arXiv subjects
Publications and source records attributed to Shengquan Xiang.
In this paper, we prove quantum ergodicity (a form of delocalization for eigenfunctions) for the Dirichlet truncations of the adjacency matrix on $\mathbb{Z}^d$. We also extend the result to the cases of finite range observables and periodic Schrödinger operators with periods of length at most two. This work partially answers a question asked by McKenzie and Sabri (Comm. Math. Phys. 403(3), 1477--1509(2023)).
We study the large deviation principle (LDP) for locally damped nonlinear wave equations perturbed by a bounded noise. When the noise is sufficiently non-degenerate, we establish the LDP for empirical distributions with lower bound of a local type. The primary challenge is the lack of compactness due to the absence of smoothing effect. This is overcome by exploiting the asymptotic compactness for the dynamics of waves, introducing the concept of asymptotic exponential tightness for random measures, and establishing a new LDP approach for random dynamical systems.
We establish a new criterion for exponential mixing of random dynamical systems. Our criterion is applicable to a wide range of systems, including in particular dispersive equations. Its verification is in nature related to several topics, i.e., asymptotic compactness in dynamical systems, global stability of evolution equations, and localized control problems. As an initial application, we exploit the exponential mixing of random nonlinear wave equations with degenerate damping, critical nonlinearity, and physically localized noise. The essential challenge lies in the fact that the weak dissipation and randomness interact in the evolution.
Fredholm-type backstepping transformation, introduced by Coron and Lü, has become a powerful tool for rapid stabilization with fast development over the last decade. Its strength lies in its systematic approach, allowing to deduce rapid stabilization from approximate controllability. But limitations with the current approach exist for operators of the form $|D_x|^α$ for $α\in (1,3/2]$. We present here a new compactness/duality method which hinges on Fredholm's alternative to overcome the $α=3/2$ threshold. More precisely, the compactness/duality method allows to prove the existence of a Riesz basis for the backstepping transformation for skew-adjoint operator verifying $α>1$, a key step in the construction of the Fredholm backstepping transformation, where the usual methods only work for $α>3/2$. The illustration of this new method is shown on the rapid stabilization of the linearized capillary-gravity water wave equation exhibiting an operator of critical order $α=3/2$.
The CIRCLES project aims to reduce instabilities in traffic flow, which are naturally occurring phenomena due to human driving behavior. These "phantom jams" or "stop-and-go waves,"are a significant source of wasted energy. Toward this goal, the CIRCLES project designed a control system referred to as the MegaController by the CIRCLES team, that could be deployed in real traffic. Our field experiment leveraged a heterogeneous fleet of 100 longitudinally-controlled vehicles as Lagrangian traffic actuators, each of which ran a controller with the architecture described in this paper. The MegaController is a hierarchical control architecture, which consists of two main layers. The upper layer is called Speed Planner, and is a centralized optimal control algorithm. It assigns speed targets to the vehicles, conveyed through the LTE cellular network. The lower layer is a control layer, running on each vehicle. It performs local actuation by overriding the stock adaptive cruise controller, using the stock on-board sensors. The Speed Planner ingests live data feeds provided by third parties, as well as data from our own control vehicles, and uses both to perform the speed assignment. The architecture of the speed planner allows for modular use of standard control techniques, such as optimal control, model predictive control, kernel methods and others, including Deep RL, model predictive control and explicit controllers. Depending on the vehicle architecture, all onboard sensing data can be accessed by the local controllers, or only some. Control inputs vary across different automakers, with inputs ranging from torque or acceleration requests for some cars, and electronic selection of ACC set points in others. The proposed architecture allows for the combination of all possible settings proposed above. Most configurations were tested throughout the ramp up to the MegaVandertest.
In this paper, we study the global controllability and stabilization problems of the harmonic map heat flow from a circle to a sphere. Combining ideas from control theory, heat flow, differential geometry, and asymptotic analysis, we obtain several important properties, such as small-time local controllability, local quantitative rapid stabilization, obstruction to semi-global asymptotic stabilization, and global controllability to geodesics. Surprisingly, due to the geometric feature of the equation we also discover the small-time global controllability between harmonic maps within the same homotopy class for general compact Riemannian manifold targets, which is to be compared with the analogous but longstanding problem for the nonlinear heat equations.
The dissipation of stop-and-go waves attracted recent attention as a traffic management problem, which can be efficiently addressed by automated driving. As part of the 100 automated vehicles experiment named MegaVanderTest, feedback controls were used to induce strong dissipation via velocity smoothing. More precisely, a single vehicle driving differently in one of the four lanes of I-24 in the Nashville area was able to regularize the velocity profile by reducing oscillations in time and velocity differences among vehicles. Quantitative measures of this effect were possible due to the innovative I-24 MOTION system capable of monitoring the traffic conditions for all vehicles on the roadway. This paper presents the control design, the technological aspects involved in its deployment, and, finally, the results achieved by the experiment.
Continuing the investigations started in the recent work [Krieger-Xiang, 2022] on semi-global controllability and stabilization of the $(1+1)$-dimensional wave maps equation with spatial domain $\mathbb{S}^1$ and target $\mathbb{S}^k$, where {\it semi-global} refers to the $2π$-energy bound, we prove global exact controllability of the same system for $k>1$ and show that the $2π$-energy bound is a strict threshold for uniform asymptotic stabilization via continuous time-varying feedback laws indicating that the damping stabilization in [Krieger-Xiang, 2022] is sharp. Lastly, the global exact controllability for $\mathbb{S}^1$-target within minimum time is discussed.
