Veli Shakhmurov
The paper is devoted to the exact controllability of a system of coupled abstract wave equations when the control is exerted on a part of the boundary by means of one control. We give a Kalman type condition and give a description of the attainable set.
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Boundary controllability for degenerate/singular hyperbolic equations in nondivergence form with drift
该研究针对带漂移项的非散度型退化/奇异波动方程的零能控问题，借助能量方法与边界能观性，给出了局域于非退化边界点的控制下边界能控性的相关条件。
We consider a degenerate wave equation with drift in presence of a leading operator which is not in divergence form. We provide some conditions for the boundary controllability of the associated Cauchy problem.
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 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 consider the exact boundary controllability and the exact boundary synchronization (by groups) for a coupled system of wave equations with coupled Robin boundary controls. Owing to the difficulty coming from the lack of regularity of the solution, we confront a bigger challenge than that in the case with Dirichlet or Neumann boundary controls. In order to overcome this difficulty, we use the regularity results of solutions to the mixed problem with Neumann boundary conditions by Lasiecka and Triggiani to get the regularity of solutions to the mixed problem with coupled Robin …
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。
We study the boundary exact controllability for the quasilinear wave equation in the higher-dimensional case. Our main tool is the geometric analysis. We derive the existence of long time solutions near an equilibrium, prove the locally exact controllability around the equilibrium under some checkable geometrical conditions. We then establish the globally exact controllability in such a way that the state of the quasilinear wave equation moves from an equilibrium in one location to an equilibrium in another location under some geometrical condition. The Dirichlet action and the Neumann action …
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 introduce some new concepts for second-order hyperbolic equations: two-point boundary value problem, global exact controllability and exact controllability. For several kinds of important linear and nonlinear wave equations arising from physics and geometry, we prove the existence of smooth solutions of the two-point boundary value problems and show the global exact controllability of these wave equations. In particular, we investigate the two-point boundary value problem for one-dimensional wave equation defined on a closed curve and prove the existence of smooth solution whi…
We propose a model using the Korteweg-de Vries $(KdV)$ equation on a finite star-shaped network. We first prove the well-posedness of the system and give some regularity results. Then we prove that the energy of the solutions of the dissipative system decays exponentially to zero when the time tends to infinity. Lastly we show an exact boundary controllability result.
In this paper, we consider the Stokes equations in a two-dimen- sional channel with periodic conditions in the direction of the channel. We establish null controllability of this system using a boundary control which acts on the normal component of the velocity only. We show null controllability of the system, subject to a constraint of zero average, by proving an observability inequality with the help of a Muntz-Szasz Theorem.
This paper is devoted to prove the exact controllability of a system of N one-dimensional coupled wave equations when the control is exerted on a part of the boundary by means of one control. We consider the case where the coupling matrix A has distinct eigenvalues. We give a Kalman condition (necessary and sufficient) and give a description, non-optimal in general, of the attainable set.
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…
On boundary controllability for the higher order nonlinear Schrödinger equation
研究有界区间上高阶非线性 Schrödinger 方程的终端超定边界控制问题，以空间导数边界条件为控制，证明小输入数据时解全局存在。
The problem of the exact bounded control of oscillations of the two-dimensional wave equation is considered. Control force is applied to the boundary of the membrane, which is located in a domain on a plane. The goal of the control is to drive the system to rest in a finite time.
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.
We study the boundary controllability problem for a multilayer Rao-Nakra sandwich beam. This beam model consists of a Rayleigh beam coupled with a number of wave equations. We consider all combinations of clamped and hinged boundary conditions with the control applied to either the moment or the rotation angle at an end of the beam. We prove that exact controllability holds provided the damping parameter is sufficiently small. In the undamped case, exact controllability holds without any restriction on the parameters in the system. In each case, optimal control time is obtained in the space of…
In this paper, we investigate the two-point boundary value problems for linear wave equation defined on a circle and prove that the equation possesses the exact controllability. We also investigate the two-point boundary value problems for the wave equation defined on a strip with Dirichlet or Neumann boundary conditions and show that the equation still possesses the exact controllability. see arXiv:0910.5782v1
We address the problem of controllability of the MHD system in a rectangular domain with a control prescribed on the side boundary. We identify a necessary and sufficient condition on the data to be null controllable, i.e., can be driven to the zero state. We also show that the validity of this condition allows the states to be stirred to each other. If the condition is not satisfied, one can move from one state to another with the help of a simple shear external magnetic force.
