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.
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.
We study fractional hypoelliptic Ornstein-Uhlenbeck operators acting on $L^2(\mathbb{R}^n)$ satisfying the Kalman rank condition. We prove that the semigroups generated by these operators enjoy Gevrey regularizing effects. Two byproducts are derived from this smoothing property. On the one hand, we prove the null-controllability in any positive time from thick control subsets of the associated parabolic equations posed on the whole space. On the other hand, by using interpolation theory, we get global $L^2$ subelliptic estimates for the these operators.
We study the null-controllability of parabolic equations associated to non-autonomous Ornstein-Uhlenbeck operators. When a Kalman type condition holds for some positive time $T>0$, these parabolic equations are shown to enjoy a Gevrey regularizing effect at time $T>0$. Thanks to this regularizing effect, we prove by adapting the Lebeau-Robbiano method that these parabolic equations are null-controllable in time greater than or equal to $T>0$ from control regions, for which null-controllability is classically known to hold in the case of the heat equation.
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…
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.
In this paper, we deal with the controllability properties of a system of $m$ coupled Stokes systems or $m$ coupled Navier-Stokes systems. We show the null-controllability of such systems in the case where the coupling is in a cascade form and when the control acts only on one of the systems. Moreover, we impose that this control has a vanishing component so that we control a $m\times N$ state (corresponding to the velocities of the fluids) by $N-1$ distributed scalar controls. The proof of the controllability of the coupled Stokes system is based on a Carleman estimate for the adjoint system.…
This paper is devoted to studying the null and approximate controllability of two linear coupled parabolic equations posed on a smooth domain of R^N (N>1) with coupling terms of zero and first orders and one control localized in some arbitrary nonempty open subset of the domain. We prove the null controllability under a new sufficient condition and we also provide the first example of a not approximately controllable system in the case where the support of one of the nontrivial first order coupling terms intersects the control domain.
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 …
This paper deals with the null controllability of a coupled parabolic system, which is Kuramoto-Sivashinsky-Korteweg-de Vries equation coupled with heat equation through first order derivative. More precisely, we prove the null controllability of the system with a single localized bilinear interior control acting on either of the components of the coupled system, and with a single periodic boundary control acting through zeroth order derivatives of either of the components. We employ the well-known moment method to study the controllability of the concerned system.
In this paper, we are concerned with the internal control of a class of one-dimensional nonlinear parabolic systems with nonlocal and weakly degenerate diffusion coefficients. Our main theorem establishes a local null controllability result with only one internal control for a system of two equations. The proof, based on the ideias developed by Fursikov and Imanivilov, is obtained from the global null controllability of the linearized system provided by \textit{Lyusternik's Inverse Mapping Theorem}. This work extends the results previously treated by the authors for just one equation. For the …
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 …
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.
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.
The purpose of this work is to investigate the exponential stability of a second order coupled wave equations by laplacian with one locally internal viscous damping. Firstly, using a unique continuation theorem combined with a Carleman estimate, we prove that our system is strongly stable without any geometric condition. Secondly, using a combination of the multiplier techniques and the frequency domain approach, we show that our system is exponentially stable under \textbf{(PMGC)} condition on the damping region without any restriction on wave propagation speed (i.e whether the two wave equat…
This article is dedicated to improve the controllability results obtained by Cerpa et al. in Commun. Contemp. Math 13 (2011) and by Micu et al. in Commun. Contemp. Math 11 (5) (2009) for a nonlinear coupled system of two Korteweg-de Vries (KdV) equations posed on a bounded interval. Initially, in Micu et al., the authors proved that the nonlinear system is exactly controllable by using four boundary controls without any restriction on the length L of the interval. Later on, in Cerpa et al., two boundary controls were considered to prove that the same system is exactly controllable for small va…
This paper is devoted to the controllability of linear systems of two coupled parabolic equations when the coupling involves a space dependent first order term. This system is set on an bounded interval, and the first equation is controlled by a force supported in a subinterval of I or on the boundary. In the case where the intersection of the coupling and control domains is nonempty, we prove null controllability at any time. Otherwise, we provide a minimal time for null controllability. Finally we give a necessary and sufficient condition for the approximate controllability. The main technic…
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.
