Exact controllability of stochastic parabolic equations with multiplicative noise
Viorel Barbu
证明了带有限个模的乘性噪声的线性和半线性随机抛物型方程是精确零能控的。
One proves that the linear and semilinear stochastic parabolic equations with a multiplicative noise with a finite number of modes are exactly null controllable.
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In this paper we consider a multidimensional semilinear reaction-diffusion equation and we obtain at any arbitrary time an approximate controllability result between nonnegative states using as control term the reaction coefficient, that is via multiplicative controls.
Carleman estimates, observability inequalities and null controllability for interior degenerate non smooth parabolic equations
给出了非光滑系数在内点退化的抛物型方程的Carleman估计、能观性不等式和零能控性结果。
Existence, Stability and Controllability of the parabolic-parabolic thermistor model
本文研究了抛物-抛物热阻模型的适定性、稳定性及可控性，通过卡尔曼估计和反函数定理证明了局部零可控性。
Null-Controllability of 1-D parabolic equations
证明了具有时间无关可测系数的一维抛物型方程的内部零能控性。
This paper deals with the analysis of the internal control with constraint of positive kind of a parabolic PDE with nonlinear diffusion when the time horizon is large enough. The minimal controllability time will be strictly positive. We prove a global steady state constrained controllability result for a quasilinear parabolic with nonlinearity in the diffusion term. Then, under suitable dissipative assumption in the system and local controllability results, we conclude the result to any initial datum and any target trajectory.
Controllability of a system of non-autonomous degenerate coupled parabolic equations
针对区域一端退化且扩散项带有时变系数的一维抛物方程证明卡尔曼估计，并利用该估计证明此类扩散算子构成的耦合系统的零能控性。
This paper deals with the hierarchical control of the parabolic equation.We use Stackelberg{Nash strategies. As usual, we consider one leader and two followers. To each leader we associate a Nash equilibrium corresponding to a bi-objective optimal control problem, then, we look for a leader that solves null controllability e with trajectories problem. We consider linear and nonlinear equations in dimension 1.
In this paper we present a null controllability result for a degenerate semilinear parabolic equation with first order terms. The main result is obtained after the proof of a new Carleman inequality for a degenerate linear parabolic equation with first order terms.
Controllability of a 2x2 parabolic system by one force with a space-dependent coupling term of order one
本文研究了两个耦合的抛物型方程线性系在耦合项含有空间依赖项时的能控性。这个系统被设置在一个有界的区间上，第一个方程由I的子区间或边界上的力控制。在耦合域和控制域的交集非空的情况下，我们证明了零能控性在任何时候。否则，我们提供一个最小的时间为零可控性。最后给出了近似能控性的一个充要条件。获得这些结果的主要技术工具是矩量法
We consider systems of parabolic equations coupled in zero order terms in a star-like or a tree-like shape, with an internal control acting in only one of the equations. We obtain local exact controllability to the stationary solutions of the system, under hypotheses concerning the supports of the coupling functions. The key point is establishing Carleman estimates with appropriate observation operators for the adjoint to the linearized system, which allows the study of the controllability problem, in either linear or nonlinear cases, in an $L^\infty$ framework.
In this paper, we analyze the null controllability property for a degenerate parabolic equation involving memory terms with a locally distributed control. We first derive a null controllability result for a nonhomogeneous degenerate heat equation via new Carleman estimates with weighted time functions that do not blow up at t= 0. Then this result is successfully used with a classical fixed point to obtain null controllability for the initial memory system.
