Genni Fragnelli, Dimitri Mugnai
We show Carleman estimates, observability inequalities and null controllability results for parabolic equations with non smooth coefficients degenerating at an interior point.
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In this paper we consider a degenerate population equation in divergence form depending on time, on age and on space and we prove a related null controllability result via Carleman estimates.
Carleman estimate for full-discrete approximations of the complex Ginzburg-Landau equation with dynamic boundary conditions and applications to controllability
本文为带动态边界条件的一维Ginzburg-Landau方程全离散逼近建立离散Carleman估计，进而推导松弛可观测性不等式，最终证明其可控性。
In this paper, we study the null controllability of weakly degenerate coupled parabolic systems with two different diffusion coefficients and one control force. To obtain this aim, we develop first new global Carleman estimates for degenerate parabolic equations with weight functions different from the ones in the previous works.
We deal with a degenerate model describing the dynamics of a population depending on time, on age and on space. We assume that the degeneracy can occur at the boundary or in the interior of the space domain and we focus on null controllability problem. To this aim, we prove first Carleman estimates for the associated adjoint problem, then, via cut off functions, we prove the existence of a null control function localized in the interior of the space domain.
In this article we consider a control problem of a linear Euler-Bernoulli damped beam equation with potential in dimension one with periodic boundary conditions. We derive a new Carleman estimate for an adjoint of the equation under consideration. Then using a well known duality argument we obtain explicitly the control function which can be used to drive the solution trajectory of the control problem to zero state.
We establish a new family of Carleman inequalities for wave operators on cylindrical spacetime domains containing a potential that is critically singular, diverging as an inverse square on all the boundary of the domain. These estimates are sharp in the sense that they capture both the natural boundary conditions and the natural $H^1$-energy. The proof is based around three key ingredients: the choice of a novel Carleman weight with rather singular derivatives on the boundary, a generalization of the classical Morawetz inequality that allows for inverse-square singularities, and the systematic…
Controllability of parabolic equations with inverse square infinite potential wells via global Carleman estimates
本文通过全局Carleman估计，首次在多维空间中解决了抛物方程在逆平方无限势阱中的可控性问题。
Observability inequality for the wave equation
本文针对波动方程，仅用Carleman估计推导可观测性不等式，为相关研究提供了新的核心工具
Based on a fundamental identity for stochastic hyperbolic-like operators, we derive in this paper a global Carleman estimate (with singular weight function) for stochastic wave equations. This leads to an observability estimate for stochastic wave equations with non-smooth lower order terms. Moreover, the observability constant is estimated by an explicit function of the norm of the involved coefficients in the equation.
In the discrete setting of one-dimensional finite-differences we prove a Carleman estimate for a semi-discretization of the parabolic operator $\partial_t-\partial_x (c\partial_x)$ where the diffusion coefficient $c$ has a jump. As a consequence of this Carleman estimate, we deduce consistent null-controllability results for classes of semi-linear parabolic equations.
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.
In this paper, we present a new Carleman estimate for the adjoint equations associated to a class of super strong degenerate parabolic linear problems. Our approach considers a standard geometric imposition on the control domain, which can not be removed in the general situations. Additionally, we also apply the aformentioned main inequality in order to investigate the null controllability of two nonlinear parabolic systems. The first application is concerned a global null controllability result obtained for some semilinear equations, relying on a fixed point argument. In the second one, a loc…
We consider a parabolic problem with degeneracy in the interior of the spatial domain, and we focus on controllability results through Carleman estimates for the associated adjoint problem. The novelty of the present paper is that the degeneracy is also in the interior of the control region, so that no previous result can be adapted to this situation.
We consider the transport equation $\ppp_t u(x,t) + H(t)\cdot \nabla u(x,t) = 0$ in $\OOO\times(0,T),$ where $T>0$ and $\OOO\subset \R^d $ is a bounded domain with smooth boundary $\ppp\OOO$. First, we prove a Carleman estimate for solutions of finite energy with piecewise continuous weight functions. Then, under a further condition which guarantees that the orbits of $H$ intersect $\ppp\OOO$, we prove an energy estimate which in turn yields an observability inequality. Our results are motivated by applications to inverse problems.