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。
We analyze controllability properties for the one-dimensional heat equation with singular inverse-square potential $$ u_t-u_{xx}-\fracμ{x^2}u=0,\;\;\; (x,t)\in(0,1)\times(0,T).$$ For any $μ<1/4$, we prove that the equation is null controllable through a boundary control $f\in H^1(0,T)$ acting at the singularity point $x=0$. This result is obtained employing the moment method by Fattorini and Russell.
Let $Ω\subset\mathbb R^N$ be a bounded open set with Lipschitz continuous boundary $Γ$. Let $γ>0$, $δ\ge 0$ be real numbers and $β$ a nonnegative measurable function in $L^\infty(Γ)$. Using some suitable Carleman estimates, we show that the linear heat equation $\partial_tu - γΔu = 0$ in $Ω\times(0,T)$ with the non-homogeneous general dynamic boundary conditions $\partial_tu_Γ -δΔ_Γu_Γ+ γ\partial_νu + βu_Γ = g$ on $Γ\times(0,T)$ is always null controllable from the boundary for every $T>0$ and initial data $(u_0,u_{Γ,0})\in L^2(Ω)\times L^2(Γ)$.
The paper deals with the controllability of a degenerate beam equation. In particular, we assume that the left end of the beam is fixed, while a suitable control $f$ acts on the right end of it. As a first step we prove the existence of a solution for the homogeneous problem, then we prove some estimates on its energy. Thanks to them we prove an observability inequality and, using the notion of solution by transposition, we prove that the initial problem is null controllable.
We obtain sharp convergence rates, using Dirichlet correctors, for solutions of wave equations in a bounded domain with rapidly oscillating periodic coefficients. The results are used to prove the exact boundary controllability that is uniform in $ε$ - the scale of the microstructure, for the projection of solutions to the subspace generated by the eigenfunctions with eigenvalues less that $Cε^{-1/2}$.
The aim of this work is to show the local null controllability of a fluid-solid interaction system by using a distributed control located in the fluid. The fluid is modeled by the incompressible Navier-Stokes system with Navier slip boundary conditions and the rigid body is governed by the Newton laws. Our main result yields that we can drive the velocities of the fluid and of the structure to 0 and we can control exactly the position of the rigid body. One important ingredient consists in a new Carleman estimate for a linear fluid-rigid body system with Navier boundary conditions. This work i…
In this paper we prove that the Ball-Marsden-Slemrod controllability obstruction also holds for nonlinear equations, with integrable bilinear controls. We first show an abstract result and then we apply it to nonlinear wave equations. The first application to the Sine-Gordon equation directly follows from the abstract result, and the second application concerns the cubic wave/Klein-Gordon equation and needs some additional work.
We consider the 1D nonlinear Schrödinger equation with bilinear control. In the case of Neumann boundary conditions, local exact controllability of this equation near the ground state has been proved by Beauchard and Laurent in arXiv:1001.3288. In this paper, we study the case of Dirichlet boundary conditions. To establish the controllability of the linearised equation, we use a bilinear control acting through four directions: three Fourier modes and one generic direction. The Fourier modes are appropriately chosen so that they satisfy a saturation property. These modes allow to control approx…
We consider nxn hyperbolic systems of balance laws in one-space dimension under the assumption that all negative (resp. positive) characteristics are linearly degenerate. We prove the local exact one-sided boundary null controllability of entropy solutions to this class of systems, which generalizes the corresponding results from the case without source terms to that with source terms. In order to apply the strategy used for conservation law, we essentially modify the constructive method by introducing two different kinds of approximate solutions to system in the forward sense and to the syste…
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.
