1 powered by PSR J2021$+$3651. In the model, the spin-down power of the pulsar is transferred to high-energy particles and magnetic field in the nebula. As the particles with an energy distribution of either a broken power-law or a power-law continually injected into the nebula, the multiband nonthermal emission is produced via synchrotron radiation and inverse Compton scattering. The spectral energy distribution of the nebula from the model with the reasonable parameters is generally consistent with the detected radio, X-ray and TeV $γ$-ray fluxes. Our study supports that the PWN has the ability to produce the TeV $γ$-rays of eHWC J2019+368, and the most energetic particles in the nebula have energies up to about $0.4$ PeV.
In this paper, we consider a kind of singular integral which can be viewed as an extension of the classical Calder$\acute{\rm o}$n-Zygmund type singular integral. We establish an estimate of the singular integral in the $L^q$ space for $1<q<\infty$. In particular, the Calder$\acute{\rm o}$n-Zygmund estimate can be recovered from our obtained estimate. The proof of our main result is via the so called "geometric approach", which was applied in \cite{CP} on the $L^q$ estimate of the elliptic equations and in \cite{LW,Wang} on a new proof of the the Calder$\acute{\rm o}$n-Zygmund estimate.
G349.7 + 00.2 is a young Galactic supernova remnant (SNR) with a mushroom morphology in radio and X-rays, and it has been detected across the entire electromagnetic spectrum from radio to high energy $γ$-rays. Moreover, the remnant is interacting with a molecular cloud based on the observations in the radio and infrared band. The reason for the formation of the periphery and the dynamical evolution of the remnant are investigated using 3D hydrodynamical (HD) simulations. Under the assumption that the supernova ejecta is evolved in the medium with a density gradient, the shell is composed of two hemispheres with different radiuses, and the smaller hemisphere is in relatively dense media. The resulting periphery of remnant is consistent with detected ones, and it can be concluded that the peculiar periphery of G349.7+00.2 can be reproduced as the remnants interacting with the medium with a density gradient.
This paper develops a full-state feedback controller that damps out oscillations in traffic density and traffic velocity whose dynamical behavior is governed by the linearized two-class Aw-Rascle (AR) model. Thereby, the traffic is considered to be in the congested regime and subdivided in two classes whereas each class represents vehicles with the same size and driver's behavior. The macroscopic second-order two-class AR model consists of four first order hyperbolic partial differential equations (PDEs) and introduces a concept of area occupancy to depict the mixed density of two-class vehicles in the traffic. Moreover, the linearized model equations show heterodirectional behavior with both positive and negative characteristic speeds in the congested regime. The control objective is to achieve convergence to a constant equilibrium in finite time. The control input is realized by ramp metering acting at the outlet of the considered track section. The backstepping method is employed to design full-state feedback for the $4\times 4$ hyperbolic PDEs. The performance of the full-state feedback controller is verified by simulation.
In this paper, we propose a control design methodology for a linearized continuum traffic model in the congested regime. The continuum traffic flow on a highway is modeled using a linearized quasilinear hyperbolic partial differential equation model known as the Aw-Rascle-Zhang (ARZ) model. The linear traffic model is augmented with a novel non-local boundary condition representing car influx due to the use of routing apps such as Google Maps and Waze. The routing apps act as real-time previews for highway traffic, introducing potentially destabilizing feedback in the app-based navigation decision process, necessitating the development of a feedback controller. We first study small-time H^1 solutions of the linearized model with the addition of the app-routing for sufficiently small initial data. We introduce an extended, multi-tiered boundary control design based upon the method of infinite-dimensional backstepping. Using an intermediate decoupling transformation, we account for the non-local boundary condition arising from routing app feedback. We study the existence of the extended backstepping method by characterizing the existence of the companion kernels associated with the backstepping method. Finally, we study the linear ARZ model with the app-routing extension under the designed feedback, and show that for sufficiently small H^1 data, the equilibrium congestion solution is exponentially stable and guarantees the existence of closed-loop solutions on the infinite time interval.
Multiband observations on the type Ia supernova remnant SN 1006 indicate peculiar properties in its morphologies of emission in the radio, optical and X-ray bands. In the hard X-rays, the remnant is bilateral with two opposite bright limbs with prominent protrusions. Moreover, a filament has been detected at the radio, optical and soft X-ray wavelengthes. The reason for these peculiar features in the morphologies of the remnant is investigated using 3D HD simulations. With the assumption that the supernova ejecta is evolved in the ambient medium with a density discontinuity, the radius of the remnant's boundary is smaller in the tenuous medium, and the shell consists of two hemispheres with different radiuses. Along particular line of sights, protrusions appear on the periphery of the remnants since the emission from the edge of the hemisphere with a larger radius is located outside that from the shell of the small hemisphere. Furthermore, the northwest filament of SN 1006 arises as a result of the intersection of the line of sight and the shocked material near the edges of the two hemispheres. It can be concluded that the features that the protrusions on the northeast and southwest limbs and the northwest filament in the morphologies of SN 1006 can be reproduced as the remnants interacting with the medium with a density discontinuity.
