Farid Ammar Khodja, Yacine Mokhtari
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
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 …
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.
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 $-approximate front tracking solutions to the mixed initial-boun…
In this paper we establish the theory on semiglobal classical solution to first order quasilinear hyperbolic systems with a kind of nonlocal boundary conditions, and based on this, the corresponding exact boundary controllability and observability are obtained by a constructive method. Moreover, with the linearized Saint-Venant system and the 1-D linear wave equation as examples, we show that the number of both boundary controls and boundary observations can not be reduced, and consequently, we conclude that the exact boundary controllability for a hyperbolic system in a network with loop can …
The general theory on exact boundary controllability for general first order quasilinear hyperbolic systems requires that the characteristic speeds of system do not vanish. This paper deals with exact boundary controllability, when this is not the case. Some important models are also shown as applications of the main result. The strategy uses the return method, which allows in certain situations to recover non zero characteristic speeds.
In this paper, we deal with the boundary controllability and boundary stabilizability of the 1D wave equation in non-cylindrical domain of the form ($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.
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.
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。
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。
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。
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.
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。
In this paper, we study the exact boundary controllability of the linear Biharmonic Schrödinger equation $i\partial_ty=-\partial_x^4y+ γ\partial_x^2y$ on a bounded domain with hinged boundary conditions and boundary control acts on the second spatial derivative at the {left} endpoint, where the parameter $γ<0$. We prove that this system is exactly controllable in time $T>0$, if and only if, the parameter $γ$ does not belong to a critical countable set of negative real numbers. The analysis in this work is based on spectral analysis together with the nonharmonic Fourier series method.
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。
We provide global and semi-global controllability results for hyperbolic conservation laws on a bounded domain, with a general (not necessarily convex)flux and a time-dependent source term acting as a control. The results are achieved for, possibly critical, both continuously differentiable states and BV states. The proofs are based on a combination of the return method and on the analysis of the Riccati equaiton for the space derivative of the solution.
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。
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。
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。
On the controllability of laminated beams with Venttsel-type boundary conditions
针对受 Venttsel 型边界条件约束的 Timoshenko 三梁层合系统，提出仅在边界施加三个控制的方案，通过伴随系统可观测性不等式与 HUM 对偶方法证明边界能控性。
Exact controllability is proven on a graph with cycle. The controls can be a mix of controls applied at the boundary and interior vertices. The method of proof first uses a dynamical argument to prove shape controllability and velocity controllability, thereby solving their associated moment problems. This enables one to solve the moment problem associated to exact controllability. In the case of a single control, either boundary or interior, it is shown that exact controllability fails.
In this work, we are interested in the small time global null controllability for the viscous Burgers' equation y_t - y_xx + y y_x = u(t) on the line segment [0,1]. The second-hand side is a scalar control playing a role similar to that of a pressure. We set y(t,1) = 0 and restrict ourselves to using only two controls (namely the interior one u(t) and the boundary one y(t,0)). In this setting, we show that small time global null controllability still holds by taking advantage of both hyperbolic and parabolic behaviors of our system. We use the Cole-Hopf transform and Fourier series to derive p…
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。
In this paper we treat controllability properties for the linear Kuramoto-Sivashinsky equation on a network with two types of boundary conditions. More precisely, the equation is considered on a star-shaped tree with Dirichlet and Neumann boundary conditions. By using the moment theory we can derive null-controllability properties with boundary controls acting on the external vertices of the tree. In particular, the controllability holds if the anti-diffusion parameter of the involved equation does not belong to a critical countable set of real numbers. We point out that the critical set for w…
In this proceeding we expose a particular case of a recent result obtained by the authors regarding the incompressible Navier-Stokes equations in a smooth bounded and simply connected bounded domain, either in 2D or in 3D, with a Navier slip-with-friction boundary condition except on a part of the boundary. This under-determination encodes that one has control over the remaining part of the boundary. We prove that for any initial data, for any positive time, there exists a weak Leray solution which vanishes at this given time.