In this paper, we recover the boundary null controllability for the degenerate heat equation by analyzing the asymptotic behavior of an eligible family of state-control pairs $((u_{\varepsilon}, h_{\varepsilon}))_{\varepsilon >0}$ solving corresponding singularly perturbed internal null controllability problems. As in other situations studied in the literature, our approach relies on Carleman estimates and meticulous weak convergence results. However, for the degenerate parabolic case, some specific trace operator inequalities must be obtained, in order to justify correctly the passage to the …
Small-time global approximate controllability for incompressible MHD with coupled Navier slip boundary conditions
本文研究二维或三维有界区域中不可压缩MHD流动的小时间全局近似可控性，通过分析耦合Navier摩擦滑移边界条件下边界层的渐近展开获得耗散性质，并证明在平面单连通区域中额外的类压力项不存在。
In this paper, we study the controllability of a Schrödinger equation with mixed boundary conditions on disjoint subsets of the boundary: dynamic boundary condition of Wentzell type, and Dirichlet boundary condition. The main result of this article is given by new Carleman estimates for the associated adjoint system, where the weight function is constructed specially adapted to the geometry of the domain. Using these estimates, we prove the exact controllability of the system with a boundary control acting only in the part of the boundary where the Dirichlet condition is imposed. Also, we obta…
In this paper, we study the local exact boundary controllability of entropy solutions to a class linearly degenerate hyperbolic systems of conservation laws with constant multiplicity. The authors prove the two-sided boundary controllability, one-sided boundary controllability and two-sided controllability with less controls, by applying the strategy used originally for classical solutions with essential modifications. Our constructive method is based on the well-posedness of semi-global solutions constructed by the limit of $ \e $-approximate front tracking solutions to the mixed initial-boun…
In this paper we discuss the controllability of two-point boundary value problem (TBVP) for one-dimensional wave equation. Some new concepts are introduced: TBVP input control problem, minimum-input solution (MS) and pre-minimum-input solution (PMS). We set the metric in $L^1$ and $L^2$ spaces on a closed set, and control the input to reach its minimum. And we mainly discuss the property of input, the existence and uniqueness of MS and PMS for $L^1$ and $L^2$ metric respectively. The minimum inputs lie on a strip in $L^1$ and PMS for $L^1$ and $L^2$ always exists. Furthermore, to construct PMS…
This article is devoted to the analysis of control properties for a heat equation with singular potential $μ/δ^2$, defined on a bounded $C^2$ domain $Ω\subset\mathbb{R}^N$, where $δ$ is the distance to the boundary function. More precisely, we show that for any $μ\leq 1/4$ the system is exactly null controllable using a distributed control located in any open subset of $Ω$, while for $μ>1/4$ there is no way of preventing the solutions of the equation from blowing-up. The result is obtained applying a new Carleman estimate.
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 …
The general theory on exact boundary controllability for general first order quasilinear hyperbolic systems requires that the characteristic speeds of system do not vanish. This paper deals with exact boundary controllability, when this is not the case. Some important models are also shown as applications of the main result. The strategy uses the return method, which allows in certain situations to recover non zero characteristic speeds.
In this paper, we study the indirect boundary stability and exact controllability of a one-dimensional Timoshenko system. In the first part of the paper, we consider the Timoshenko system with only one boundary fractional damping. We first show that the system is strongly stable but not uniformly stable. Hence, we look for a polynomial decay rate for smooth initial data. Using frequency domain arguments combined with the multiplier method, we prove that the energy decay rate depends on coefficients appearing in the PDE and on the order of the fractional damping. Moreover, under the equal speed…
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。
In this paper, we deal with the boundary controllability and boundary stabilizability of the 1D wave equation in non-cylindrical domain of the form ($α(t)<x<β(t)$). By using the characteristics method, we prove under a natural assumption on the boundary functions that the 1D wave equation is controllable and stabilizable from one side of the boundary. Furthermore, the control function and the decay rate of solution are given explicitly
Boundary null controllability of the heat equation with Wentzell boundary condition and Dirichlet control
针对带Wentzell边界条件与Dirichlet控制的线性热方程，本文在边界控制系统框架下证明适定性，并借助Sturm–Liouville谱分析和矩方法建立任意正时刻的零能控性，且用惩罚HUM逼近最小能量控制以验证理论结果。
The main purpose of this article is to prove a logarithmic convexity estimate for the solution of a linear heat equation subject to dynamic boundary conditions in a bounded convex domain. As an application, we prove the impulsive null approximate controllability for an impulsive heat equation with dynamic boundary conditions.