In this paper we develop a boundary state feedback control law for a traffic flow network system in its most fundamental form: one incoming and one outgoing road connected by a junction. The macroscopic traffic dynamics on each road segment are governed by Aw-Rascle-Zhang (ARZ) model, consisting of second-order nonlinear partial differential equations (PDEs) for traffic density and velocity. Different equilibrium road conditions are considered for the connected segments. For stabilization of the stop-and-go traffic congestion on the two roads, we consider a ramp metering located at the connecting junction. The traffic flow rate entering from the on-ramp to the mainline junction is actuated. The objective is to simultaneously stabilize the upstream and downstream traffic to a given spatially-uniform constant steady-state. We design a full state feedback control law for this under-actuated network of two systems of two hetero-directional linear first-order hyperbolic PDEs interconnected through the boundary condition (junction). The exponential stability is validated by numerical simulation.
In this paper, we are concerned with regularity of suitable weak solutions of the 3D Navier-Stokes equations in Lorentz spaces. We obtain $\varepsilon$-regularity criteria in terms of either the velocity, the gradient of the velocity, the vorticity, or deformation tensor in Lorentz spaces. As an application, this allows us to extend the result involving Leray's blow up rate in time, and to show that the number of singular points of weak solutions belonging to $ L^{p,\infty}(-1,0;L^{q,l}(\mathbb{R}^{3})) $ and $ {2}/{p}+{3}/{q}=1$ with $3<q<\infty$ and $q\leq l <\infty$ is finite.
The early acceleration of protons and electrons in the nonrelativistic collisionless shocks with three obliquities are investigated through 1D particle-in-cell simulations. In the simulations, the charged particles possessing a velocity of $0.2\, c$ flow towards a reflecting boundary, and the shocks with a sonic Mach number of $13.4$ and a Alfven Mach number of $16.5$ in the downstream shock frame are generated. In these quasi-parallel shocks with the obliquity angles $θ= 15^\circ$, $30^\circ$, and $45^\circ$, some of the protons and the electrons can be injected into the acceleration processes, and their downstream spectra in the momentum space show a power law tail at a time of $1.89\times10^5 ω_{\rm pe}^{-1}$, where $ω_{\rm pe}$ is the electron plasma frequency. Moreover, the charged particles reflected at the shock excite magnetic waves upstream of the shock. The shock drift acceleration is more prominent with a larger obliquity angle for the shocks, but the accelerated particles diffuse parallel to the shock propagation direction more easily to participate in the diffusive shock acceleration. At the time still in the early acceleration stage, more energetic protons and electrons appear in the downstream of the shock for $θ= 15^\circ$ compared with the other two obliquities; moreover, in the upstream region, the spectrum of the accelerated electrons is the hardest for $θ_{\rm nB} = 45^\circ$ among the three obliquities, whereas the proton spectra for $θ_{\rm nB} = 15^\circ$ and $45^\circ$ are similar as a result of the competition of the effectiveness of the shock drift acceleration and the diffusive shock acceleration.
The Gamma-Ray Integrated Detectors (GRID) is a space mission concept dedicated to monitoring the transient gamma-ray sky in the energy range from 10 keV to 2 MeV using scintillation detectors onboard CubeSats in low Earth orbits. The primary targets of GRID are the gamma-ray bursts (GRBs) in the local universe. The scientific goal of GRID is, in synergy with ground-based gravitational wave (GW) detectors such as LIGO and VIRGO, to accumulate a sample of GRBs associated with the merger of two compact stars and study jets and related physics of those objects. It also involves observing and studying other gamma-ray transients such as long GRBs, soft gamma-ray repeaters, terrestrial gamma-ray flashes, and solar flares. With multiple CubeSats in various orbits, GRID is unaffected by the Earth occultation and serves as a full-time and all-sky monitor. Assuming a horizon of 200 Mpc for ground-based GW detectors, we expect to see a few associated GW-GRB events per year. With about 10 CubeSats in operation, GRID is capable of localizing a faint GRB like 170817A with a 90% error radius of about 10 degrees, through triangulation and flux modulation. GRID is proposed and developed by students, with considerable contribution from undergraduate students, and will remain operated as a student project in the future. The current GRID collaboration involves more than 20 institutes and keeps growing. On August 29th, the first GRID detector onboard a CubeSat was launched into a Sun-synchronous orbit and is currently under test.
This paper develops boundary observer for estimation of congested freeway traffic states based on Aw-Rascle-Zhang (ARZ) partial differential equations (PDE) model. Traffic state estimation refers to acquisition of traffic state information from partially observed traffic data. This problem is relevant for freeway due to its limited accessibility to real-time traffic information. We propose a model-driven approach in which estimation of aggregated traffic states in a freeway segment are obtained simply from boundary measurement of flow and velocity without knowledge of the initial states. The macroscopic traffic dynamics is represented by the ARZ model, consisting of $2 \times 2$ coupled nonlinear hyperbolic PDEs for traffic density and velocity. Analysis of the linearized ARZ model leads to the study of a hetero-directional hyperbolic PDE model for congested traffic regime. Using spatial transformation and PDE backstepping method, we construct a boundary observer consisting of a copy of the nonlinear plant with output injections from boundary measurement errors. The output injection gains are designed for the estimation error system so that the exponential stability of the error system in the $L^2$ norm and finite-time convergence to zero are guaranteed. Numerical simulations are conducted to validate the boundary observer design for estimation of the nonlinear ARZ model. In data validation, we calibrate model parameters of the ARZ model and then use vehicle trajectory data to test the performance of the observer design.
