Neumann Boundary Control Research Articles

Neumann boundary control for degenerate wave equations in time-varying domains

Degenerate wave equation with Neumann boundary control in time-varying domains is considered. The fixed endpoint has a Dirichlet-type boundary condition, a Neumann boundary control applied to the moving end. By using the Hilbert Uniqueness Method, we establish the exact boundary controllability for the degenerate wave equation and obtain estimation for the control time.

Boundary controllability for a 1D degenerate parabolic equation with drift, a singular potential, and a Neumann boundary condition

We prove the null controllability of a one-dimensional degenerate parabolic equation with drift and a singular potential. Here, we consider a weighted Neumann boundary control at the left endpoint, where the potential arises. We use a spectral decomposition of a suitable operator, defined in a weighted Sobolev space, and the moment method by Fattorini and Russell to obtain an upper estimate of the cost of controllability. We also obtain a lower estimate of the cost of controllability using a representation theorem for analytic functions of exponential type.

Numerical analysis of a Neumann boundary control problem with a stochastic parabolic equation

Numerical analysis of a Neumann boundary control problem with a stochastic parabolic equation

Exponential Tracking and Disturbance Rejection for Nonlinear Reaction Convection Diffusion Equations via Boundary Control

This paper studies the problem of exponential tracking and disturbance rejection for nonlinear reaction convection diffusion equations via Dirichlet or Neumann boundary control. Disturbances are present both in the equation and on the boundary. The tracking problem is simply a nonlinear dynamic regulator problem. If an initial condition is not compatible with a reference, the nonlinear dynamic regulator problem does not have a solution. Thus, we split the tracking problem into two problems to relax the compatibility condition. We choose one problem to be exponentially stable and leave everything else in a dynamic regulator problem. Then a controller is just a solution component of the dynamic regulator problem, while the tracking error is given by the solution of the exponentially stable problem. The controller can be obtained by integrating the dynamic regulator equation and using the reference condition. The dynamic regulator problem with the obtained controller becomes an initial boundary value problem. We then transfer the initial boundary value problem into an equivalent integral equation and prove that the integral equation has a unique global solution if the nonlinearity is globally Lipschitz. In addition, a numerical example is given to illustrate the theoretical results.

Existence of Optimal Control for Dirichlet Boundary Optimization in a Phase Field Problem

Phase field modeling finds utility in various areas. In optimization theory in particular, the distributed control and Neumann boundary control of phase field models have been investigated thoroughly. Dirichlet boundary control in parabolic equations is commonly addressed using the very weak formulation or an approximation by Robin boundary conditions. In this paper, the Dirichlet boundary control for a phase field model with a non-singular potential is investigated using the Dirichlet lift technique. The corresponding weak formulation is analyzed. Energy estimates and problem-specific embedding results are provided, leading to the existence and uniqueness of the solution for the state equation. These results together show that the control to state mapping is well defined and bounded. Based on the preceding findings, the optimization problem is shown to have a solution.

Optimal control problems of parabolic fractional Sturm-Liouville equations in a star graph

<p style='text-indent:20px;'>In the present paper we deal with parabolic fractional initial-boundary value problems of Sturm–Liouville type in an interval and in a general star graph. We first give several existence, uniqueness and regularity results of weak and very-weak solutions. We prove the existence and uniqueness of solutions to a quadratic boundary optimal control problem and provide a characterization of the optimal contol via the Euler–Lagrange first order optimality conditions. We then investigate the analogous problems for a fractional Sturm–Liouville problem in a general star graph with mixed Dirichlet and Neumann boundary controls. The existence and uniqueness of minimizers, and the characterization of the first order optimality conditions are obtained in a general star graph by using the method of Lagrange multipliers.</p>

On the small-time local controllability of a KdV system for critical lengths

This paper is devoted to the local null-controllability of the nonlinear KdV equation equipped the Dirichlet boundary conditions using the Neumann boundary control on the right. Rosier proved that this KdV system is small-time locally controllable for all noncritical lengths and that the uncontrollable space of the linearized system is of finite dimension when the length is critical. Concerning critical lengths, Coron and Crépeau showed that the same result holds when the uncontrollable space of the linearized system is of dimension 1; later Cerpa, and then Cerpa and Crépeau, established that the local controllability holds at a finite time for all other critical lengths. In this paper, we prove that, for a class of critical lengths, the nonlinear KdV system is not small-time locally controllable.

