Exact Boundary Controllability Research Articles

Exact Boundary Controllability and Energy Decay for a System of Wave Equations Linearly Coupled

The aim of this paper is to study an exact boundary controllability problem and local energy decay for a system of m linearly coupled wave equations . We obtain control time near of the optimal time with control of Neuman type acting on all boundary of the considered domain. We also show that the local energy decays at a rate $$t^{-n}$$ , where n is the spatial dimension of the domain where the initial data are supported.

Exact boundary controllability and exact boundary synchronization for a coupled system of wave equations with coupled Robin boundary controls

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 [J. Differ. Equ. 94 (1991) 112–164] to get the regularity of solutions to the mixed problem with coupled Robin boundary conditions. Thus we show the exact boundary controllability of the system, and by a method of compact perturbation, we obtain the non-exact boundary controllability of the system with fewer boundary controls on some special domains. Based on this, we further study the exact boundary synchronization (by groups) for the same system, the determination of the exactly synchronizable state (by groups), as well as the necessity of the compatibility conditions of the coupling matrices.

Numerical approximation of the averaged controllability for the wave equation with unknown velocity of propagation

We propose a numerical method to approximate the exact averaged boundary control of a family of wave equations depending on an unknown parameter σ. More precisely the control, independent of σ, that drives an initial data to a family of final states at time t = T, whose average in σ is given. The idea is to project the control problem in the finite dimensional space generated by the first N eigenfunctions of the Laplace operator. When applied to a single (nonparametric) wave equation, the resulting discrete control problem turns out to be equivalent to the Galerkin approximation proposed by F. Bourquin et al. [C.R. Acad. Sci. Paris 313 I (1991) 757–760]. We give a convergence result of the discrete controls to the continuous one. The method is illustrated with several examples in 1-d and 2-d in a square domain and allows us to give some conjectures on the averaged controllability for the continuous problem.

On the Exact Boundary Control for the Linear Klein-Gordon Equation in Non-cylindrical Domains

The purpose of this paper is to study an exact boundary controllability problem in noncylindrical domains for the linear Klein-Gordon equation. Here, we work near of the extension techniques presented By J. Lagnese in [12] which is based in the Russell’s controllability method. The control time is obtained in any time greater then the value of the diameter of the domain on which the initial data are supported. The control is square integrable and acts on whole boundary and it is given by conormal derivative associated with the above-referenced wave operator.

On the Exact Boundary Control for the Linear Klein-Gordon Equation in Non-cylindrical Domains

The purpose of this paper is to study an exact boundary controllability problem in noncylindrical domains for the linear Klein-Gordon equation. Here, we work near of the extension techniques presented By J. Lagnese in [12] which is based in the Russell’s controllability method. The control time is obtained in any time greater then the value of the diameter of the domain on which the initial data are supported. The control is square integrable and acts on whole boundary and it is given by conormal derivative associated with the above-referenced wave operator.

Exact Boundary Controllability for the Spatial Vibration of String with Dynamical Boundary Conditions

This paper deals with the spatial vibration of an elastic string with masses at the endpoints. The authors derive the corresponding quasilinear wave equation with dynamical boundary conditions, and prove the exact boundary controllability of this system by means of a constructive method with modular structure.

Boundary controllability of a one-dimensional phase-field system with one control force

In this paper, we present some controllability results for linear and nonlinear phase-field systems of Caginalp type considered in a bounded interval of R when the scalar control force acts on the temperature equation of the system by means of the Dirichlet condition on one of the endpoints of the interval. In order to prove the linear result we use the moment method providing an estimate of the cost of fast controls. Using this estimate and following the methodology developed in [19], we prove a local exact boundary controllability result to constant trajectories of the nonlinear phase-field system. To the authors' knowledge, this is the first nonlinear boundary controllability result in the framework of non-scalar parabolic systems, framework in which some “hyperbolic” behaviors could arise.