Controllability issues for parabolic-elliptic systems involving nonlocal couplings
本文研究含非局部积分耦合的偏微分方程系统，分别证明非线性抛物-椭圆系统可借助 Carleman 估计实现控制，并揭示一维线性非局部抛物-抛物系统的可控性依赖耦合结构且可能失效。
Partial null controllability of parabolic linear systems
研究了n阶抛物型线性系的部分零能控性问题。给定RN中的一个有界区域，我们研究了在一个非空开子集中m个局部化控制只控制解的p个分量（p，m < n）的效果.本文的第一个主要结果是当耦合矩阵和控制矩阵为常数时的一个充要条件。第二个结果提供了，在第一步，部分零可控性的充分条件时，矩阵只依赖于时间。在第二步中，通过一个部分控制的2 × 2抛物系统的例子，我们将提供部分零可控性的积极和消极的结果时，系数是空间依赖的。
Null controllability for degenerate parabolic equations with a nonlocal space term
该研究针对带非局部空间项的两类退化热方程，通过分析相关非齐次方程的适定性、伴随问题Carleman估计及零能控性，结合Kakutani不动点定理，证明了原非局部问题的零能控性。
We prove the null controllability of a one dimensional degenerate parabolic equation with drift and a singular potential. We study the case the potential arises at the left end point and the weighted Dirichlet boundary control is located at this point. We get a spectral decomposition of a suitable operator, defined in a weighted Sobolev space, involving Bessel functions and their zeros, then we use the moment method by Fattorini and Russell to obtain an upper estimate of the cost of controllability. We also obtain a lower estimate of the cost of controllability by using a representation theore…
Controllability of a Semilinear System of Parabolic Equations with Nonlocal Terms
本文研究了具有非局部项的半线性抛物型方程组的可控性问题，通过结合不动点定理和线性化动力学估计，建立了在固定时间内的局部零可控性，并拓展了线性系统的可控性范围。
Positive and negative results on the internal controllability of parabolic equations coupled by zero and first order terms
研究了R^N（N>1）中光滑区域上两个线性耦合抛物型方程的零能控性和近似能控性，其中耦合项为零阶和一阶，控制项位于区域的任意非空子集中.在一个新的充分条件下证明了零能控性，并给出了非近似能控系统的第一个例子，其中一个非平凡的一阶耦合项的支集与控制域相交.
Carleman estimates for semi-discrete parabolic operators with a discontinuous diffusion coefficient and application to controllability
在一维有限差分的离散设置中，我们证明了抛物算子$\partial_t-\partial_x (c\partial_x)$的半离散化的Carleman估计，其中扩散系数$c$具有跳跃.作为这个Carleman估计的结果，我们推导出一致的零可控性的半线性抛物型方程类的结果。
Approximate and null controllability of a parabolic system with coupling terms of order one
本文研究具有一阶耦合项抛物型系统的近似与零能控性，基于已有结果推导出控制域非区间情况下的一般能控性条件。
Null controllability for parabolic equations with dynamic boundary conditions of reactive-diffusive type
证明了具有表面扩散型动态边界条件的线性和半线性热方程的零能控性。结果是基于一个新的Carleman估计这种类型的边界条件。
Null controllability of a parabolic system with a cubic coupling term
本文考虑一组两个抛物型方程，其中一个方程中有强迫项，另一个方程中有立方耦合项。我们证明了系统是局部零能控的。
Null controllability of quasilinear parabolic equations with gradient dependent coefficients
本文研究含梯度相关系数的拟线性抛物方程零能控性，借助线性问题的正则控制与不动点策略证明近似零能控，并对含抛物 $p$-Laplacian 的情形证明任意小时间精确零能控，且用有限元数值验证。
We study the null-controllability of some hypoelliptic quadratic parabolic equations posed on the whole Euclidean space with moving control supports, and provide necessary or sufficient geometric conditions on the moving control supports to ensure null-controllability. The first class of equations is the one associated to non-autonomous Ornstein-Uhlenbeck operators satisfying a generalized Kalman rank condition. In particular, when the moving control supports comply with the flow associated to the transport part of the Ornstein-Uhlenbeck operators, a necessary and sufficient condition for null…
Boundary controllability for a 1D degenerate parabolic equation with drift and a singular potential and a Neumann boundary condition
本文研究具有漂移项和奇异位势的一维退化抛物型方程，通过算子谱分解与矩方法证明了其在加权Neumann边界控制下的零能控性，并给出了能控性代价的上下界估计。
In this paper we prove a uniform controllability result for a fourth order parabolic partial differential equation which includes a transport term, when the coefficients of higher order terms vanish. We prove the null controllability of the system with a single boundary control, and also it is obtained that the cost of the control decreases to zero, under the hypothesis of the control time being large enough. Moreover, we prove that, if the control time is small enough, the cost of controllability increases to infinity.
