Duy Phan, Sérgio S. Rodrigues
The feedback exponential stabilization to trajectories for semilinear parabolic equations in a given bounded domain is addressed. The controls take values in a finite-dimensional space and are supported in a small region. Both internal and boundary controls are considered.
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Stabilization of equilibrium solution to parabolic like equations via proportional boundary feedbacks.
This work studies the stabilization for a periodic parabolic system under perturbations in the system conductivity. A perturbed system does not have any periodic solution in general. However, we will prove that the perturbed system can always be pulled back to a periodic system after imposing a control from a fixed finite dimensional subspace.
The aim of this work is to design an explicit finite dimensional boundary feedback controller of sampled-data form for locally exponentially stabilizing the equilibrium solutions to semilinear parabolic equations. The feedback controller is expressed in terms of the eigenfunctions corresponding to unstable eigenvalues of the linearized equation. This stabilizing procedure is applicable for any sampling rate, not necessary to be small enough, and it tends to the continuous-times version when the sampling period tends to zero.
The present work is concerned with the stabilization of a general class of time-varying linear parabolic equations by means of a finite-dimensional receding horizon control (RHC). The stability and suboptimality of the unconstrained receding horizon framework is studied. The analysis allows the choice of the squared $\ell_1$-norm as control cost. This leads to a nonsmooth infinite-horizon problem which provides stabilizing optimal controls with a low number of active actuators over time. Numerical experiments are given which validate the theoretical results and illustrate the qualitative diffe…
In this paper, we prove the stabilizability of abstract Parabolic Integro-Differential Equations (PIDE) in a Hilbert space with decay rate $e^{-γt} $ for certain $γ> 0,$ by means of a finite dimensional controller in the feedback form. We determine a linear feedback law which is obtained by solving an algebraic Riccati equation. To prove the existence of the Riccati operator, we consider a linear quadratic optimal control problem with unbounded observation operator. The abstract theory of stabilization developed here is applied to specific problems related to viscoelastic fluids, e.g. Oldroyd …
Stabilization of a class of time-varying parabolic equations with uncertain input data using Receding Horizon Control (RHC) is investigated. The diffusion coefficient and the initial function are prescribed as random fields. We consider both cases, uniform and log-normal distributions of the diffusion coefficient. The controls are chosen to be finite-dimensional and enter into the system as a linear combination of finitely many indicator functions (actuators) supported in open subsets of the spatial domain. Under suitable regularity assumptions, we study the expected (averaged) stabilizability…
Observer design and boundary output feedback stabilization for semilinear parabolic system over general multidimensional domain
本文提出了一种基于观测器和有限维反馈控制器的边界输出反馈控制策略，用于解决一般多维域上半线性抛物型系统的稳定化问题。
An oblique projections based feedback stabilizability result in the literature is extended to a larger class of reaction-convection terms. A discussion is presented including a comparison between explicit oblique projections base feedback controls and Riccati based feedback controls. Advantages and limitations of each type of feedback are addressed as well as their finite-elements implementation. Results of numerical simulations are presented comparing their stabilizing performances for the case of time-periodic dynamics. It is shown that the solution of the periodic Riccati based feedback can…
Stabilization of nonautonomous linear parabolic equations with inputs subject to time-delay
本研究通过李恩伯格观测器和预测控制方法，解决非自主抛物方程在输入存在时间延迟时的稳定化问题。
This work concerns the internal stabilization of underactuated linear systems of $m$ heat equations in cascade, where the control is placed internally in the first equation only and the diffusion coefficients are distinct. Combining the modal decomposition method with a recently introduced state-transformation approach for observation problems, a proportional-type stabilizing control is given explicitly. It is based on a transformation for the ODE system corresponding to the comparatively unstable modes into a target one, where calculation of the stabilization law is independent of the arbitra…
It is shown that an internal control based on a moving indicator function is able to stabilize the state of parabolic equations evolving in rectangular domains. For proving the stabilizability result, we start with a control obtained from an oblique projection feedback based on a finite number of static actuators, then we used the continuity of the state when the control varies in relaxation metric to construct a switching control where at each given instant of time only one of the static actuators is active, finally we construct the moving control by traveling between the static actuators. Nu…
In this paper we consider feedback stabilization for parabolic variational inequalities of obstacle type with time and space depending reaction and convection coefficients and show exponential stabilization to nonstationary trajectories. Based on a Moreau-Yosida approximation, a feedback operator is established using a finite (and uniform in the approximation index) number of actuators leading to exponential decay of given rate of the state variable. Several numerical examples are presented addressing smooth and nonsmooth obstacle functions.