Asymptotic stability of the exact boundary controllability of nodal profile for 1D quasi‐linear wave equations

In this paper, we consider the asymptotic stability of the exact boundary controllability of nodal profile for 1D quasi‐linear wave equations. First, for 1D quasi‐linear hyperbolic systems with zero eigenvalues, we establish the existence and uniqueness of semiglobal classical solution to the one‐sided mixed initial‐boundary value problem on a semibounded initial axis and discuss the asymptotic behavior of the corresponding solutions under different hypotheses on the initial data. Based on these results, we obtain the asymptotic stability of the exact boundary controllability of nodal profile for 1D quasi‐linear wave equations on a semibounded time interval.

Asymptotic stability of the exact boundary controllability of nodal profile for quasilinear hyperbolic systems

In this paper, we consider the asymptotic stability of the exact boundary controllability of nodal profile for quasilinear hyperbolic systems. We will prove that if the nodal profile and the given boundary function possess an exponential or polynomial decaying property, then the boundary control function and the solution to the corresponding mixed initial-boundary value problem will possess the same decaying property.

Boundary Exact Controllability of the Heat Equation in 1D by Strategic Actuators and a Linear Surjective Compact Operator

In this paper we show a boundary result of controllability by a new approach using a linear, continuous and surjective operator built from the solution of the heat system. And, subsequently, the border exact controllability of the 1D heat equation through a compactness criterion and the use of strategic zone actuators were established.

Uniform indirect boundary controllability of semi-discrete <inline-formula><tex-math id="M1">\begin{document}$ 1 $\end{document}</tex-math></inline-formula>-<inline-formula><tex-math id="M2">\begin{document}$ d $\end{document}</tex-math></inline-formula> coupled wave equations

<p style='text-indent:20px;'>In this paper, we treat the problem of uniform exact boundary controllability for the finite-difference space semi-discretization of the <inline-formula><tex-math id="M3">\begin{document}$ 1 $\end{document}</tex-math></inline-formula>-<inline-formula><tex-math id="M4">\begin{document}$ d $\end{document}</tex-math></inline-formula> coupled wave equations with a control acting only in one equation. First, we show how, after filtering the high frequencies of the discrete initial data in an appropriate way, we can construct a sequence of uniformly (with respect to the mesh size) bounded controls. Thus, we prove that the weak limit of the aforementioned sequence is a control for the continuous system. The proof of our results is based on the moment method and on the construction of an explicit biorthogonal sequence.

Energy decay and control for a system of strings elastically connected in parallel

In the present paper, we study energy decay and exact boundary controllability for a system of n one‐dimensional linear wave equations coupled in parallel. The control obtained is a square integrable of the Neuman type for initial data with finite energy. The controllability time is near optimal value. We treat the case of control in the whole boundary and also in part of it.

Controllability of the Navier–Stokes Equation in a Rectangle with a Little Help of a Distributed Phantom Force

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 in Guerrero et al. (C R Math Acad Sci Paris 343(9):573–577, 2006), Guerrero et al. (J Math Pures Appl (9) 98(6):689–709, 2012) where the distributed force is small only in a negative Sobolev space. It is a further step towards an answer to Lions’ question in Lions (Exact controllability for distributed systems. Some trends and some problems. In: Applied and industrial mathematics (Venice, 1989), vol. 56, Math. Appl., pp. 59–84. Kluwer Academic Publications, Dordrecht, 1991) about the small-time global exact boundary controllability of the Navier–Stokes equation with the no-slip boundary condition, for which no distributed force is allowed. Our analysis relies on the well-prepared dissipation method already used in Marbach (J Math Pures Appl (9) 102(2):364–384, 2014) for Burgers and in Coron et al. (J Eur Math Soc, 2016) for Navier–Stokes in the case of the Navier slip-with-friction boundary condition. In order to handle the larger boundary layers associated with the no-slip boundary condition, we perform a preliminary regularization into analytic functions with arbitrarily large analytic radius and prove a long-time nonlinear Cauchy–Kovalevskaya estimate relying only on horizontal analyticity, in the spirit of Chemin (Le système de Navier–Stokes incompressible soixante dix ans après Jean Leray. In: Actes des Journées Mathématiques à la Mémoire de Jean Leray, vol. 9, Séminar Congress, pp. 99–123. Société Mathématique de France, Paris (2004), 2004), Zhang and Zhang (J Funct Anal 270(7):2591–2615, 2016).