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 well-posedness and exact controllability of a physical model for a food extrusion process in the isothermal case. The model expresses the mass balance in the extruder chamber and consists of a hyperbolic Partial Differential Equation (PDE) and a nonlinear Ordinary Differential Equation (ODE) whose dynamics describes the evolution of a moving interface. By suitable change of coordinates and fixed point arguments, we prove the existence, uniqueness and regularity of the solution, and finally the exact controllability of the coupled system.
In this paper, we study the simultaneous controllability of wave systems in an open domain of R d , d $\in$ N *. We obtain a partial controllability result on a co-finite dimensional space for wave equations coupled by a single control function. We use microlocal defect measures and the unique continuation property of eigenfunctions to prove that an appropriate observability inequality holds for wave equations with space varying and different speeds coupled by a single control function. For the unique continuation property of eigenfunctions, we construct a counterexample to show that in some m…
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…
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 first give an algebraic characterization of uniqueness of continuation for a coupled system of wave equations with coupled Robin boundary conditions. Then, the approximate boundary controllability and the approximate boundary synchronization by groups for a coupled system of wave equations with coupled Robin boundary controls are developed around this fundamental characterization.
This paper is devoted to the study of the null and approximate controllability for some classes of linear coupled parabolic systems with less controls than equations. More precisely, for a given bounded domain in R^N, we consider a system of m linear parabolic equations (m > 2) with coupling terms of first and zero order, and m-1 controls localized in some arbitrary nonempty open subset. In the case of constant coupling coefficients, we provide a necessary and sufficient condition to obtain the null or approximate controllability in arbitrary small time. In the case m = 2 and N = 1, we also gi…
We study the controllability of a coupled system of linear parabolic equations, with non-negativity constraint on the state. We establish two results of controllability to trajectories in large time: one for diagonal diffusion matrices with an "approximate" nonnegativity constraint, and a another stronger one, with "exact" nonnegativity constraint, when all the diffusion coefficients are equal and the eigenvalues of the coupling matrix have nonnegative real part. The proofs are based on a "staircase" method. Finally, we show that state-constrained controllability admits a positive minimal time…
Let the matrix operator $L = D\partial_{xx} + q(x)A_0$, with $D = diag(1, ν)$, $ν \neq 1$, $q \in L^{\infty} (0, $π$)$, and $A_0$ is a Jordan block of order 1. We analyze the boundary null controllability for system $y_t - Ly = 0$. When $ ν\notin \mathbb{Q} ^*_+ $ and $q(x) = 1$, $x $\in$ (0, $π$)$, there exists a family of root vectors of $(L * , D(L *)) $ forming a Riesz basis, moreover, F. Ammar Khodja, A.Benabdallah, M.Gonzalez-Burgos, L.Teresa, show the existence of a minimal time of control depending on condensation of eigenvalues of $(L^* , D(L^*))$. But there exists $q \in L^{\infty} (…
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…
Studies on the Rao-Nakra Sandwich Beam: Well-Posedness, Dynamics, and Controllability
Universidade Federal Fluminense（弗鲁米嫩塞联邦大学）；Universidade Federal de Pernambuco（伯南布哥联邦大学）；Kuwait University（科威特大学）；Fundação Getúlio Vargas（巴西得恩利奥·瓦加斯基金会）
In this paper, we prove the local-controllability to positive constant trajectories of a nonlinear system of two coupled ODE equations, posed in the one-dimensional spatial setting, with nonlocal spatial nonlinearites, and using only one localized control with a moving support. The model we deal with is derived from the well-known nonlinear reaction-diffusion Gray-Scott model when the diffusion coefficient of the first chemical species $d_u$ tends to $0$ and the diffusion coefficient of the second chemical species ${d_v}$ tends to $+ \infty$. The strategy of the proof consists in two main step…
