Michael Malisoff, Ludovic Rifford, Eduardo Sontag
We study nonlinear systems with observation errors. The main problem addressed in this paper is the design of feedbacks for globally asymptotically controllable (GAC) control affine systems that render the closed loop systems input to state stable with respect to actuator errors. Extensions for fully nonlinear GAC systems with actuator errors are also discussed.
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In the present article we study the stabilization of first-order linear integro-differential hyperbolic equations. For such equations we prove that the stabilization in finite time is equivalent to the exact controllability property. The proof relies on a Fredholm transformation that maps the original system into a finite-time stable target system. The controllability assumption is used to prove the invertibility of such a transformation. Finally, using the method of moments, we show in a particular case that the controllability is reduced to the criterion of Fattorini.
Given a linear control system in a Hilbert space with a bounded control operator, we establish a characterization of exponential stabilizability in terms of an observability inequality. Such dual characterizations are well known for exact (null) controllability. Our approach exploits classical Fenchel duality arguments and, in turn, leads to characterizations in terms of observability inequalities of approximately null controllability and of $α$-null controllability. We comment on the relationships between those various concepts, at the light of the observability inequalities that characterize…
Controllability and Stabilization of a Wave-Heat Cascade System
该研究针对带Neumann波控制的波-反应-扩散PDE级联系统，通过对偶问题可观性不等式建立可控性，设计显式输出反馈策略实现镇定，利用了Riesz谱算子性质。
We prove global internal controllability in large time for the nonlinear Schrodinger equation on a bounded interval with periodic, Dirichlet or Neumann conditions. Our strategy combines stabilization and local controllability near 0. We use Bourgain spaces to prove this result on L2. We also get a regularity result about the control if the data are assumed smoother.
In this paper, we discuss our recent works on the null-controllability, the exact controllability, and the stabilization of linear hyperbolic systems in one dimensional space using boundary controls on one side for the optimal time. Under precise and generic assumptions on the boundary conditions on the other side, we first obtain the optimal time for the null and the exact controllability for these systems for a generic source term. We then prove the null-controllability and the exact controllability for any time greater than the optimal time and for any source term. Finally, for homogeneous …
This paper studies the stabilization for a kind of linear and impulse control systems in finite-dimensional spaces, where impulse instants appear periodically. We present several characterizations on the stabilization; show how to design feedback laws; and provide locations for impulse instants to ensure the stabilization. In the proofs of these results, we set up a discrete LQ problem; derived a discrete dynamic programming principle, built up a variant of Riccati's equation; applied repeatedly the Kalman controllability decomposition; and used a controllability result built up in [17].
Global controllability and stabilization of the wave maps equation from a circle to a sphere
本文研究$(1+1)$维波映射方程，证明目标为$\mathbb{S}^k$（$k>1$）时的全局精确能控性，并确立$2\pi$能量界是连续时变反馈一致渐近镇定的严格阈值。
We prove global internal controllability in large time for the nonlinear Schrödinger equation on some compact manifolds of dimension 3. The result is proved under some geometrical assumptions : geometric control and unique continuation. We give some examples where they are fulfilled on $\Tot$, $S^3$ and $S^2\times S^1$. We prove this by two different methods both inherently interesting. The first one combines stabilization and local controllability near 0. The second one uses successive controls near some trajectories. We also get a regularity result about the control if the data are assumed s…
We prove that the thickness property is a necessary and sufficient geometric condition that ensures the (rapid) stabilization or the approximate null-controllability with uniform cost of a large class of evolution equations posed on the whole space $\mathbb R^n$. These equations are associated with operators of the form $F(\vert D_x\vert)$, the function $F:[0,+\infty)\rightarrow\mathbb R$ being continuous and bounded from below. We also provide explicit feedbacks and constants associated with these stabilization properties. The notion of thickness is known to be a necessary and sufficient cond…
Null controllability and finite-time stabilization in minimal time of one-dimensional first-order $2 \times 2$ linear hyperbolic systems
该研究针对一维一阶$2×2$线性双曲系统，结合反步方法与Titchmarsh卷积定理，确定了其零能控性与有限时间镇定的最小时间并证明该时间不可更优。
Global controllability to harmonic maps of the heat flow from a circle to a sphere
本文研究从圆周到球面的调和映射热流的全局能控性与镇定问题，结合控制理论与微分几何等方法，证明了小时间局部能控性及同伦类内的全局能控性等重要性质。
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 $k=1$.
