Exact Boundary Controllability Research Articles

On the Exact Boundary Controllability of Semilinear Wave Equations

On the Exact Boundary Controllability of Semilinear Wave Equations

Controllability and stabilization of a degenerate/singular Schrödinger equation

The aim of this paper is to prove controllability and stabilization properties for a degenerate and singular Schrödinger equation with degeneracy and singularity occurring at the boundary of the spatial domain. We first address the boundary control problem. In particular, by combining multiplier techniques and compactness-uniqueness argument, we prove direct and inverse inequalities for the associated adjoint system. Consequently, via the Hilbert Uniqueness Method, we deduce exact boundary controllability for the control system under consideration in any time T>0. Moreover, we investigate the stabilization problem for this class of equations in the range of subcritical coefficients of the singular potential. By introducing a suitable linear boundary feedback, we prove that the solution decays exponentially in an appropriate energy space.

Numerical approximation of the boundary control for the wave equation in a square domain with a spectral collocation method

We propose a spectral collocation method to approximate the exact boundary control of the wave equation in a square domain. The idea is to introduce a suitable approximate control problem that we solve in the finite-dimensional space of polynomials of degree N∈N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N\\in {\\mathbb {N}}$$\\end{document} in space. We prove that we can choose a sequence of controls fN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f^N$$\\end{document} associated with the approximate control problem in such a way that they converge, as N→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N\\rightarrow \\infty $$\\end{document}, to a control of the continuous wave equation. Unlike other numerical approximations tried in the literature, this one does not require regularization techniques and can be easily adapted to other equations and systems where the controllability of the continuous model is known. The method is illustrated with several examples in 1-d and 2-d in a square domain. We also give numerical evidence of the highly accurate approximation inherent to spectral methods.

Exact Boundary Controllability of Nodal Profile for Nonautonomous Quasilinear Hyperbolic Systems

Exact Boundary Controllability of Nodal Profile for Nonautonomous Quasilinear Hyperbolic Systems

Exact boundary controllability of nodal profile for the system of one-dimensional adiabatic flow

In this paper, we study the exact boundary controllability of nodal profile for the system of one-dimensional adiabatic flow by means of the specificity of the entropy S which is constant about the time t. By a constructive method, we realize the exact boundary controllability of nodal profile by controlling the velocity or the pressure on the side without the nodal profile. Moreover, for the case of the infinite time interval, under the suitable assumption, we can obtain that if the nodal profile and the given boundary function possess an exponential or polynomial decaying property, the boundary control possesses the same exponential or polynomial decaying property.

Exact internal controllability and exact internal synchronization for a kind of first order hyperbolic system

In this paper we investigate the exact controllability and exact synchronization in L2 space for a 1-D first order linear hyperbolic system with internal controls located on some part of the domain. Based on the exact boundary controllability theory, we first use the constructive method to establish the exact controllability by internal controls for a time reversible system with inhomogeneous boundary conditions. The method is then adapted to prove the exact internal null controllability for a general system with homogeneous boundary conditions. These results can be then used to establish the exact internal synchronization for the system, and related subjects are further studied.

Existence of unattainable states for Schrödinger type flows on the half-line

Abstract We prove that the solutions of the Schrödinger and biharmonic Schrödinger equations do not have the exact boundary controllability property on the half-line by showing that the associated adjoint models lack observability. We consider the framework of $L^2$ boundary controls with data spaces $H^{-1}(\mathbb{R}_+)$ and $H^{-2}(\mathbb{R}_+)$ for the classical and biharmonic Schrödinger equations, respectively. The lack of controllability on the half-line contrasts with the corresponding dynamics on a finite interval for a similar regularity setting. Our proof is based on an argument that uses the sharp fractional time trace estimates for solutions of the adjoint models. We also make several remarks on the connection of controllability and temporal regularity of spatial traces.