In the paper, the problems of controllability and approximate controllability are studied for the control system $w_t=\frac{1}ρ\left(kw_x\right)_x+γw$, $\left.\left(\sqrt{\frac{k}ρ}w_x\right)\right|_{x=0}=u$, $x>0$, $t\in(0,T)$, where $u$ is a control, $u\in L^\infty(0,T)$. It is proved that each initial state of the control system is approximately controllable to any target state in a given time $T>0$. To obtain this result, the transformation operator generated by the equation data $ρ$, $k$, $γ$ is applied. The results are illustrated by examples.
In this paper, we prove a logarithmic convexity that reflects an observability estimate at a single point of time for 1-D heat equation with dynamic boundary conditions. Consequently, we establish the impulse approximate controllability for the impulsive heat equation with dynamic boundary conditions. Moreover, we obtain an explicit upper bound of the cost of impulse control. At the end, we give a constructive algorithm for computing the impulsive control of minimal $L^2$-norm. We also present some numerical tests to validate the theoretical results and show the efficiency of the designed algo…
Boundary approximate controllability of some linear parabolic systems
本文研究两类线性抛物型系统的边界近似可控性，提出充分条件并证明一维单控制下的必要性，给出矩形域刻画，并构造分布可控但边界不可控的级联系统。
Boundary Effects on the Controllability of Coupled KdV Systems
本文研究有界区间上耦合KdV系统的精确边界可控性，通过线性化系统的对偶方法、Paley-Wiener谱分析及压缩映射定理，在最多四个控制输入下建立了完整系统的局部可控性。
In this paper, we deal with the boundary controllability of a one-dimensional degenerate and singular wave equation with degeneracy and singularity occurring at the boundary of the spatial domain. Exact boundary controllability is proved in the range of subcritical/critical potentials and for sufficiently large time, through a boundary controller acting away from the degenerate/singular point. By duality argument, we reduce the problem to an observability estimate for the corresponding adjoint system, which is proved by means of the multiplier method and a new special Hardy-type inequality.
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。
In this paper, we are concerned with the regional boundary controllability of the Riemann-Liouville time fractional diffusion systems of order $α\in (0,1]$. The characterizations of strategic actuators are established when the systems studied are regionally boundary controllable. The determination of control to achieve regional boundary controllability with minimum energy is explored. We also show a connection between the regional internal controllability and regional boundary controllability. Several useful results for the optimal control from an implementation point of view are presented in …
In this paper we consider a viscoelastic plate (linear viscoelasticity of the Maxwell-Boltzmann type) and we compare its controllability properties with the (known) controllability of a purely elastic plate (the control acts on the boundary displacement or bending). By combining operator and moment methods, we prove that the viscoelastic plate inherits the controllability properties of the purely elastic plate.
In this paper, we consider the approximate controllability of partial differential equations with time derivatives of non-integer order via boundary control. We first show the unique existence of the solution under smooth boundary conditions using the eigenfunction expansion. Next we also study the dual system and apply their results to prove (i) the regularity of the solution with non-smooth boundary conditions and (ii) approximate controllability.
In this paper we will generalize the Kalman rank condition for the null controllability to $n$-coupled linear degenerate parabolic systems with constant coefficients, diagonalizable diffusion matrix, and $m$-controls. For that we prove a global Carleman estimate of the solution of a scalar $2n$-order equation then we infer from it an observability inequality for the corresponding adjoint system, and thus the null controllability.
Controllability of diffusive Lotka-Volterra strongly competitive systems under boundary constrained controls
该研究针对一维带Dirichlet边界约束的扩散型Lotka-Volterra强竞争系统，证明可引导至弱竞争物种灭绝稳态，而种间竞争失衡时无法实现强竞争物种消失的可控性，还构造屏障解解释障碍并通过数值模拟验证结果。
One proves that the moving interface of a two-phase Stefan problem on $\ooo\subset\rr^d$, $d=1,2,3,$ is controllable at the end time $T$ by a Neumann boundary controller $u$. The phase-transition region is a mushy region $\{σ^u_t;\ 0\le t\le T\}$ of a modified Stefan problem and the main result amounts to saying that, for each Lebesque measurable set $\ooo^*$ with positive measure, there is $u\in L^2((0,T)\times\pp\ooo)$ such that $\ooo^*\subsetσ^u_T.$ To this aim, one uses an optimal control approach combined with Carleman's inequality and the Kakutani fixed point theorem.