In this Note, we review the main existing results, methods, and some key open problems on the controllability of nonlinear hyperbolic and parabolic equations. Especially, we describe our recent universal approach to solve the local controllability problem of quasilinear time-reversible evolution equations, which is based on a new unbounded perturbation technique. It is also worthy to mention that the technique we develop can also be applied to other problems for quasilinear equations, say local existence, stabilization, etc.
Data-Driven Stabilization Using Prior Knowledge on Stabilizability and Controllability
该研究针对线性时不变系统的数据驱动镇定问题，引入能稳性、能控性先验知识拓展数据信息性概念，证明能稳性先验可放宽镇定充要条件，并提出配套的线性矩阵不等式形式的控制设计方法。
The purpose of this paper is to present a universal approach to the study of controllability/observability problems for infinite dimensional systems governed by some stochastic/deterministic partial differential equations. The crucial analytic tool is a class of fundamental weighted identities for stochastic/deterministic partial differential operators, via which one can derive the desired global Carleman estimates. This method can also give a unified treatment of the stabilization, global unique continuation, and inverse problems for some stochastic/deterministic partial differential equation…
Exact Controllability and Stabilization of the Wave Equation
本文基于作者1989年在巴西UFRJ的课程笔记，系统介绍波动方程精确可控性与镇定理论的基本工具，并补充耗散波动方程衰减的研究成果，为年轻研究者提供简明参考。
Mean-Field Controllability and Decentralized Stabilization of Markov Chains, Part II: Asymptotic Controllability and Polynomial Feedbacks
本文研究了有限状态空间中群体代理连续时间马尔可夫链的均场反馈稳定化方法，证明了强连通支持概率分布可被时间不变输入稳定，并展示了渐近可控性及去中心化密度反馈律的存在性。
Stabilization of a chain of 3 hyperbolic PDEs with 2 inputs in arbitrary position
本文研究了由两个控制输入在任意节点处驱动的三耦合超几何偏微分方程链的稳定化问题，提出了一种统一框架下的回溯变换和积分差分方程（IDE）改写方法。
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^…
This paper deals with the stabilization problem for nonlinear control-affine systems with the use of oscillating feedback controls. We assume that the local controllability around the origin is guaranteed by the rank condition with Lie brackets of length up to 3. This class of systems includes, in particular, mathematical models of rotating rigid bodies. We propose an explicit control design scheme with time-varying trigonometric polynomials whose coefficients depend on the state of the system. The above coefficients are computed in terms of the inversion of the matrix appearing in the control…
Stabilization and Optimal Control of an Interconnected $n + m$ Hetero-directional Hyperbolic PDE-SDE System
针对通过异向双曲 PDE 控制的线性 SDE 耦合系统，本文利用 backstepping 将其转化为带随机漂移的输入延迟 SDE，给出能控性与方差结构下界，并设计均值驱动、方差有界的控制器及可达任意下界上方方差的最优控制。
Mean-Field Controllability and Decentralized Stabilization of Markov Chains, Part I: Global Controllability and Rational Feedbacks
本文研究了连续时间马尔可夫链的可控性和稳定化性质，通过转移率作为控制参数。证明了在强连通图下，小时间局部和全局可控性，以及存在去中心化线性反馈法稳定概率密度。
Experiment design using prior knowledge on controllability and stabilizability
本文提出利用系统可控性和可稳定性先验知识设计输入信号，以生成适合识别和稳定化的数据，并讨论了最短实验设计方法。
The main purpose of this paper is to show the global stabilization and exact controllability properties for a fourth order nonlinear fourth order nonlinear Schrödinger system: $$i\partial_tu +\partial_x^2u-\partial_x^4u=λ|u|^2u,$$ on a periodic domain $\mathbb{T}$ with internal control supported on an arbitrary sub-domain of $\mathbb{T}$. More precisely, by certain properties of propagation of compactness and regularity in Bourgain spaces, for the solutions of the associated linear system, we show that the system is globally exponentially stabilizable. This property together with the local exa…