Nash-Stackelberg controllability for coupled systems of degenerate equations in non-cylindrical domains
本文研究了非圆柱域中退化方程耦合系统的纳什-斯塔克尔伯格可控性，通过分层零可控性方法和Carleman估计实现局部可控性。
We establish new Carleman estimates for the wave equation, which we then apply to derive novel observability inequalities for a general class of linear wave equations. The main features of these inequalities are that (a) they apply to a fully general class of time-dependent domains, with timelike moving boundaries, (b) they apply to linear wave equations in any spatial dimension and with general time-dependent lower-order coefficients, and (c) they allow for significantly smaller time-dependent regions of observations than allowed from existing Carleman estimate methods. As a standard applicat…
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.
Carleman Inequalities for the Heat Equation with Fourier Boundary Conditions: Applications to Null Controllability Problems
本文为带有傅里叶边界条件的热方程建立了卡尔门不等式，并通过耦合抛物方程解决零可控性问题，提出迭代数值方法。
The goal of this article is to show a local exact controllability to smooth (C2) trajectories for the 2-d density dependent incompressible Navier-Stokes equations. Our controllability result requires some geometric condition on the ow of the target trajectory, which is remanent from the transport equation satisfied by the density. The proof of this result uses a fixed point argument in suitable spaces adapted to a Carleman weight function that follows the ow of the target trajectory. Our result requires the proof of new Carleman estimates for heat and Stokes equations.
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。
New global Carleman estimates and null controllability for a stochastic Cahn-Hilliard type equation
本文针对半线性项含一、二阶导数的随机 Cahn-Hilliard 型方程，通过建立新的全局 Carleman 估计并结合不动点方法，证明了其零能控性。
A Carleman estimate and an energy method for a first-order symmetric hyperbolic system
本文针对一阶对称双曲系统，在系数矩阵满足正性条件时建立Carleman估计，并由此导出由边界数据观测初值的$L^2$可观测性估计。
In this paper, we study some controllability and observability properties for a coupled system of time-discrete fourth- and second-order parabolic equations. This system can be regarded as a simplification of the well-known stabilized Kumamoto-Sivashinsky equation. Unlike the continuous case, we can prove only a relaxed observability inequality which yields a $ϕ(\triangle t)$-controllability result. This result tells that we cannot reach exactly zero but rather a small target whose size goes to 0 as the discretization parameter $\triangle t$ goes to 0. The proof relies on a known Carleman esti…
In this paper we prove an observability inequality for a degenerate transport equation. First we introduce a local in time Carleman estimate for the degenerate equation, then we apply it to obtain a global in time observability inequality by using also an energy estimate.
In this article, we present a novel Carleman estimate for ultrahyperbolic operators, in $ \mathbb{R}^m_t \times \mathbb{R}^n_x $. Then, we use a special case of this estimate to obtain improved observability results for wave equations with time-dependent lower order terms. The key improvements are: (1) we obtain smaller observation regions compared to standard Carleman estimate results, and (2) we also prove observability when the observation point lies inside the domain. Finally, as a corollary of the observability result, we obtain improved interior controllability for the wave equation.
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…
Observability and controllability for Schrödinger equations in the semi-periodic setting
本文研究波导半周期流形上 Schrödinger 方程的能观性与能控性，先分析线性算子，再结合 HUM 估计、Bourgain 空间和不动点方法证明三次 NLS 的局部精确能控性。
In the paper, we show a global Carleman estimate for the non-local heat equation. To be more precise, let $Ω\subset\RR^d$ be a bounded domain and $\CO\subsetΩ$ an open subdomain, $s\in(0,1)$. We show that there exist constants $C_1,C_2,r_0, T_0>0$ and a weight function $α:Ω\to(0,\infty)$ such that any solution $u$ of %consider the following system % \begin{eqnarray}\label{oben1} \left\{ \begin{array}{rcl} \timed u(x,t)+(-\De)^s u (x,t) &=&f(x,t) \quad\mbox{for}\quad (x,t)\in \Om \times (0,\infty), \\ u(x,t) &=& 0 \quad\mbox{for}\quad(x,t)\in \partial \Om \times (0,\infty), \end{array}\right. \…
Carleman estimates for parabolic equations with super strong degeneracy in a set of positive measure
本文针对具有超强退化性的线性抛物方程建立新Carleman估计，在控制域包含退化集条件下，进而获得零可控性结果。
The approach to Lipschitz stability for uniformly parabolic equations introduced by Imanuvilov and Yamamoto in 1998, based on Carleman estimates, seems hard to apply to the case of Grushin-type operators of interest to this paper. Indeed, such estimates are still missing for parabolic operators degenerating in the interior of the space domain. Nevertheless, we are able to prove Lipschitz stability results for inverse source problems for such operators, with locally distributed measurements in arbitrary space dimension. For this purpose, we follow a mixed strategy which combines the appraoch du…
We consider a linear Korteweg-de Vries equation on a bounded domain with a left Dirichlet boundary control.The controllability to the trajectories of such a system was proved in the last decade by using Carleman estimates.Here, we go a step further by establishing the exact controllability in a space of analytic functions with the aid of the flatness approach.
