Igor Romanov, Alexey Shamaev
We will consider exact controllability of the distributed system governed by the wave equation with memory. It will be proved that this mechanical system can be driven to rest in finite time, the absolute value of the distributed control function being bounded. In this case the memory kernel is a linear combination of exponentials.
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The paper deals with controllability problem for a distributed system governed by the two-dimensional Gurtin-Pipkin equation. We consider a system with compactly supported distributed control and show that if the memory kernel is a twice continuously differentiable function, such that its Laplace transformation has at least one non-zero root, then the system cannot be driven to the equilibrium in a finite time.
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 study the local controllability properties of 2D and 3D bio-mimetic swimmers employing the change of their geometric shape to propel themselves in an incompressible fluid described by Navier-Stokes equations. It is assumed that swimmers' bodies consist of finitely many parts, identified with the fluid they occupy, that are subsequently linked by the rotational and elastic internal forces. These forces are explicitly described and serve as the means to affect the geometric configuration of swimmers' bodies. Similar models were previously investigated in [6]-[13].
Controllability for a non-local formulation of surface gravity waves
针对二维不可压缩无黏流体的非局部晃荡问题，论文通过建立唯一延拓结果，证明了内部局部源控制可实现系统的近似可控性。
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…
In this note, we give an elementary proof of the lack of null controllability for the heat equation on the half line by employing the machinery inherited by the unified transform, known also as the Fokas method. This approach also extends in a uniform way to higher dimensions and different initial-boundary value problems governed by the heat equation, suggesting a novel methodology for studying problems related to controllability.
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…
On the controllability of the Navier-Stokes equation in a rectangle, with a little help of a distributed phantom force
本文概述了作者关于矩形区域中Navier-Stokes方程能控性的研究结果，借助分布虚拟力给出核心思想的粗略证明，并引导读者参阅原始论文获取完整技术细节。
Bilinear control to trajectories of 1D degenerate parabolic equations in moving domains
本文研究了在时间演化域中退化抛物方程的局部可控性，通过双线性控制实现正轨迹的精确控制，并结合局部反演方法和特定估计进行分析。
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 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。
We consider the 2D incompressible Navier-Stokes equation in a rectangle with the usual no-slip boundary condition prescribed on the upper and lower boundaries. We prove that for any positive time, for any finite energy initial data, there exist controls on the left and right boundaries and a distributed force, which can be chosen arbitrarily small in any Sobolev norm in space, such that the corresponding solution is at rest at the given final time. Our work improves earlier results where the distributed force is small only in a negative Sobolev space. It is a further step towards an answer to …
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.
First, let $u_{g}$ be the unique solution of an elliptic variational inequality with source term $g$. We establish, in the general case, the error estimate between $u_{3}(μ)=μu_{g_{1}}+ (1-μ)u_{g_{2}}$ %(the convex combination of two solutions) and $u_{4}(μ)=u_{μg_{1}+ (1-μ) g_{2}}$ %(the solution corresponding to the convex combination of two data) for $μ\in [0, 1]$. Secondly, we consider a family of distributed optimal control problems governed by elliptic variational inequalities over the internal energy $g$ for each positive heat transfer coefficient $h$ given on a part of the boundary of …
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.
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.