The Exact Boundary Controllability of Nodal Profile for Entropy Solutions to 1-D Quasilinear Hyperbolic Systems of Conservation Laws with Linearly Degenerate Negative (Resp., Positive) Characteristic Fields

The Exact Boundary Controllability of Nodal Profile for Entropy Solutions to 1-D Quasilinear Hyperbolic Systems of Conservation Laws with Linearly Degenerate Negative (Resp., Positive) Characteristic Fields

Ground state solitary waves local controllability for the nonlinear focusing Schrödinger equation in the mass critical and mass slightly subcritical case

In this paper, we prove the boundary exact local controllability around ground state solitary waves with different scaling for the mass critical and mass slightly subcritical nonlinear Schrödinger equation. More precisely, we prove that there exists a control driving the solution from the vicinity of Qλ1 to the vicinity of Qλ2, for λ1,λ2>0, with a minimal time of controllability. In the mass slightly subcritical case, we also prove the null controllability for initial data starting in a neighborhood of the ground state Qλ, for λ>0, with a minimal time of controllability. The control strategy deployed in this article is non-destructive as the solution remains close to the ground state for all time.

Exact controllability to the trajectories of the one-phase Stefan problem

This paper deals with the boundary exact controllability to the trajectories of the one-phase Stefan problem in one spatial dimension. This is a free-boundary problem that models solidification and melting processes. We prove the local exact controllability to (smooth) trajectories. To this purpose, we first reformulate the problem as the local null controllability of a coupled PDE-ODE system with distributed controls. Then, a new Carleman inequality for the adjoint of the linearized PDE-ODE system, coupled on the boundary through nonlocal in space and memory terms, is presented. This leads to the null controllability of an appropriate linear system. Finally, the result is obtained via local inversion, by using Lyusternik-Graves' Theorem. As a byproduct of our approach, we find that some parabolic equations which contains memory terms located on the boundary are null-controllable.

Hierarchical exact controllability of hyperbolic equations and Dunford-Pettis' theorem

This paper deals with the Stackelberg-Nash strategies for boundary control problems for linear and semilinear wave equations. Assuming that we can act on the system through a hierarchy of controls, to each leader, we associate a Nash equilibrium (two followers) corresponding to a bi-objective optimal control problem. Then we look for a leader that solves an exact boundary controllability problem. We mainly consider the case of a semilinear hyperbolic PDE where all the controls (leader and followers) act on small parts of the boundary. In view of the lack of regularity, the arguments usually invoked to prove controllability do not work here. Accordingly, we carry out a fixed-point approach based on weak compactness in $L^1$. To this purpose, we apply Dunford-Pettis' Theorem. We also consider the case where the followers act in small subsets of the domain.

Constrained exact boundary controllability of a semilinear model for pipeline gas flow

AbstractWhile the quasilinear isothermal Euler equations are an excellent model for gas pipeline flow, the operation of the pipeline flow with high pressure and small Mach numbers allows us to obtain approximate solutions by a simpler semilinear model. We provide a derivation of the semilinear model that shows that the semilinear model is valid for sufficiently low Mach numbers and sufficiently high pressures. We prove an existence result for continuous solutions of the semilinear model that takes into account lower and upper bounds for the pressure and an upper bound for the magnitude of the Mach number of the gas flow. These state constraints are important both in the operation of gas pipelines and to guarantee that the solution remains in the set where the model is physically valid. We show the constrained exact boundary controllability of the system with the same pressure and Mach number constraints.

Exact Boundary Controllability of Nodal Profile for Quasilinear Hyperbolic Systems and Its Asymptotic Stability

Exact Boundary Controllability of Nodal Profile for Quasilinear Hyperbolic Systems and Its Asymptotic Stability

On the exact controllability for the Benney-Luke equation in a bounded domain

We study the exact boundary controllability of the nonlinear - Benney-Luke equation at the right endpoint of a bounded domain, applying the control first to the first spatial derivative and to the second spatial derivative. We use the Hilbert Uniqueness Method to get the exact controllability of the linearized problem. From this fact and the Banach contraction fixed point principle, we establish the exact boundary controllability for the nonlinear Benney-Luke equation in bounded domains, for sufficiently small initial and final states.