This work is devoted to the stabilization of parabolic systems with a finite-dimensional control subjected to a constant delay. Our main result shows that the Fattorini-Hautus criterion yields the existence of such a feedback control, as in the case of stabilization without delay. The proof consists in splitting the system into a finite dimensional unstable part and a stable infinite-dimensional part and to apply the Artstein transformation on the finite-dimensional system to remove the delay in the control. Using our abstract result, we can prove new results for the stabilization of parabolic…
We consider the semilinear parabolic equation of normal type connected with the 3D Helmholtz equation with periodic boundary condition. The problem of stabilization to zero of the solution for normal parabolic equation with arbitrary initial condition by starting control is studied. This problem is reduced to establishing three inequalities connected with starting control, one of which has been proved previously. The proof for the other two is given here.
This paper deals with the rapid stabilization of a degenerate parabolic equation with a right Dirich-let control. Our strategy consists in applying a backstepping strategy, which seeks to find an invertible transformation mapping the degenerate parabolic equation to stabilize into an exponentially stable system whose decay rate is known and as large as we desire. The transformation under consideration in this paper is Fredholm. It involves a kernel solving itself another PDE, at least formally. The main goal of the paper is to prove that the Fredholm transformation is well-defined, continuous …
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。
Rapid boundary stabilization of 1D nonlinear parabolic equations
本文采用模态分解法构造仅改变不稳定特征值的反馈控制，建立一维非线性抛物方程的局部快速边界镇定，并利用耗散性得到 Burgers、Allen-Cahn 等系统的全局快速镇定结果。
Quantitative rapid stabilization for parabolic equations via the linear quadratic theory
针对非自伴抛物方程，利用线性二次型理论和定量可观测性不等式，实现定量快速镇定，并揭示二者等价性；推广至Navier-Stokes方程及自伴情形下的有限维反馈律。
It is shown that an explicit oblique projection nonlinear feedback controller is able to stabilize semilinear parabolic equations, with time-dependent dynamics and with a polynomial nonlinearity. The actuators are typically modeled by a finite number of indicator functions of small subdomains. No constraint is imposed on the sign of the polynomial nonlinearity. The norm of the initial condition can be arbitrarily large, and the total volume covered by the actuators can be arbitrarily small. The number of actuators depend on the operator norm of the oblique projection, on the polynomial degree …
It is shown that an oblique projection based feedback control is able to stabilize the state of the Kuramoto-Sivashinsky equation, evolving in rectangular domains, to a given time-dependent trajectory. The number of actuators is finite and consists of a finite number of indicator functions supported in small subdomains. Simulations are presented, in the one-dimensional case, showing the stabilizing performance of the feedback control.
A Decomposition Method for Finite-Time Stabilization of Bilinear Systems with Applications to Parabolic and Hyperbolic Equations
本文针对双线性系统提出一种基于分解的有限时间稳定化方法，将标称系统分裂为两个子系统以简化分析，并利用 Lyapunov 函数和观测条件推导出稳定化结果，最后通过抛物型和双曲型无穷维系统实例验证了其有效性。
Feedback stabilization of some fourth-order nonlinear parabolic equations with saturated controls
本文研究了四阶非线性抛物型方程在饱和反馈控制下的稳定化问题，通过谱分析和模态分解等方法实现局部指数稳定化。
Input-to-state stabilization of an ODE cascaded with a parabolic equation involving Dirichlet-Robin boundary disturbances
本文研究了在存在Dirichlet-Robin边界扰动和域内扰动的情况下，ODE与抛物型PDE级联系统的输入到状态稳定化问题，采用背向设计方法和广义Lyapunov方法进行稳定性分析。
Input-to-state stabilization of $1$-D parabolic equations with Dirichlet boundary disturbances under boundary fixed-time control
本文针对带Dirichlet边界扰动的1-D抛物方程，采用反步法与分裂技术设计边界反馈控制器，实现输入-状态稳定与固定时间稳定，并用广义Lyapunov方法分析，数值仿真验证有效性。