The aim of this work is to consider the controllability problem of the linear system associated to Korteweg-de Vries Burgers equation posed in the whole real line. We obtain a sort of exact controllability for solutions in $L^2_{loc}(\R^2)$ by deriving an internal observability inequality and a Global Carlemann estimate. Following the ideas contained in \cite{rosier2000}, the problem is reduced to prove an approximate theorem.
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。
We consider a parabolic problem with degeneracy in the interior of the spatial domain and Neumann boundary conditions. In particular, we will focus on the well-posedness of the problem and on Carleman estimates for the associated adjoint problem. The novelty of the present paper is that for the first time it is considered a problem with an interior degeneracy and Neumann boundary conditions so that no previous result can be adapted to this situation. As a consequence new observability inequalities are established.
We consider the heat equation with dynamic bounary conditions involving gradient terms in a bounded domain. In this paper we study the cost of approximate controllability for this equation. Combining new developed Carleman estimates and some optimization techniques, we obtain explicit bounds of the minimal norm control. We consider the linear and the semilinear cases.
By mean of generalized Fourier series and Parseval's equality in weighted $L^{2}$--spaces, we derive a sharp energy estimate for the wave equation in a bounded interval with a moving endpoint. Then, we show the observability, in a sharp time, at each of the endpoints of the interval. The observability constants are explicitly given. Using the Hilbert Uniqueness Method we deduce the exact boundary controllability of the wave equation.
In this paper, we deal with the null controllability of a population dynamics model with an interior degenerate diffusion. To this end, we proved first a new Carleman estimate for the full adjoint system and afterwards we deduce a suitable observability inequality which will be needed to establish the existence of a control acting on a subset of the space which lead the population to extinction in a finite time.
Null controllability of n-dimensional parabolic equations degenerated on partial boundary
本文针对部分边界退化且边界非光滑的高维抛物型方程，通过构造特殊权函数并转化退化区域积分项，建立了可用的Carleman估计，从而证明了相应方程的零能控性，推广了已有结果并给出新例子。
Local null controllability of the complete N-dimensional Ladyzhenskaya-Boussinesq model
研究完全Ladyzhenskaya-Boussinesq系统的局部和大时间零可控性，采用Carleman估计和Liusternik反向映射定理，处理速度和温度方程的非线性项。
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。
Change of regularity in controllability and observability of systems of wave equations
本文构造了不同 Dirichlet 边界条件下波动方程组的解，证明了各正则性水平的边界能观性与精确能控性等价，核心工具为基于微局部分析的 Neumann 迹时间导数椭圆性。
We study the wave equation in an interval with two linearly moving endpoints. We give the exact solution by a series formula, then we show that the energy of the solution decay at the rate $1/t$. We also establish observability results, at one or two endpoints, in a sharp time. Moreover, using the Hilbert uniqueness method, we derive exact boundary controllability results.
Controllability and observability are important properties of a distributed paramater systems.The equivalence between the notion of exact observability and exact controllability holds in general. In this work, we define a new notion of controllability say weak which is related to some weak observability inequality and we give the equivalence between.