Sufficient conditions for the controllability of wave equations with a transmission condition at the interface

We consider waves travelling in two different mediums each endowed with a different constant speed of propagation. At the interface between the two mediums, the refraction of the rays of the optic geometry obeys the Snell’s law. We provide sufficient conditions on the geometry’ of the mediums and on the speed of propagation for the exact boundary controllability.

Exact controllability for a degenerate and singular wave equation with moving boundary

<p style='text-indent:20px;'>This paper is concerned with the exact boundary controllability for a degenerate and singular wave equation in a bounded interval with a moving endpoint. By the multiplier method and using an adapted Hardy-poincaré inequality, we prove direct and inverse inequalities for the solutions of the associated adjoint equation. As a consequence, by the Hilbert Uniqueness Method, we deduce the controllability result of the considered system when the control acts on the moving boundary. Furthermore, improved estimates of the speed of the moving endpoint and the controllability time are obtained.</p>

Positive and negative exact controllability results for the linear Biharmonic Schrödinger equation

In this paper, we study the exact boundary controllability of the linear Biharmonic Schrödinger equation on a bounded domain with hinged boundary conditions and boundary control acts on the second spatial derivative at the left endpoint. We prove that this system is exactly controllable at any positive time $ T $, if and only if, the group velocity dispersion (GVD) coefficient 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.

Regularity results for degenerate wave equations in a neighborhood of the boundary

In this paper we establish some regularity results concerning the behavior of weak solutions and very weak solutions of the degenerate wave equation near the boundary. For the nondegenerate case, the correponding results were originally obtained by Fabre and Puel (J. of Diff. Eq. 106, 1993). This kind of results is closely related to the exact boundary controllability for the wave equation as the limit of internal controllability.

Exact boundary controllability of 1D semilinear wave equations through a constructive approach

The exact controllability of the semilinear wave equation $$y_{tt}-y_{xx}+ f(y)=0$$ , $$x\in (0,1)$$ assuming that f is locally Lipschitz continuous and satisfies the growth condition $$\limsup _{\vert r\vert \rightarrow \infty } \vert f(r)\vert /(\vert r\vert \ln ^{p}\vert r\vert )\leqslant \beta $$ for some $$\beta $$ small enough and $$p=2$$ has been obtained by Zuazua (Ann Inst H Poincaré Anal Non Linéaire 10(1):109–129, 1993). The proof based on a non-constructive fixed point arguments makes use of precise estimates of the observability constant for a linearized wave equation. Under the above asymptotic assumption with $$p=3/2$$ , by introducing a different fixed point application, we present a simpler proof of the exact boundary controllability which is not based on the cost of observability of the wave equation with respect to potentials. Then, assuming that f is locally Lipschitz continuous and satisfies the growth condition $$\limsup _{\vert r\vert \rightarrow \infty } \vert f^\prime (r)\vert /\ln ^{3/2}\vert r\vert \leqslant \beta $$ for some $$\beta $$ small enough, we show that the above fixed point application is contracting yielding a constructive method to approximate the controls for the semilinear equation. Numerical experiments illustrate the results. The results can be extended to the multi-dimensional case and for nonlinearities involving the gradient of the solution.

Global piecewise classical solutions to quasilinear hyperbolic systems on a tree-like network

In this paper, we discuss the existence, uniqueness and asymptotic stability of global piecewise [Formula: see text] solution to the mixed initial-boundary value problem for 1-D quasilinear hyperbolic systems on a tree-like network. Under the assumption of boundary dissipation, when the given boundary and interface functions possess suitably small [Formula: see text] norm, we obtain the existence and uniqueness of global piecewise [Formula: see text] solution. Moreover, when they further possess a polynomial or exponential decaying property with respect to [Formula: see text], then the corresponding global piecewise [Formula: see text] solution possesses the same or similar decaying property. These results will be used to show the asymptotic stability of the exact boundary controllability of nodal profile on a tree-like network.