Input-to-State Stabilization of 1-D Parabolic PDEs under Output Feedback Control
针对含时变系数与边界扰动的一维抛物型 PDE，本文用时不变核和 backstepping 设计基于观测器的输出反馈控制器，并以广义 Lyapunov 方法证明闭环及误差系统在空间 $L^\infty$ 范数下的 ISS。
Given a nonstationary trajectory of the Navier-Stokes system, a finite-dimensional feedback boundary controller stabilizing locally the system to the given trajectory is derived. Moreover the controller is supported in a given open subset of the boundary of the domain containing the fluid. In a first step a controller is derived that stabilizes the linear Oseen-Stokes system "around the given trajectory" to zero; for that a corollary of a suitable truncated boundary observability inequality, the regularizing property for the system, and some standard techniques of the optimal control theory ar…
We derive a saturated feedback control, which locally stabilizes a linear reaction-diffusion equation. In contrast to most other works on this topic, we do not assume the Lyapunov stability of the uncontrolled system and consider general unstable systems. Using Lyapunov methods, we provide estimates for the region of attraction for the closed-loop system, given in terms of linear and bilinear matrix inequalities. We show that our results can be used with distributed as well as scalar boundary control, and with different types of saturations. The efficiency of the proposed method is demonstrate…
For an arbitrary parameter $p\in [1,+\infty]$, we consider the problem of exponential stabilization in the spatial $L^{p}$-norm, and $W^{1,p}$-norm, respectively, for a class of anti-stable linear parabolic PDEs with space-time-varying coefficients in the absence of a Gevrey-like condition, which is often imposed on time-varying coefficients of PDEs and used to guarantee the existence of smooth (w.r.t. the time variable) kernel functions in the literature. Then, based on the obtained exponential stabilities, we show that the solution of the considered system depends continuously on the $L^{p}$…
Successive design of backstepping observers for parabolic PDE-ODE systems and its duality to state feedback stabilization
本文针对严格前馈型抛物型PDE-ODE系统提出连续反步观测器设计，该设计与同类型系统的多步状态反馈设计对偶，可将误差动力学转化为指数稳定的级联子系统。
It is shown that there exist a finite number of indicator functions, which allow us to track an arbitrary given trajectory of the Schlögl model, by means of an explicit saturated feedback control input whose magnitude is bounded by a constant independent of the given targeted trajectory. Simulations are presented showing the stabilizing performance of the explicit feedback constrained control. Further, such performance is compared to that of a receding horizon constrained control minimizing the classical energy cost functional.
Stabilization of Parabolic Time-Varying PDEs using Certified Reduced-Order Receding Horizon Control
针对线性时变抛物型PDE，提出基于降阶模型的有限维滚动时域控制方法，证明指数稳定性与次优性，并引入Galerkin模型降阶与后验误差分析，实现自适应控制与可计算性能界。
Small-Gain-Based Boundary Feedback Design for Global Exponential Stabilization of 1-D Semilinear Parabolic PDEs
本文提出了一种用于1-D半线性抛物型PDEs边界反馈稳定器设计的新方法，基于小增益论证，适用于满足线性增长条件的非线性项，包括非局部项，并展示了反馈设计在抛物型情况下的根本限制。
Finite-Dimensional Feedback Stabilization of Nonautonomous Stochastic Parabolic Equations
研究非自治随机抛物方程的有限维反馈镇定，通过局部指标型执行器的有限线性组合及斜投影构造反馈律，在一定假设下证明闭环系统适定性，实现指数均方镇定及几乎必然镇定，数值实验展示相关影响。
Stabilization of a Wave-Heat Cascade System
本文提出了一种基于有限维动态输出反馈的波-热级联系统稳定化方法，通过Riesz谱算子结构实现任意指数衰减。
We consider the problem of stabilization to zero of semilinear normal parabolic equations connected with the 3D Helmholtz system with periodic boundary conditions and arbitrary initial datum. This problem was previously studied in \cite{FSh16}. As it was recently revealed, the control function suggested in that work contains a term impeding transference the stabilization construction on the 3D Helmholtz system. The main concern of this article is to prove that this term is not necessary for the stabilization result, and therefore the control function can be changed by a proper way.