We investigate the stability and stabilization of the cubic focusing Klein-Gordon equation around static solutions on the closed ball of radius L in $\mathbb{R}^3$. First we show that the system is linearly unstable near the static solution $u\equiv 1$ for any dissipative boundary condition $u_t+ au_ν=0, a\in (0, 1)$. Then by means of boundary controls (both open-loop and closed-loop) we stabilize the system around this equilibrium exponentially under the condition $\sqrt{2}L\neq \tan \sqrt{2}L$. Furthermore, we show that the equilibrium can be stabilized with any rate less than $ \frac{\sqrt{2}}{2L} \log{\frac{1+a}{1-a}}$, provided $(a,L)$ does not belong to a certain zero set. This rate is sharp.
We prove the semi-global controllability and stabilization of the $(1+1)$-dimensional wave maps equation with spatial domain $\mathbb{S}^1$ and target $\mathbb{S}^k$. First we show that damping stabilizes the system when the energy is strictly below the threshold $2π$, where harmonic maps appear as obstruction for global stabilization. Then, we adapt an iterative control procedure to get low-energy exact controllability of the wave maps equation. This result is optimal in the case $k=1$.
Traffic waves, known also as stop-and-go waves or phantom hams, appear naturally as traffic instabilities, also in confined environments as a ring-road. A multi-population traffic is studied on a ring-road, comprised of drivers with stable and unstable behavior. There exists a critical penetration rate of stable vehicles above which the system is stable, and under which the system is unstable. In the latter case, stop-and-go waves appear, provided enough cars are on the road. The critical penetration rate is explicitly computable, and, in reasonable situations, a small minority of aggressive drivers is enough to destabilize an otherwise very stable flow. This is a source of instability that a single population model would not be able to explain. Also, the multi-population system can be stable below the critical penetration rate if the number of cars is sufficiently small. Instability emerges as the number of cars increases, even if the traffic density remains the same (i.e. number of cars and road size increase similarly). This shows that small experiments could lead to deducing imprecise stability conditions.
In this paper, we study, in the semiclassical sense, the global approximate controllability in small time of the quantum density and quantum momentum of the 1-D semiclassical cubic Schrödinger equation with two controls between two states with positive quantum densities. We first control the asymptotic expansions of the zeroth and first order of the physical observables via Agrachev-Sarychev's method. Then we conclude the proof through techniques of semiclassical approximation of the nonlinear Schrödinger equation.
We study the rapid stabilization of the heat equation on the 1-dimensional torus using the backstepping method with a Fredholm transformation. We prove that, under some assumption on the control operator, two scalar controls are necessary and sufficient to get controllability and rapid stabilization. This classical framework allows us to present the backstepping method with Fredholm transformations on Laplace operators in a sharp functional setting, which is the main objective of this work. Finally, we prove that the same Fredholm transformation also leads to the local rapid stability of the viscous Burgers equation.
In this article we study the so-called water tank system. In this system, the behavior of water contained in a 1-D tank is modelled by Saint-Venant equations, with a scalar distributed control. It is well-known that the linearized systems around uniform steady-states are not controllable, the uncontrollable part being of infinite dimension. Here we will focus on the linearized systems around non-uniform steady states, corresponding to a constant acceleration of the tank. We prove that these systems are controllable in Sobolev spaces, using the moments method and perturbative spectral estimates. Then, for steady states corresponding to small enough accelerations, we design an explicit Proportional Integral feedback law (obtained thanks to a well-chosen dynamic extension of the system) that stabilizes these systems exponentially with arbitrarily large decay rate. Our design relies on feedback equivalence/backstepping.
We provide explicit time-varying feedback laws that locally stabilize the two dimensional internal controlled incompressible Navier-Stokes equations in arbitrarily small time. We also obtain quantitative rapid stabilization via stationary feedback laws, as well as quantitative null controllability with explicit controls having $e^{C/T}$ costs.
The null controllability of the heat equation is known for decades [19,23,30]. The finite time stabilizability of the one dimensional heat equation was proved by Coron--Nguyên [13], while the same question for high dimensional spaces remained widely open. Inspired by Coron--Trélat [14] we find explicit stationary feedback laws that quantitatively exponentially stabilize the heat equation with decay rate $λ$ and $Ce^{C\sqrtλ}$ estimates, where Lebeau--Robbiano's spectral inequality [30] is naturally used. Then a piecewise controlling argument leads to null controllability with optimal cost $Ce^{C/T}$, as well as finite time stabilization.
The controllability of the linearized KdV equation with right Neumann control is studied in the pioneering work of Rosier [25]. However, the proof is by contradiction arguments and the value of the observability constant remains unknown, though rich mathematical theories are built on this totally unknown constant. We introduce a constructive method that gives the quantitative value of this constant.
We study the exponential stabilization problem for a nonlinear Korteweg-de Vries equa- tion on bounded interval in cases where the linearized control system is not controllable. The system has Dirichlet boundary conditions at the end-points of the interval, a Neumann nonhomogeneous boundary condition at the right end-point which is the control. We build a class of time-varying feedback laws for which the solutions of the closed-loop systems with small initial data decay exponentially to 0. We present also results on the well-posedness of the closed-loop systems for general time-varying feedback laws.