The problem of state-feedback stabilizability of discrete-time nonlinear systems has been considered in this note. Two assertions have been proved. First, if the system is $N$-step controllable to the origin, then there is a state feedback control law for which the trajectory of the closed-loop system converges to the origin in $N$ steps. Second, if the system is asymptotically controllable to the origin and satisfies the controllability rank condition at the origin, then there is a state feedback control law for which the trajectory of the closed-loop system converges to the origin in finite …
We consider a controlled Schrödinger equation with a dipolar and a polarizability term, used when the dipolar approximation is not valid. The control is the amplitude of the external electric field, it acts non linearly on the state. We extend in this infinite dimensional framework previous techniques used by Coron, Grigoriu, Lefter and Turinici for stabilization in finite dimension. We consider a highly oscillating control and prove the semi-global weak $H^2$ stabilization of the averaged system using a Lyapunov function introduced by Nersesyan. Then it is proved that the solutions of the Sch…
Coupled linear Schrödinger equations: Control and stabilization results
本文研究一维Kirchhoff边界条件下耦合线性薛定谔方程组的控制与镇定，通过新Carleman估计证明零边界能控性，并利用Gramian方法和反馈律证明系统指数衰减。
This paper focuses on the problem of constructing time-varying feedback laws that asymptotically stabilize a given part of the state variables for nonlinear control-affine systems. It is assumed that the class of systems under consideration satisfies nonlinear controllability conditions with respect to the stabilizable variables. Under these assumptions, a time-periodic feedback control is constructed explicitly by using the inversion of the matrix composed of the control vector fields and their Lie brackets. The proposed control design scheme is applied to solving the leader-following problem…
This paper continues our study of the interconnection between controllability and mixing properties of random dynamical systems. We begin with an abstract result showing that the approximate controllability to a point and a local stabilisation property imply the uniqueness of a stationary measure and exponential mixing in the dual-Lipschitz metric. This result is then applied to the 2D Navier-Stokes system driven by a random force acting through the boundary. A by-product of our analysis is the local exponential stabilisation of the boundary-driven Navier-Stokes system by a regular boundary co…
For a class of linear switched systems in continuous time a controllability condition implies that state feedbacks allow to achieve almost sure stabilization with arbitrary exponential decay rates. This is based on the Multiplicative Ergodic Theorem applied to an associated system in discrete time. This result is related to the stabilizability problem for linear persistently excited systems.
This paper is the first part of a project devoted to studying the interconnection between controllability properties of a dynamical system and the large-time asymptotics of trajectories for the associated stochastic system. It is proved that the approximate controllability to a given point and the solid controllability from the same point imply the uniqueness of a stationary measure and exponential mixing in the total variation metric. This result is then applied to random differential equations on a compact Riemannian manifold. In the second part, we shall replace the solid controllability by…
We are motivated by the problem of control for a non-homogeneous elastic string with memory. We reduce the problem of controllability to a non-standard moment problem. The solution of the latter problem is based on an auxiliary Riesz basis property result for a family of functions quadratically close to the nonharmonic exponentials. This result requires the detailed analysis of an integro-differential equation and is of interest in itself for Function Theory. Controllability of the string implies observability of a dual system.