This paper is devoted to study exact controllability of two one-dimensional coupled wave equations with first-order coupling terms with coefficients depending on space and time. We give a necessary and sufficient condition for both exact controllability in high frequency in the general case and the unique continuation in the cascade case.
An explicit saturating set consisting of eigenfunctions of Stokes operator in general 3D Cylinders is proposed. The existence of saturating sets implies the approximate controllability for Navier--Stokes equations in $\rm 3D$ Cylinders under Lions boundary conditions.
In this paper, we are concerned with the boundary controllability of heat equation with dynamic boundary conditions. More precisely, we prove that the equation is null controllable at any positive time by means of a boundary control supported on an arbitrary subboundary. The proof of the main result combines a new boundary Carleman estimate and some regularity estimates for the adjoint system, with an explicit dependence with respect to the final time. This technique allows us to overcome a new difficulty that arises when absorbing a normal derivative term.
We consider the linear Zakharov-Kuznetsov equation on a rectangle with a left Dirich-let boundary control. Using the flatness approach, we prove the null controllability of this equation and provide a space of analytic reachable states.
Controllability and Inverse Problems for Hyperbolic and Dispersive Equations with Dynamic Boundary Conditions
本综述梳理带动态边界条件的双曲与色散方程可控性及反问题的研究进展，阐明Carleman估计在建立可控性与推导反问题稳定性估计中的应用，指出相关挑战并展望未来方向。
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。
In this work, we investigate the small-time global exact controllability of the Navier-Stokes equation, both towards the null equilibrium state and towards weak trajectories. We consider a viscous incompressible fluid evolving within a smooth bounded domain, either in 2D or in 3D. The controls are only located on a small part of the boundary, intersecting all its connected components. On the remaining parts of the boundary, the fluid obeys a Navier slip-with-friction boundary condition. Even though viscous boundary layers appear near these uncontrolled boundaries, we prove that small-time glob…
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 study the exact boundary controllability of the linear fourth-order Schrödinger equation, with variable physical parameters and clamped boundary conditions on a bounded interval. The control acts on the first spatial derivative at the left endpoint. We prove that this control system is exactly controllable at any time $T>0$. The proofs are based on a detailed spectral analysis and on the use of nonharmonic Fourier series.
This paper deals with the hierarchical control of the anisotropic heat equation with dynamic boundary conditions and drift terms. We use the Stackelberg-Nash strategy with one leader and two followers. To each fixed leader, we find a Nash equilibrium corresponding to a bi-objective optimal control problem for the followers. Then, by some new Carleman estimates, we prove a null controllability result.
Boundary null controllability of degenerate wave equation as the limit of internal controllability
通过分析内部零能控性问题的状态-控制对族，建立显式依赖于ε的观测不等式，证明退化波动方程边界零能控性。
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。
We prove controllability results for abstract systems of weakly coupled $N$ evolution equations in cascade by a reduced number of boundary or locally distributed controls ranging from a single up to $N-1$ controls. We give applications to cascade coupled systems of $N$ multi-dimensional-hyperbolic, parabolic and diffusive equations.
Local null controllability of a cubic Ginzburg-Landau equation with dynamic boundary conditions
本文研究了具有动态边界条件的立方Ginzburg-Landau方程，通过单一控制、线性化、对偶方法、Carleman估计及反函数定理，证明了该非线性系统的局部零能控性。
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.
Global small-time approximate null and Lagrangian controllability of the viscous non-resistive MHD system in a $3D$ domain with Navier type boundary conditions
本文研究三维有界区域内无磁扩散粘性 MHD 系统，借助带快速时间尺度的控制与多尺度渐近展开，在光滑解中证明全局短时间近似零能控性和拉格朗日能控性。
Exact boundary controllability of the 3D incompressible ideal MHD system
本文证明了三维不可压缩理想MHD系统在部分边界上给定法向速度与磁场法向分量时的精确边界可控性，并首次给出了不使用Elsasser变量的局部适定性证明，同时简化了二维可控性证明。
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。
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