In this paper we study the boundary controllability for a system of two coupled degenerate/singular parabolic equations with a control acting on only one equation. We analyze both approximate and null boundary controllability properties. Besides, we provide an estimate on the null-control cost. The proofs are based on the use of the moment method by Fattorini and Russell together with some results on biorthogonal families.
This paper is devoted to study boundary controllability of the Korteweg-de Vries equation posed on a finite interval, in which, because of the third-order character of the equation, three boundary conditions are required to secure the well-posedness of the system. We consider the cases where one, two, or all three of those boundary data are employed as boundary control inputs. The system is first linearized around the origin and the corresponding linear system is shown to be exactly boundary controllable if using two or three boundary control inputs. In the case where only one control input is…
Exact boundary controllability and stabilizability of a degenerated Timoshenko beam
本文研究退化蒂莫申科梁的边界可控性与稳定性，通过能量乘子技术和HUM方法，证明了非退化端的反馈控制可实现指数稳定。
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 …
Minimal time for the exact controllability of one-dimensional first-order linear hyperbolic systems by one-sided boundary controls
本研究聚焦单侧边界控制下的一维一阶线性双曲系统精确能控最小时间问题，通过引入典范$UL$-分解与紧致性-唯一性方法，推导出关于系统所有耦合参数的显式易算最小时间公式。
This paper focuses on boundary approximate controllability under positivity constraints of a wide range of infinite-dimensional control systems. We develop frequency domain controllability criteria. Firstly, we derive a controllability result under positivity constraints on the control for such systems. Then, and more importantly, we provide a necessary and sufficient condition for controllability under positivity constraints on the control and the state. The obtained results are applied to the controllability of transportation and heat conduction networks. In particular, provided that the und…
Null controllability of two kinds of coupled parabolic systems with switching control
本文研究含退化与非退化方程的两类切换控制耦合抛物系统，借助时间可测子集上的能观性不等式、HUM 与谱不等式证明其零能控性。
In this paper we study exact boundary controllability for a linear wave equation with strong and weak interior degeneration of the coefficient in the principle part of the elliptic operator. The objective is to provide a well-posedness analysis of the corresponding system and derive conditions for its controllability through boundary actions. Passing to a relaxed version of the original problem, we discuss existence and uniqueness of solutions, and using the HUM method we derive conditions on the rate of degeneracy for both exact boundary controllability and the lack thereof.
In this paper, we study the small-time global null controllability of the generalized Burgers' equations $y_t + γ|y|^{γ-1}y_x-y_{xx}=u(t)$ on the segment $[0,1]$. The scalar control $u(t)$ is uniform in space and plays a role similar to the pressure in higher dimension. We set a right Dirichlet boundary condition $y(t,1)=0$, and allow a left boundary control $y(t,0)=v(t)$. Under the assumption $γ>3/2$ we prove that the system is small-time global null controllable. Our proof relies on the return method and a careful analysis of the shape and dissipation of a boundary layer.
We investigate the controllability of quantum electrons trapped in a two-dimensional device, typically a MOS field-effect transistor. The problem is modeled by the Schrödinger equation in a bounded domain coupled to the Poisson equation for the electrical potential. The controller acts on the system through the boundary condition on the potential, on a part of the boundary modeling the gate. We prove that, generically with respect to the shape of the domain and boundary conditions on the gate, the device is controllable. We also consider control properties of a more realistic nonlinear version…
An observability estimate for the wave equation and applications to the Neumann boundary controllability for semi-linear wave equations
该研究给出带势一维波动方程的边界能观性结果，借助Schauder不动点论证推导满足最优增长条件的半线性波动方程Neumann边界控制的存在性，构造极小化序列并通过数值实验验证，拓展了前人相关工作。
Boundary null-controllability for the beam equation with classical structural damping
研究了具有经典结构阻尼的梁方程的边界零控制性，证明了对于ρ≤2的所有情况和ρ>2的几乎所有情况的可控性，以及Neumann控制的类似结果。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。