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 consider a $2\times 2$ system of parabolic equations with first and zeroth coupling and establish a Carleman estimate by extra data of only one component without data of initial values. Then we apply the Carleman estimate to inverse problems of determining some or all of the coefficients by observations in an arbitrary subdomain over a time interval of only one component and data of two components at a fixed positive time $θ$ over the whole spatial domain. The main results are Lipschitz stability estimates for the inverse problems. For the Lipschitz stability, we have to assume some non-deg…
In this paper we consider the null controllability for a population model depending on time, on space and on age. Moreover, the diffusion coefficient degenerate at the boundary of the space domain. The novelty of this paper is that for the first time we consider the presence of a memory term, which makes the computations more difficult. However, under a suitable condition on the kernel we deduce a null controllability result for the original problem via new Carleman estimates for the adjoint problem associated to a suitable nonhomogeneous parabolic equation.
We consider the observability problem for non-autonomous evolution systems (i.e., the operators governing the system depend on time). We introduce an averaged Hautus condition and prove that for skew-adjoint operators it characterizes exact observability. Next, we extend this to more general class of operators under a growth condition on the associated evolution family. We give an application to the Schrödinger equation with time dependent potential and the damped wave equation with a time dependent damping coefficient.
Observability and controllability for the Schrödinger equation on polyhedra
本文证明凸多面体上具有H^s(s>d/2)正则性初值的薛定谔方程，在边界奇异集任意非空开邻域为控制区域时可观测且可控，方法基于预解估计与HUM论证。
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 establish the theory on semiglobal classical solution to first order quasilinear hyperbolic systems with a kind of nonlocal boundary conditions, and based on this, the corresponding exact boundary controllability and observability are obtained by a constructive method. Moreover, with the linearized Saint-Venant system and the 1-D linear wave equation as examples, we show that the number of both boundary controls and boundary observations can not be reduced, and consequently, we conclude that the exact boundary controllability for a hyperbolic system in a network with loop can …
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。
In this paper we consider a cascade system in non divergence form which models the interaction between two different species, the first one can be seen as a predator and the other as a prey. Both of them depend on time, on age and on space. Moreover, the diffusion coefficients degenerate at the boundary of domain. We study, in particular, null controllability of the system via the observability inequality for the non homogeneous adjoint problem, which is deduced by Carleman estimates.
We obtain a novel interior control result for wave equations on time dependent domains. This is done by deriving a suitable Carleman estimate and proving the corresponding observability inequality. We consider the wave equation with time dependent lower order coefficients, without any time analyticity assumptions. Moreover, we obtain improved control regions when compared with standard Carleman methods.
Controllability of a Fluid-Structure Interaction System Governed by the Heat and Damped Beam Equations
本文研究热方程与阻尼梁方程耦合的二维流固耦合系统的零可控性与可观测性，通过证明全局Carleman估计并明确阻尼依赖性，以及建立梁方程在不同阻尼参数下的不等式，为系统控制提供了理论基础。
We prove null controllability for linear and semilinear heat equations with dynamic boundary conditions of surface diffusion type. The results are based on a new Carleman estimate for this type of boundary conditions.
We prove the exact controllability of linear KP-I equation if the control input is added on a vertical domain. More generally, we have obtained the least dispersion needed to insure observability for fractional linear KP I equation.
This paper is concerned with the control properties of the Korteweg-de Vries (KdV) equation posed on a bounded interval with a distributed control. When the control region is an arbitrary open subdomain, we prove the null controllability of the KdV equation by means of a new Carleman inequality. As a consequence, we obtain a regional controllability result, the state function being controlled on the left part of the complement of the control region. Finally, when the control region is a neighborhood of the right endpoint, an exact controllability result in a weighted L2 space is also establish…
We are mainly interested in extending the known results on ob-servability inequalities and stabilization for the Schr{ö}dinger equation to the magnetic Schr{ö}dinger equation. That is in presence of a magnetic potential. We establish observability inequalities and exponential stabilization by extending the usual multiplier method, under the same geometric condition to that needed for the Schr{ö}dinger equation. We also prove, with the help of elliptic Carleman inequalities, logarithmic stabilization results through a resolvent estimate. Although the approach is classical, these results on loga…
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.
Stackelberg-Nash strategy for the null controllability of semilinear degenerate equations in non-cylindrical domains
本研究针对非柱形域内一维半线性退化抛物方程的零能控问题，采用Stackelberg-Nash策略，结合退化非自治系统的Carleman不等式与Liusternik逆函数定理，证明了该系统的局部零能控性。
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