Null-controllability for a fourth order parabolic equation under general boundary conditions
本文研究一般边界条件下四阶内部控制抛物方程的可零控性，通过Carleman不等式推导插值不等式，进而得到谱不等式，最终建立可零控性结果。
Over the past two decades, the controllability of several examples of parabolic-hyperbolic systems has been investigated. The present article is the beginning of an attempt to find a unified framework that encompasses and generalizes the previous results. We consider constant coefficients heat-transport systems with coupling of order zero and one, with a locally distributed control in the source term, posed on the one dimensional torus. We prove the null-controllability, in optimal time (the one expected because of the transport component) when there is as much controls as equations. When the …
Local controllability of free boundary three-dimensional semilinear radial parabolic equations
针对具有Stefan边界条件和径向对称数据的自由边界三维半线性热方程，通过简化为一维形式并适配Carleman不等式，结合Schauder不动点定理证明其局部零可控性，为该问题在一维以上空间维度提供首个可控性结果。
Boundary null controllability of a class of 2-d degenerate parabolic PDEs
本文首次研究方形区域上退化抛物方程的边界零能控性，结合矩方法与Lebeau-Robbiano策略，分析退化边界子集上的控制，并建立Kalman秩条件刻画耦合系统，推广至N维。
Local null controllability for degenerate parabolic equations with nonlocal term
我们建立了以下非线性抛物型方程的局部零能控性结果：$$u_t-\left(b\left(x,\int_0^1u \ \right)u_x \right)_x+f(t,x,u)=hχ_ω,\ (t,x)\in (0,T)\times (0,1) $$其中$b(x,r)=\ell(r)a(x)$是一个分离变量的函数，定义了一个算子，该算子在$x=0$退化，并具有一个非局部项。我们的方法依赖于Liusternik的逆映射定理的应用程序，需要证明一个合适的Carleman估计。
Controllability for forward stochastic parabolic equations with dynamic boundary conditions without extra forces
本文研究动态边界条件下前向随机抛物方程的可控性，通过谱不等式和迭代策略建立可观测性，进而证明零可控性和近似可控性。
Controllability and Inverse Problems for Parabolic Systems with Dynamic Boundary Conditions
本文是一篇综述，梳理带动态边界条件的抛物型系统零可控性与反问题的研究成果，阐释Carleman估计等经典方法的推广应用，分析其相较静态边界条件的难点，并给出结论与开放问题。
In this paper, we prove the small-time global null-controllability of forward (resp. backward) semilinear stochastic parabolic equations with globally Lipschitz nonlinearities in the drift and diffusion terms (resp. in the drift term). In particular, we solve the open question posed by S. Tang and X. Zhang, in 2009. We propose a new twist on a classical strategy for controlling linear stochastic systems. By employing a new refined Carleman estimate, we obtain a controllability result in a weighted space for a linear system with source terms. The main novelty here is that the Carleman parameter…
In this paper, we prove a Carleman estimate for fully-discrete approximations of parabolic operators in which the discrete parameters $h$ and $\triangle t$ are connected to the large Carleman parameter. We use this estimate to obtain relaxed observability inequalities which yield, by duality, controllability results for fully-discrete linear and semilinear parabolic equations.
Let us consider a nonlinear degenerate reaction-diffusion equation with application to climate science. After proving that the solution remains nonnegative at any time, when the initial state is nonnegative, we prove the approximate controllability between nonnegative states at any time via multiplicative controls, that is, using as control the reaction coefficient.
Approximate controllability for nonlinear degenerate parabolic problems with bilinear control
本文研究了非线性退化抛物型Cauchy-Neumann问题的全局近似乘能控性。首先，我们将得到嵌入结果加权索伯列夫空间，已被证明是决定性的非线性退化问题达到适定性。然后，我们证明了上述系统可以引导在$L^2$从任何非零，非负的初态到任何期望的非负目标状态的任何邻域的双线性分段静态控制。此外，我们放宽了对初始日期的符号约束，推广了上述结果。
Null controllability for stochastic parabolic equations with Robin boundary conditions
本文研究带线性 Robin 边界条件且含零阶、一阶有界系数项的正反向线性随机抛物型方程，通过对偶方法为负 Sobolev 空间中的伴随方程弱解建立两个新的全局 Carleman 估计，从而证明其零能控性。
Indirect controllability of some linear parabolic systems of m equations with m-1 controls involving coupling terms of zero or first order
本文研究了几类控制少于方程的线性耦合抛物系统的零能控性和近似能控性。更精确地说，对于R^N中的一个给定的有界区域，我们考虑一个由m个线性抛物型方程（m > 2）组成的方程组，方程组具有一阶和零级耦合项，m-1个控制局部化在某个任意的非空开子集上。在常耦合系数的情形下，给出了在任意小时间内获得零能控性或近似能控性的充要条件。在m = 2，N = 1的情形下，对于依赖于空间和时间变量的一般系数，只要耦合项的支集与控制域相交，我们也给出了在任意小时间内获得零能控性或近似能控性的一般充分条件.结果得到感谢虚拟控制方法与代数方法和一些适当的Carleman估计。
Inverse source problem and null controllability for multidimensional parabolic operators of Grushin type
Imanuvilov和Yamamoto于1998年提出的基于Carleman估计的一致抛物型方程Lipschitz稳定性方法似乎很难适用于本文感兴趣的Grushin型算子的情况。事实上，这样的估计仍然缺少抛物算子退化的内部空间域。尽管如此，我们能够证明Lipschitz稳定性结果的逆源问题，这样的运营商，在任意空间维度的局部分布的测量。为此，我们遵循一个混合策略，它结合了由于勒博和Robbiano的apraoch，依赖于傅立叶分解，与Carleman不等式的热方程的非光滑系数（解决的傅立叶模式）。作为推论，我们得到了多维Grushin型抛物型方程的可观测性的一个直接证明，它等价于具有局部分布控制的零能控性.