Stabilization to trajectories of nonisothermal Cahn-Hilliard equations
本文证明非等温Cahn-Hilliard方程对时间依赖轨迹的半全局指数镇定，提出基于斜投影的显式反馈控制，并验证等温情形及数值性能。
Exponential stabilization of infinite-dimensional systems by finite-dimensional controllers
本文研究无限维系统（抽象柯西问题）通过有限维控制器的反馈镇定，提出新的输入-输出增益证明闭环稳定性，并针对拟有限系统给出反馈可镇定的充要条件，验证于抛物和双曲方程。
This paper is concerned with the output feedback boundary stabilization of general 1-D reaction diffusion PDEs in the presence of an arbitrarily large input delay. We consider the cases of Dirichlet/Neumann/Robin boundary conditions for the both boundary control and boundary condition. The boundary measurement takes the form of a either Dirichlet or Neumann trace. The adopted control strategy is composed of a finite-dimensional observer estimating the first modes of the PDE coupled with a predictor to compensate the input delay. In this context, we show for any arbitrary value of the input del…
Differentiability of the Value Function of Semilinear Parabolic Infinite Time Horizon Optimal Control Problems on $L^2(Ω)$ under Control Constraints
本文建立了一个抽象框架，证明了在控制范数约束下，半线性抛物型无限时间horizon最优稳定化问题值函数的局部连续可微性，并保证其经典意义上满足 Hamilton-Jacobi-Bellman 方程。
The aim of this work is to design an explicit finite dimensional boundary feedback controller for locally exponentially stabilizing the equilibrium solutions to Fisher's equation in both $L^2(0,1)$ and $H^1(0,1)$. The feedback controller is expressed in terms of the eigenfunctions corresponding to unstable eigenvalues of the linearized equation. This stabilizing procedure is applicable for any level of instability, which extends the result of \cite{02} for nonlinear parabolic equations. The effectiveness of the approach is illustrated by a numerical simulation.
This paper is addressed to a stabilization problem of a system coupled by a wave and a Euler-Bernoulli plate equation. Only one equation is supposed to be damped. Under some assumption about the damping and the coupling terms, it is shown that sufficiently smooth solutions of the system decay logarithmically at infinity without any geometric conditions on the effective damping domain. The proofs of these decay results rely on the interpolation inequalities for the coupled elliptic-parabolic systems and make use of the estimate of the resolvent operator for the coupled system. The main tools to…
Non-Convex Global Optimization as an Optimal Stabilization Problem: Dynamical Properties
该研究通过最优控制理论处理非凸全局优化问题，证明折扣控制问题的（准）最优轨迹可全局实际渐近收敛到全局极小值点集，无需求解遍历型HJB方程，并针对三种问题形式给出结果。
The present paper addresses the topic of boundary output feedback stabilization of parabolic-type equations, governed by linear differential operators which can be diagonalized by the introduction of adequate weighting functions (by means of the Sturm-Liouville method), and which evolve in bounded spatial domains that are subsets of $\mathbb{R}^d,\ d=1,2,3$. Combining ideas inspired by \cite{lhachemi2022finite} for the boundary output feedback control of 1-D parabolic PDEs and \cite{munteanu2019boundary} for the state feedback control of multi-D parabolic PDEs, we report in this paper an outpu…
Stabilization of underactuated linear coupled reaction-diffusion PDEs via distributed or boundary actuation
本文提出一种基于模态分解和广义Sylvester方程的构造性镇定算法，解决具有不同扩散系数的级联线性反应-扩散PDE系统的欠驱动镇定问题，适用于分布式和边界控制。
In this paper, we investigate the problem of semi-global minimal time robust stabilization of analytic control systems with controls entering linearly, by means of a hybrid state feedback law. It is shown that, in the absence of minimal time singular trajectories, the solutions of the closed-loop system converge to the origin in quasi minimal time (for a given bound on the controller) with a robustness property with respect to small measurement noise, external disturbances and actuator noise.
Input-to-State Stabilization of a Coupled ODE-PDE System with Time-Varying Coefficients via Composite Boundary Control
本文提出一种复合边界反馈控制律，通过解析增益函数和时变Volterra核显式设计，并构造非二次Lyapunov函数和广义Lyapunov泛函，实现了具有时变系数的耦合ODE-抛物型PDE系统的输入到状态稳定性，数值仿真验证了有效性。
We consider a one-dimensional controlled reaction-diffusion equation, where the control acts on the boundary and is subject to a constant delay. Such a model is a paradigm for more general parabolic systems coupled with a transport equation. We prove that this is possible to stabilize (in H 1 norm) this process by means of an explicit predictor-based feedback control that is designed from a finite-dimensional subsystem. The implementation is very simple and efficient and is based on standard tools of pole-shifting. Our feedback acts on the system as a finite-dimensional predictor. We compare o…
Exponential Selection and Feedback Stabilization of Invariant Subspaces of Quantum Trajectories
该研究证明量子轨迹会以期望指数速度收敛至最小不变子空间，基于子空间概率分布可辨识性完成证明，并提出可实现目标子空间指数级定向收敛的反馈控制策略。
When applying methods of optimal control to motion planning or stabilization problems, some theoretical or numerical difficulties may arise, due to the presence of specific trajectories, namely, singular minimizing trajectories of the underlying optimal control problem. In this article, we provide characterizations for singular trajectories of control-affine systems. We prove that, under generic assumptions, such trajectories share nice properties, related to computational aspects; more precisely, we show that, for a generic system -- with respect to the Whitney topology --, all nontrivial sin…
This paper studies the controllability problem of a parabolic system of chemotaxis. The local exact controllability to trajectories of the system imposed one control force only is obtained by applying Kakutani's fixed point theorem combined with the null controllability of the associated linearized parabolic system. The control function is shown to be in $L^\infty(Q)$, which is estimated by using the methods of maximal regularity and $L^p$-$L^q$ estimates of parabolic equations.