Asymptotic behaviour of a linearized water waves system in a rectangle1

We consider the asymptotic behaviour of small-amplitude gravity water waves in a rectangular domain where the water depth is much smaller than the horizontal scale. The control acts on one lateral boundary, by imposing the horizontal acceleration of the water along that boundary, as a scalar input function u. The state z of the system consists of two functions: the water level ζ along the top boundary, and its time derivative ∂ ζ ∂ t . We prove that the solution of the water waves system converges to the solution of the one dimensional wave equation with Neumann boundary control, when taking the shallowness limit. Our approach is based on a special change of variables and a scattering semigroup, which provide the possiblity to apply the Trotter–Kato approximation theorem. Moreover, we use a detailed analysis of Fourier series for the dimensionless version of the partial Dirichlet to Neumann and Neumann to Neumann operators.

Prescribed-time stabilisation for uncertain reaction-diffusion equations with Neumann boundary control

We study the Neumann boundary feedback stabilisation for reaction-diffusion equations with matched boundary uncertainties (including internal uncertainty and external disturbance) within a prescribed time, where the convergence time is independent of initial conditions and can be prescribed a priori. We first estimate the uncertainties within the prescribed time by constructing an estimator and then propose the prescribed-time boundary control law by utilising backstepping transformation with a time-varying kernel. Neumann boundary control is verified to be uniformly bounded.

Optimal control of fractional Sturm–Liouville wave equations on a star graph

ABSTRACT In the present paper, we are concerned with a fractional wave equation of Sturm–Liouville type in a general star graph. We first give several existence, uniqueness and regularity results of weak solutions for the one-dimensional case using the spectral theory; we prove the existence and uniqueness of solutions to a quadratic boundary optimal control problem and provide a characterization of the optimal control via the Euler–Lagrange first-order optimality conditions. We then investigate the analogous problems for a fractional Sturm–Liouville problem in a general star graph with mixed Dirichlet and Neumann boundary conditions and controls of the velocity. We show the existence and uniqueness of minimizers, and by using the first-order optimality conditions with the Lagrange multipliers, we are able to characterize the optimal controls.

Analysis and Computations of Optimal Control Problems for Boussinesq Equations

The main purpose of engineering applications for fluid with natural and mixed convection is to control or enhance the flow motion and the heat transfer. In this paper, we use mathematical tools based on optimal control theory to show the possibility of systematically controlling natural and mixed convection flows. We consider different control mechanisms such as distributed, Dirichlet, and Neumann boundary controls. We introduce mathematical tools such as functional spaces and their norms together with bilinear and trilinear forms that are used to express the weak formulation of the partial differential equations. For each of the three different control mechanisms, we aim to study the optimal control problem from a mathematical and numerical point of view. To do so, we present the weak form of the boundary value problem in order to assure the existence of solutions. We state the optimization problem using the method of Lagrange multipliers. In this paper, we show and compare the numerical results obtained by considering these different control mechanisms with different objectives.

Local a posteriori error estimates for boundary control problems governed by nonlinear parabolic equations

We derive local a posteriori error estimates of finite element approximation to nonlinear parabolic boundary control problems in a bounded convex domain with Lipschitz boundary. We apply piecewise linear and continuous finite elements to approximate the state and co-state variables, and the control variable is approximated by using the piecewise constant functions. The discrete-in-time scheme is based on the backward Euler method. We derive a reliable type local a posteriori error estimates for Neumann boundary control problems in the L2([0,T];L2(∂Ω))-norm. The main feature of our estimators is that they are of local character in the sense that the leading terms of the estimators depend on the small neighbourhood of the boundary. These new local a posteriori error bounds can be used to study the behaviour of the state and co-state variables near the boundary. The derived error indicators will provide the necessary feedback for the adaptive mesh refinements in the finite element method. We report numerical tests to illustrate the effectiveness of our error indicators.

From low to high-and lower-optimal regularity of the SMGTJ equation with Dirichlet and Neumann boundary control, and with point control, via explicit representation formulae