This paper develops boundary feedback control laws in order to damp out traffic oscillations in the congested regime of the linearized two-class Aw-Rascle (AR) traffic model. The macroscopic second-order two-class AR traffic model consists of four hyperbolic partial differential equations (PDEs) describing the dynamics of densities and velocities on freeway. The concept of area occupancy is used to express the traffic pressure and equilibrium speed relationship yielding a coupling between the two classes of vehicles. Each vehicle class is characterized by its own vehicle size and driver's behavior. The considered equilibrium profiles of the model represent evenly distributed traffic with constant densities and velocities of both classes along the investigated track section. After linearizing the model equations around those equilibrium profiles, it is observed that in the congested traffic one of the four characteristic speeds is negative, whereas the remaining three are positive. Backstepping control design is employed to stabilize the $4 \times 4$ heterodirectional hyperbolic PDEs. The control input actuates the traffic flow at outlet of the investigated track section and is realized by a ramp metering. A full-state feedback is designed to achieve finite time convergence of the density and velocity perturbations to the equilibrium at zero. This result is then combined with an anti-collocated observer design in order to construct an output feedback control law that damps out stop-and-go waves in finite time by measuring the velocities and densities of both vehicle classes at the inlet of the investigated track section. The performance of the developed controllers is verified by simulation.
This paper develops output feedback boundary control to mitigate traffic congestion of a unidirectional two-lane freeway segment. The macroscopic traffic dynamics are described by the Aw-Rascle-Zhang (ARZ) model respectively for both the fast and slow lanes. The traffic density and velocity of each of the two lanes are governed by coupled $2 \times 2$ nonlinear hyperbolic partial differential equations (PDEs). Lane-changing interactions between the two lanes lead to exchanging source terms between the two pairs second-order PDEs. Therefore, we are dealing with $4 \times 4$ nonlinear coupled hyperbolic PDEs. Based on driver's preference for the slow and fast lanes, a reference system of lane-specific uniform steady states in congested traffic is chosen. To stabilize traffic densities and velocities of both lanes to the steady states, two distinct variable speed limits (VSL) are applied at outlet boundary, controlling the traffic velocity of each lane. Using backstepping transformation, we map the coupled heterodirectional hyperbolic PDE system into a cascade target system, in which traffic oscillations are damped out through actuation of the velocities at the downstream boundary. Two full-state feedback boundary control laws are developed. We also design a collocated boundary observer for state estimation with sensing of densities at the outlet. Output feedback boundary controllers are obtained by combining the collocated observer and full-state feedback controllers. The finite time convergence to equilibrium is achieved for both the controllers and observer designs. Numerical simulations validate our design in two different traffic scenarios.
This paper develops boundary observer for estimation of congested freeway traffic states based on Aw-Rascle-Zhang(ARZ) partial differential equations (PDE) model. Traffic state estimation refers to acquisition of traffic state information from partially observed traffic data. This problem is relevant for freeway due to its limited accessibility to real-time traffic information. We propose a boundary observer design so that estimates of aggregated traffic states in a freeway segment are obtained simply from boundary measurement of flow and velocity. The macroscopic traffic dynamics is represented by the ARZ model, consisting of $2 \times 2$ coupled nonlinear hyperbolic PDEs for traffic density and velocity. Analysis of the linearized ARZ model leads to the study of a hetero-directional hyperbolic PDE model for congested traffic regime. Using spatial transformation and PDE backstepping method, we construct a boundary observer with a copy of the nonlinear plant and output injection of boundary measurement errors. The output injection gains are designed for the error system of the linearized ARZ model so that the exponential stability of error system in the $L^2$ norm and finite-time convergence to zero are guaranteed. Simulations are conducted to validate the boundary observer design for nonlinear ARZ model without knowledge of initial conditions.
This paper examines the uniqueness of weak solutions to the d-dimensional magnetohydrodynamic (MHD) equations with the fractional dissipation $(-Δ)^αu$ and without the magnetic diffusion. Important progress has been made on the standard Laplacian dissipation case $α=1$. This paper discovers that there are new phenomena with the case $α<1$. The approach for $α=1$ can not be directly extended to $α<1$. We establish that, for $α<1$, any initial data $(u_0, b_0)$ in the inhomogeneous Besov space $B^σ_{2,\infty}(\mathbb R^d)$ with $σ> 1+\frac{d}{2}-α$ leads to a unique local solution. For the case $α\ge 1$, $u_0$ in the homogeneous Besov space $\mathring B^{1+\frac{d}{2}-2α}_{2,1}(\mathbb R^d)$ and $b_0$ in $ \mathring B^{1+\frac{d}{2}-α}_{2,1}(\mathbb R^d)$ guarantees the existence and uniqueness. These regularity requirements appear to be optimal.