We study the global approximate controllability properties of a one dimensional semilinear reaction-diffusion equation governed via the coefficient of the reaction term. It is assumed that both the initial and target states admit no more than finitely many changes of sign. Our goal is to show that any target state $ u^*\in H_0^1 (0,1)$, with as many changes of sign in the same order as the given initial data $ u_0\in H^1_0(0,1)$, can be approximately reached in the $ L^2 (0,1)$-norm at some time $T>0$. Our method employs shifting the points of sign change by making use of a finite sequence of …
In this paper, we study the exact controllability and stabilization of a system of two wave equations coupled by velocities with an internal, local control acting on only one equation. We distinguish two cases. In the first one, when the waves propagate at the same speed: using a frequency domain approach combined with multiplier technique, we prove that the system is exponentially stable when the coupling region satisfies the geometric control condition GCC. Following a result of Haraux ([11]), we establish the main indirect observability inequality. This results leads, by the HUM method, to …
Existence and uniqueness of mild solutions and evolution operators for a class of non-autonomous conformable fractional semi-linear systems and Their Exact Null Controllability
本文研究非自治适形分数阶半线性系统，建立演化算子及温和解的存在唯一性充分条件，并基于温和解分析系统的精确零能控性。
Non null controllability of Stokes equations with memory
本文研究带记忆项的三维Stokes方程的零能控性，构造出使零能控性不成立的初始条件，并证明该否定结果对边界控制和分布控制均成立。
Local exact controllability of the 1D NLS (subject to zero boundary conditions) with distributed control is shown to hold in a $H^1$--neighbourhood of the nonlinear ground state. The Hilbert Uniqueness Method (HUM), due to J.-L. Lions, is applied to the linear control problem that arises by linearization around the ground state. The application of HUM crucially depends on the spectral properties of the linearized NLS operator which are given in detail.
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 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…
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。
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。
The symmetric regularized long wave system (SRLW) is a model for the weakly nonlinear ion acoustic and space-charge waves, which was introduced by C. Seyler and D. Fenstermacher. In this paper, we investigated the wellposedness and controllability properties of the generalized symmetric regularized long wave system (g-SRLW) in different structures (periodic and bounded domains). Firstly, the wellposedness and the exact controllability results for both linear and nonlinear g-SRLW system posed on the one-dimensional torus are obtained under the effect of a distributed moving control. Second, we …
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。
Uniqueness for the Schrödinger equation with an inverse square potential and application to controllability and inverse problems
本文利用将 Schrödinger 方程转化为椭圆方程的技术，在无观测区域几何限制下证明反平方势奇异方程的精确唯一性，并据此建立近似可控性与反源问题唯一性。
Small-time global exact controllability to the trajectories for the viscous Boussinesq system
本文针对二维和三维光滑有界域内带特定边界条件的粘性Boussinesq系统，通过区域延拓、借鉴Navier-Stokes方程能控性策略及线性化加Carleman估计三步法，证明了边界小区域控制下的轨迹小时间全局精确能控性。
The goal of the present article is to study controllability properties of mixed systems of linear parabolic-transport equations, with possibly non-diagonalizable diffusion matrix, on the one-dimensional torus. The equations are coupled by zero or first order coupling terms, with constant coupling matrices, without any structure assumptions on them. The distributed control acts through a constant matrix operator on the system, so that there might be notably less controls than equations, encompassing the case of indirect and simultaneous controllability. More precisely, we prove that in small ti…
In this paper we study the global approximate multiplicative controllability for nonlinear degenerate parabolic Cauchy problems. In particular, we consider a one-dimensional semilinear degenerate reaction-diffusion equation in divergence form governed via the coefficient of the \-reaction term (bilinear or multiplicative control). The above one-dimensional equation is degenerate since the diffusion coefficient is positive on the interior of the spatial domain and vanishes at the boundary points. Furthermore, two different kinds of degenerate diffusion coefficient are distinguished and studied …
The exact distributed controllability of the semilinear wave equation $\partial_{tt}y-Δy + g(y)=f \,1_ω$ posed over multi-dimensional and bounded domains, assuming that $g\in C^1(\mathbb{R})$ satisfies the growth condition $\limsup_{r\to \infty} g(r)/(\vert r\vert \ln^{1/2}\vert r\vert)=0$ has been obtained by Fu, Yong and Zhang in 2007. The proof based on a non constructive Leray-Schauder fixed point theorem makes use of precise estimates of the observability constant for a linearized wave equation. Assuming that $g^\prime$ does not grow faster than $β\ln^{1/2}\vert r\vert$ at infinity for $β…
A Lebeau--Robbiano approach to the controllability of the Boussinesq system with a reduced number of controls
本文采用Lebeau--Robbiano方法，克服Boussinesq系统耦合结构带来的困难，证明其在控制量减少时的局部零可控性，控制量小时间代价符合预期抛物阶。
In this paper local exact controllability to the trajectories for the one-dimensional monodomain equations with the FitzHugh-Nagumo and Rogers-McCulloch ionic models using distributed controls with a moving support is investigated. In a first step a new Carleman inequality for the adjoint of the linearized monodomain equations, under assumptions on the movement of the control region, is presented. It leads to null controllability at any positive time. Subsequently, a local result concerning the exact controllability to the trajectories for the nonlinear monodomain equations is deduced.