On classical solutions in the stabilization problem for nonholonomic control systems with time-varying feedback laws
本文研究了非holonomic控制系统稳定问题，通过构造时间变化反馈律并提出经典解收敛的充分条件，结合数值模拟验证了理论结果。
Global stabilization of the cubic defocusing nonlinear Schrödinger equation on the torus
本文通过Bourgain空间适定性、含三次项的Carleman估计及Morawetz乘子法，证明环面上三次散焦薛定谔方程的全局指数镇定，并给出零可控最小时间上界。
The goal of this work is to prove global controllability and stabilization properties for the fractional Schrödinger equation on $d$-dimensional compact Riemannian manifolds without boundary $(M,g)$. To prove our main results we use techniques of pseudo-differential calculus on manifolds. More precisely, by using microlocal analysis, we are able to prove propagation of regularity which together with the so-called Geometric Control Condition and Unique Continuation Property help us to prove global control results for the system under consideration. As a main novelty this manuscript presents the…
Finite-time stabilization via impulse control of degenerate singular parabolic equations
本文研究退化奇异抛物型方程的冲击可控性，通过现代框架实现有限时间稳定化，并提供解的指数衰减估计。核心方法结合对数凸性估计与特定谱性质，证明了系统最小范数冲击控制的存在性和唯一性。
Boundary output feedback stabilization of a cascade of N heat equations
本文针对边界耦合的N热方程级联系统，通过谱分析与模态能控能观性研究，基于广义特征向量的Riesz基性质，在两种构型下实现了L²和H¹范数下的输出反馈镇定。
Well-posedness and controllability of hyperbolic boundary control systems on a one-dimensional spatial domain
本文研究一维空间域上一类边界控制与观测系统的适定性，给出充要条件，并证明适定性与完全控制观测蕴含精确可控性与可观测性，以欧拉-伯努利梁模型为例。
Hybrid Minimum-Seeking in Synergistic Lyapunov Functions: Robust Global Stabilization under Unknown Control Directions
针对仿射控制系统中控制方向动态不确定性与拓扑障碍导致无光滑全局控制李雅普诺夫函数的问题，基于混合动态包含李括号平均结果提出混合极小值搜索反馈律，实现鲁棒全局实用稳定性，应用于流形镇定与车辆避障并验证有效性。
In this expository paper, which covers material presented at the NATO Advanced Study Institute "Nonlinear Analysis, Differential Equations, and Control" (Montreal, Jul/Aug 1998), we deal with several questions related to stability and stabilization of nonlinear finite-dimensional continuous-time systems. We review the basic problem of feedback stabilization, placing an emphasis upon relatively new areas of research which concern stability with respect to "noise" (such as errors introduced by actuators or sensors). The table of contents is as follows: Review of Stability and Asymptotic Controll…
Exponential mixing for the stochastic Allen--Cahn equation with localized white noise
本文研究了由局部白噪声驱动的1D随机Allen-Cahn方程，证明了相关马尔可夫过程具有唯一不变测度和指数混合性。主要挑战在于噪声的局部性质与系统非平凡全局动力学的相互作用，通过PDE控制理论中的稳定化和非线性方程的全局稳态可控性来克服。
Stabilization and control of the nonlinear plate equation
本文基于非线性板方程的线性可观察性，提出了一种新的唯一延续性质，实现了半全局稳定性和精确控制，通过分析非线性能量衰减和控制属性。
It is well-known that the controllability of finite-dimensional nonlinear systems can be established by showing the controllability of the linearized system. However, this classical result does not generalize to infinite-dimensional nonlinear systems. In this paper, we restrict ourselves to semilinear infinite-dimensional systems, and show that the exact controllability of the linearized system implies exact controllability of the nonlinear system. The restrictions concerning the nonlinear operator are similar to those that can be found in the literature about the linearized stability analysis…
This paper is concerned with the local output feedback stabilization of a nonlinear Kuramoto-Sivashinsky equation. The control is located at the boundary of the domain while the measurement is selected as a Neumann trace. This choice of system output requires the study of the system trajectories in $H^2$-norm. Moreover, the choice of the actuation/sensing scheme is discussed and adapted in function of the parameters of the plant in order to avoid the possible loss of controllability/observability property of certain eigenvalues of the underlying operator. This leads in certain cases to a multi…