Local controllability of a free-boundary problem for a class of one-dimensional degenerate parabolic equations
本文研究了一类一维退化抛物方程自由边界问题的局部可控性，证明了在初始状态足够小的情况下，存在控件可将系统状态在指定时间驱动到零。
Robust Stackelberg controllability for a parabolic equation
本文的目的是执行一个Stackelberg策略来控制抛物型方程。我们有一个控制，\textit{the leader}，负责一个空的可控性属性;此外，我们有一个控制\textit{the follower}，解决了鲁棒控制的目标。这意味着，我们寻求一个成本泛函的鞍点。以这种方式，跟随器控制对广泛类别的外部干扰不敏感。据我们所知，将鲁棒性与Stackelberg策略相结合的想法在文献中是新的
Hierarchical local null controllability for a non-degenerate quasi-linear parabolic system
该论文针对有界区间上的非退化拟线性抛物系统，采用含领导者与追随者控制的Stackelberg-Nash策略，先证明线性系统的Carleman不等式，再通过Liusternik反函数定理得到拟线性系统的局部零能控性。
Stackelberg-Nash null controllability for stochastic parabolic equations
本文研究了含梯度项的随机抛物型方程的分层控制问题，采用双领导者与双跟随者的Stackelberg-Nash策略，通过Carleman估计建立能观性不等式，实现了耦合前向-后向随机系统的零能控性。
Null-controllability for some linear parabolic systems with controls acting on different parts of the domain and its boundary
本文研究常系数线性抛物系统在多个控制作用于不同子域及边界时的零能控性，证明了一维Kalman秩条件，并建立了Carleman估计。
Analyticity of Exponential Dirichlet Series and Applications to the Approximate Controllability of Parabolic Equations
本文研究一类指数 Dirichlet 级数的解析性、幂级数系数及余项估计，并将其用于矩量法，证明线性抛物方程在局部分布或集中控制下的近似能控性。
We consider a fully-discrete approximations of 1-D heat equation with dynamic boundary conditions for which we provide a controllability result. The proof of this result is based on a relaxed observability inequality for the corresponding adjoint system. This is done by using a suitable Carleman estimate for such models where the discrete parameters $h$ and $Δt$ are connected to the one of the large Carleman parameters.
Boundary controllability for a 1D degenerate parabolic equation with a Robin boundary condition
本文研究左端点带加权 Robin 边界条件且势函数具有奇异性的一维退化抛物型方程，利用奇异 Sturm-Liouville 理论、Fourier-Dini 展开和矩方法证明零可控性，并给出可控性代价的上下界估计。
In this paper, we consider forward stochastic nonlinear parabolic equations, with a control localized in the drift term. Under suitable assumptions, we prove the small-time global null-controllability, with a truncated nonlinearity. We also prove the statistical local null-controllability of the true system. The proof relies on a precise estimation of the cost of null-controllability of the stochastic heat equation and on an adaptation of the source term method to the stochastic setting. The main difficulty comes from the estimation of the nonlinearity in the fixed point argument due to the la…
Null-controllability of non-autonomous Ornstein-Uhlenbeck equations
研究了一类非自治Ornstein-Uhlenbeck算子所对应的抛物型方程的零能控性。当Kalman型条件对某个正时间$T>0$成立时，这些抛物型方程在时间$T>0$具有Gevrey正则化效应.由于这种正则化效应，我们证明了通过适应Lebeau-Robbiano方法，这些抛物型方程是零可控的时间大于或等于$T>0$从控制区域，零可控性是经典已知的热方程的情况下举行。
In this paper, we consider a parabolic PDE on a torus of arbitrary dimension. The nonlinear term is a smooth function of polynomial growth of any degree. In this general setting, the corresponding Cauchy problem is not necessarily well posed. We show that the equation in question is approximately controllable by only a finite number of Fourier modes. This result is proved by using some ideas from the geometric control theory introduced by Agrachev and Sarychev.