In this paper, we consider a nonlinear system of two parabolic equations, with a distributed control in the first equation and an odd coupling term in the second one. We prove that the nonlinear system is small-time locally null-controllable. The main difficulty is that the linearized system is not null-controllable. To overcome this obstacle, we extend in a nonlinear setting the strategy introduced by the first author that consists in constructing odd controls for the linear heat equation. The proof relies on three main steps. First, we obtain from the classical L^2 parabolic Carleman estimat…
Exact controllability to the trajectories of the one-phase Stefan problem
本文针对一维单相Stefan自由边界问题，通过将其转化为耦合PDE-ODE系统的零能控问题，结合新的Carleman不等式与Liusternik-Graves定理，证明了光滑轨迹的局部精确能控性，并附带得到一类带边界记忆项抛物方程的零能控性结果。
For $α\in (0,2)$ we study the null controllability of the parabolic operator $$Pu= u_t - (\vert x \vert ^αu_x)_x\qquad (1<x<1),$$ which degenerates at the interior point $x=0$, for locally distributed controls acting only one side of the origin (that is, on some interval $(a,b)$ with $0<a<b<1$). Our main results guarantees that $P$ is null controllable if and only if it is weakly degenerate, that is, $α\in (0,1)$. So, in order to steer the system to zero, one needs controls to act on both sides of the point of degeneracy in the strongly degenerate case $α\in [1,2)$. Our approach is based on sp…
We analyze here different types of fractional differential equations, under the assumption that their fractional order $ν\in (0,1] $ is random\ with probability density $n(ν).$ We start by considering the fractional extension of the recursive equation governing the homogeneous Poisson process $N(t),t>0.$\ We prove that, for a particular (discrete) choice of $n(ν)$, it leads to a process with random time, defined as $N(% \widetilde{\mathcal{T}}_{ν_{1,}ν_{2}}(t)),t>0.$ The distribution of the random time argument $\widetilde{\mathcal{T}}_{ν_{1,}ν_{2}}(t)$ can be expressed, for any fixed $t$, in …
Local exact controllability to the trajectories of the convective Brinkman-Forchheimer equations
本文通过全局 Carleman 估计和逆映射定理，证明了对流 Brinkman-Forchheimer 方程在任意时间 T>0 的轨道局部精确可控性。
Null controllability of the parabolic spherical Grushin equation
研究抛物型格鲁辛方程在球面上的零可控性，通过构造控制区域证明在足够大时间下系统可被可控，利用Carleman估计和Hardy-Poincaré不等式得出正可控性结果，同时通过球面谐波函数证明负可控性结果。
In this paper, an abstract framework for the error analysis of discontinuous finite element method is developed for the distributed and Neumann boundary control problems governed by the stationary Stokes equation with control constraints. {\it A~priori} error estimates of optimal order are derived for velocity and pressure in the energy norm and the $L^2$-norm, respectively. Moreover, a reliable and efficient {\it a~posteriori} error estimator is derived. The results are applicable to a variety of problems just under the minimal regularity possessed by the well-posedness of the problem. In par…
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