Endpoint Asymptotics of Optimal Stabilization Prefactors
本文研究线性控制系统最优镇定前因子在衰减率端点处的渐近阶，证明其由可控性指标或不可控部分最大Jordan块尺寸决定，并可由渐近阶反推该结构不变量。
A Linear Test for Global Nonlinear Controllability
该研究探讨无漂移非线性控制仿射系统中次拉普拉斯算子可逆性与可控性的逆关联，证明算子可逆可推出邻域可达集满测度的弱可控性，提出用谱隙定义可控度，并结合最优传输理论给出两种证明视角。
Studies on the Rao-Nakra Sandwich Beam: Well-Posedness, Dynamics, and Controllability
本文研究了线性Rao-Nakra类型夹层梁的适定性、稳定性和边界可控性，通过半群理论和Kato定理证明了解的存在性和唯一性，并利用Lyapunov方法证明了系统能量指数衰减，同时通过HUM方法证明了系统的可控性。
The solution of the global controllability problem is obtained for a class of the triangular systems of O.D.E. that are not feedback linearizable. The introduced class is a generalization of the classes of triangular systems investigated before. The solution of the problem is based on the approach proposed in another work [18] devoted to the triangular systems of the Volterra equations and written jointly with W.H. Schmidt by the current authors. This yields the same properties of the considered class of triangular systems as those established in [18] for the Volterra systems. As well as in [1…
The goal of this paper is to provide a tutorial on the so-called informativity framework for direct data-driven analysis and control. This framework achieves certified data-based analysis and control by assessing system properties and determining controllers for sets of systems unfalsified by the data. We will first introduce the informativity approach at an abstract level. Thereafter, we will report case studies where we highlight the strength of the framework in the context of various problems involving both noiseless and noisy data. In particular, we will treat controllability and stabiliza…
$H^2$ Stabilization of the $2$-D and $3$-D Heat Equation via Modal Decomposition
本文通过模态分解控制器证明了二维和三维热方程的H¹和H²指数稳定性，解决了高维情况下状态收敛性问题。
Rapid Fredholm stabilization of the Kuramoto--Sivashinsky equation with unrestricted, spatially-varying anti-diffusion
针对具有空间变化反扩散的 Kuramoto--Sivashinsky 方程，提出双输入 Fredholm 反步设计，利用预反馈和曲率输入实现快速镇定，并通过神经算子近似增益，数值验证了约 0.1% 的误差和稳定效果。
Minimal Experiments for Robust Stabilization: Information, Spectral Geometry, and Duration
本研究证明，在广泛线性系统中，最短实验序列（精确状态$mn+1$步，噪声状态$m(n+1)$步）可达到最佳实验误差容限的固定比例，但缓慢驱动系统需$1/h$量级持续时间才能恢复该比例。
Backstepping based controller and observer models were designed for higher order linear and nonlinear Schrödinger equations on a finite interval in Part I of this study where the controller was assumed to be acting from the left endpoint of the medium. In this companion paper, we further the analysis by considering boundary controller(s) acting at the right endpoint of the domain. It turns out that the problem is more challenging in this scenario as the associated boundary value problem for the backstepping kernel becomes overdetermined and lacks a smooth solution. The latter is essential to s…
Existing smart composite piezoelectric beam models in the literature mostly ignore the electro-magnetic interactions and adopt the linear elasticity theory. However, these interactions substantially change the controllability and stabilizability at the high frequencies, and linear models fail to represent and predict the governing dynamics since mechanical nonlinearities are pronounced in certain applications such as energy harvesting. In this paper, first, a consistent variational approach is used by considering nonlinear elasticity theory to derive equations of motion for a single-layer piez…
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