Boundary controllability for a fourth order degenerate parabolic equation with a singular potential
本文研究带奇异势的一维四阶退化抛物型方程在左端点边界控制下的零能控性，利用 Bessel 函数谱分解、半群理论和矩方法给出能控性代价的上下界估计。
Controllability for a 2x2 nonlinear degenerate parabolic system via one boundary control force
本文研究两个退化抛物方程组成的非线性系统，通过单一边界控制力，利用矩方法和迭代法证明了局部精确边界可控性到零，并给出了控制成本估计。
Carleman Estimates and null controllability of coupled degenerate systems
本文研究具有两个不同扩散系数和一个控制力的弱退化耦合抛物系统的零能控性。为了实现这一目标，我们发展了第一个新的整体Carleman估计的退化抛物方程的权函数不同于以前的作品。
Observability and Null Controllability for Parabolic Systems with Delay and Memory
本文研究了同时包含记忆和延迟效应的发展方程的零能控性问题，采用延迟与记忆型零能控性概念，利用对偶论证、Carleman估计和移动控制策略推导出实现能控性的充分条件，并通过数值模拟验证了理论结果。
Small-time global controllability of a class of bilinear fourth-order parabolic equations
本文研究了一类双线性四阶抛物方程的小时间全局可控性，通过几何控制方法和固定点论证，实现了状态的精确控制。
Null controllability of one-dimensional parabolic equations by the flatness approach
考虑一维线性抛物型方程，其系数与空间有关，可测且可退化或奇异，考虑其两端的广义Robin-Neumann边界条件，利用平坦化方法证明了具有一个边界控制的零能控性，该方法将控制和相应的轨迹显式地表示为级数.该方法特别适用于（可能退化或奇异的）热方程$L^p$和$a(x)\textgreater{}0$，对于a.e. $x\in (0,1)$和$a+1/a \in L^1(0,1)$，或具有平方反比势的热方程$u\_{xx}+(μ / |x|^2)u-u\_t=0$和$μ\ge 1/4$。
We study the global approximate controllability of the reaction-diffusion equation in a parallelpiped $ Ω= (a_1,b_1 ) \times \ldots (a_n,b_n) \subset R^n $, governed by a multiplicative control in a reaction term. It is assumed that the initial state $ u_0 $ admits zeros only on the intersections of $ Ω$ with finitely many hyperplanes, parallel to the sides of $ Ω$, and that $ u_0$ changes its sign after crossing such hyperplanes (we further refer to them as the "hyperplanes of change of sign" or "zero hyperplanes"). This paper can be viewed as a continuation of work presented in \cite{CanKh, …
In a separable Hilbert space $X$, we study the linear evolution equation \begin{equation*} u'(t)+Au(t)+p(t)Bu(t)=0, \end{equation*} where $A$ is an accretive self-adjoint linear operator, $B$ is a bounded linear operator on $X$, and $p\in L^2_{loc}(0,+\infty)$ is a bilinear control. We give sufficient conditions in order for the above control system to be locally controllable to the ground state solution, that is, the solution of the free equation ($p\equiv0$) starting from the ground state of $A$. We also derive global controllability results in large time and discuss applications to paraboli…
This paper deals with a multi-objective control problem for a class of nonlocal parabolic equations, where the non-locality is expressed through an integral kernel. We present the Stackelberg strategy that combines the concepts of controllability to trajectories with optimal control. The strategy involves two controls: a main control (the leader) and a secondary control (the follower). The leader solves a controllability to trajectories problem which consists to drive the state of the system to a prescribed target at a final time while the follower solves an optimal control problem which consi…
Unique continuation and approximate controllability for a degenerate parabolic equation
针对一维弱退化抛物方程，通过建立局部Carleman估计证明解在退化集上消失且共法向导数为零时必恒为零，并推导Dirichlet边界条件下的近似可控性。
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