Exact boundary observability and controllability of the wave equation in an interval with two moving endpoints

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.

On the simplicity of eigenvalues of two nonhomogeneous Euler-Bernoulli beams connected by a point mass

In this paper we consider a linear system modeling the vibrations of two nonhomogeneous Euler-Bernoulli beams connected by a point mass. This system is generated by the following equationsρ(x)ytt(t,x)+(σ(x)yxx(t,x))xx−(q(x)yx(t,x))x=0,t>0,x∈(−1,0)∪(0,1),Mytt(t,0)=(Ty(t,x))|x=0−−(Ty(t,x))|x=0+,t>0, with hinged boundary conditions at both ends, where M>0, ρ(x)>0,σ(x)>0, q(x)≥0 and Ty=(σ(x)yxx)x−q(x)yx for x∈(−1,0)∪(0,1). We prove that all the associated eigenvalues (λn)n≥1 are algebraically simple, furthermore the corresponding eigenfunctions (ϕn)n≥1 satisfy ϕn′(−1)Tϕn(−1)>0 and ϕn′(1)Tϕn(1)<0 for all n≥1. These results are essential for the proof of the exact boundary controllability problem related to this system by means of the non-harmonic Fourier series method.

Exact boundary controllability and its applications for a coupled system of quasilinear wave equations with dynamical boundary conditions

A constructive method with modular structure is used to obtain the local exact boundary controllability for a coupled system of quasilinear wave equations with local or non-local dynamical boundary conditions. Some applications are given for the chain-like or star-like system of strings coupled via elastic springs, and of strings coupled via Maxwell-type and Kelvin-type viscoelastic springs, respectively.

Stability and exact controllability of a Timoshenko system with only one fractional damping on the boundary

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 propagation condition, we obtain the optimal polynomial energy decay rate. In the second part of this paper, we study the indirect boundary exact controllability of the Timoshenko system with mixed Dirichlet–Neumann boundary conditions and boundary control. Using non-harmonic analysis, we first establish a weak observability inequality, which depends on the ratio of the waves propagation speeds. Next, using the HUM method, we prove that the system is exactly controllable in appropriate spaces and that the control time can be small.

Exact boundary controllability of nodal profile for Saint-Venant system on a network with loops

The exact boundary controllability for hyperbolic systems can not be realized generally on a network with loops (see [16]). In this paper we consider the exact boundary controllability of nodal profile on a network with loops. Precisely speaking, on a network with a triangle-like loop, when nodal profiles are given at various kinds of nodes, different constructive methods can be used to get the corresponding exact boundary controllability of nodal profile for Saint-Venant system by means of boundary controls acting on suitable nodes, respectively. This reveals that the exact boundary controllability of nodal profile is quite different from the usual exact boundary controllability, and has relatively distinctive behaviors and characters.

Exact controllability of the wave equation with time-dependent and variable coefficients

This paper deals with boundary exact controllability for the dynamics governed by the wave equation with variable coefficients in time and space, subject to Dirichlet or Neumann boundary controls. The observability inequalities are established by the Riemannian geometry method under some geometric conditions.

Exact Boundary Controllability on a Planar Tree-Like Network of Vibrating Strings with Dynamical Boundary Conditions

Exact Boundary Controllability on a Planar Tree-Like Network of Vibrating Strings with Dynamical Boundary Conditions