This paper considers the backstepping design of state feedback controllers for coupled linear parabolic partial integro-differential equations (PIDEs) of Volterra-type with distinct diffusion coefficients, spatially-varying parameters and mixed boundary conditions. The corresponding target system is a cascade of parabolic PDEs with local couplings allowing a direct specification of the closed-loop stability margin. The determination of the state feedback controller leads to kernel equations, which are a system of coupled linear second-order hyperbolic PIDEs with spatially-varying coefficients …
Output Regulation for Linear Parabolic Systems Using Finite-Dimensional Tracking-Error-Based Control
研究一维扩散反应系统在干扰和参考信号由不稳定外系统生成时的输出调节问题，提出基于有限维跟踪误差的调节器设计方法，通过推导组合装置转化问题，利用模态分解法刻画可观测性，证明观测器增益性质并给出LMI条件，分析了含未知时变测量延迟的情况。
A learning approach for optimal feedback gains for nonlinear continuous time control systems is proposed and analysed. The goal is to establish a rigorous framework for computing approximating optimal feedback gains using neural networks. The approach rests on two main ingredients. First, an optimal control formulation involving an ensemble of trajectories with 'control' variables given by the feedback gain functions. Second, an approximation to the feedback functions via realizations of neural networks. Based on universal approximation properties we prove the existence and convergence of opti…
Motion planning and stabilization of nonholonomic systems using gradient flow approximations
本文针对时变非线性控制仿射系统，提出基于振荡输入和梯度流近似的统一控制方案，解决非完整系统的轨迹跟踪与镇定问题，并给出渐近行为分析及避障应用。
This paper considers the backstepping state feedback control of coupled linear parabolic PDEs with spatially varying coefficients and bilateral actuation. By making use of the folding technique, a system representation with unilateral actuation is obtained, allowing to apply the standard backstepping transformation. To ensure the regularity of the solution, the folded system is subject to unusual folding boundary conditions, which lead to additional boundary couplings between the PDEs. Therefore, the solution of the corresponding kernel equations determining the transformations is a very chall…
Finite-Dimensional RHC Control of Linear Time-Varying Parabolic PDEs: Stability Analysis and Model-Order Reduction
本文针对线性时变抛物型偏微分方程，提出有限维滚动时域控制方法，采用平方l1范数代价实现少量执行器下的镇定，并结合数据驱动模型降阶加速计算，通过数值实验验证了方法的有效性。
Trajectory stabilization of nonlocal continuity equations by localized controls
本文研究非局部连续性方程在局部控制下的轨迹镇定，给出基于 Wasserstein 距离的误差改善定义，并证明轨迹穿过控制区域且满足正则性条件时可实现镇定。
Stabilizability of parabolic equations by switching controls based on point actuators
针对非自治抛物型方程，本文提出基于有限个 Dirac delta 点执行器、每时刻至多激活一个的切换控制，并通过无限时域最优控制与滚动时域框架实现镇定。
A class of optimal control problems of hybrid nature governed by semilinear parabolic equations is considered. These problems involve the optimization of switching times at which the dynamics, the integral cost, and the bounds on the control may change. First- and second-order optimality conditions are derived. The analysis is based on a reformulation involving a judiciously chosen transformation of the time domains. For autonomous systems and time-independent integral cost, we prove that the Hamiltonian is constant in time when evaluated along the optimal controls and trajectories. A numerica…
Quantitative rapid boundary stabilization via modal decomposition and its application to the Allen-Cahn equation
研究一维艾伦 - 卡恩方程定量快速稳定化，用模态分解法明确反馈律和稳定成本与衰减率关系，构造反馈律并结合频率李雅普诺夫方法得稳定估计，还构建分段反馈律实现零可控性和有限时间稳定化。
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