<p style='text-indent:20px;'>We consider the linear third order (in time) PDE known as the SMGTJ-equation, defined on a bounded domain, under the action of either Dirichlet or Neumann boundary control <inline-formula><tex-math id="M1">\begin{document}$ g $\end{document}</tex-math></inline-formula>. Optimal interior and boundary regularity results were given in [<xref ref-type="bibr" rid="b1">1</xref>], after [<xref ref-type="bibr" rid="b41">41</xref>], when <inline-formula><tex-math id="M2">\begin{document}$ g \in L^2(0, T;L^2(\Gamma)) \equiv L^2(\Sigma) $\end{document}</tex-math></inline-formula>, which, moreover, in the canonical case <inline-formula><tex-math id="M3">\begin{document}$ \gamma = 0 $\end{document}</tex-math></inline-formula>, were expressed by the well-known explicit representation formulae of the wave equation in terms of cosine/sine operators [<xref ref-type="bibr" rid="b19">19</xref>], [<xref ref-type="bibr" rid="b17">17</xref>], [<xref ref-type="bibr" rid="b24">24</xref>,Vol Ⅱ]. The interior or boundary regularity theory is however the same, whether <inline-formula><tex-math id="M4">\begin{document}$ \gamma = 0 $\end{document}</tex-math></inline-formula> or <inline-formula><tex-math id="M5">\begin{document}$ 0 \neq \gamma \in L^{\infty}(\Omega) $\end{document}</tex-math></inline-formula>, since <inline-formula><tex-math id="M6">\begin{document}$ \gamma \neq 0 $\end{document}</tex-math></inline-formula> is responsible only for lower order terms. Here we exploit such cosine operator based-explicit representation formulae to provide optimal interior and boundary regularity results with <inline-formula><tex-math id="M7">\begin{document}$ g $\end{document}</tex-math></inline-formula> "smoother" than <inline-formula><tex-math id="M8">\begin{document}$ L^2(\Sigma) $\end{document}</tex-math></inline-formula>, qualitatively by one unit, two units, etc. in the Dirichlet boundary case. To this end, we invoke the corresponding results for wave equations, as in [<xref ref-type="bibr" rid="b17">17</xref>]. Similarly for the Neumann boundary case, by invoking the corresponding results for the wave equation as in [<xref ref-type="bibr" rid="b22">22</xref>], [<xref ref-type="bibr" rid="b23">23</xref>], [<xref ref-type="bibr" rid="b37">37</xref>] for control smoother than <inline-formula><tex-math id="M9">\begin{document}$ L^2(0, T;L^2(\Gamma)) $\end{document}</tex-math></inline-formula>, and [<xref ref-type="bibr" rid="b44">44</xref>] for control less regular in space than <inline-formula><tex-math id="M10">\begin{document}$ L^2(\Gamma) $\end{document}</tex-math></inline-formula>. In addition, we provide optimal interior and boundary regularity results when the SMGTJ equation is subject to interior point control, by invoking the corresponding wave equations results [<xref ref-type="bibr" rid="b42">42</xref>], [<xref ref-type="bibr" rid="b24">24</xref>,Section 9.8.2].</p>

The Continuous Classical Boundary Optimal Control of Triple Nonlinear Elliptic Partial Differential Equations with State Constraints

Our aim in this work is to study the classical continuous boundary control vector problem for triple nonlinear partial differential equations of elliptic type involving a Neumann boundary control. At first, we prove that the triple nonlinear partial differential equations of elliptic type with a given classical continuous boundary control vector have a unique "state" solution vector, by using the Minty-Browder Theorem. In addition, we prove the existence of a classical continuous boundary optimal control vector ruled by the triple nonlinear partial differential equations of elliptic type with equality and inequality constraints. We study the existence of the unique solution for the triple adjoint equations related with the triple state equations.
 The Fréchet derivative is obtained. Finally we prove the theorems of both the necessary and sufficient conditions for optimality of the triple nonlinear partial differential equations of elliptic type through the Kuhn-Tucker-Lagrange's Multipliers theorem with equality and inequality constraints.

Boundary null-controllability of 1-D coupled parabolic systems with Kirchhoff-type conditions

The main concern of this article is to investigate the boundary controllability of some $$2\times 2$$ one-dimensional parabolic systems with both the interior and boundary couplings: The interior coupling is chosen to be linear with constant coefficient while the boundary one is considered by means of some Kirchhoff-type condition at one end of the domain. We consider here the Dirichlet boundary control acting only on one of the two state components at the other end of the domain. In particular, we show that the controllability properties change depending on which component of the system the control is being applied. Regarding this, we point out that the choices of the interior coupling coefficient and the Kirchhoff parameter play a crucial role to deduce the positive or negative controllability results. Further to this, we pursue a numerical study based on the well-known penalized HUM approach. We make some discretization for a general interior-boundary coupled parabolic system, mainly to incorporate the effects of the boundary couplings into the discrete setting. This allows us to illustrate our theoretical results as well as to experiment some more examples which fit under the general framework, for instance a similar system with a Neumann boundary control on either one of the two components.

Residual-type a posteriori error analysis of HDG methods for Neumann boundary control problems

We study a posteriori error analysis of linear-quadratic boundary control problems under bilateral box constraints on the control which acts through a Neumann-type boundary condition. We adopt the hybridizable discontinuous Galerkin method as the discretization technique, and the flux variables, the scalar variables, and the boundary trace variables are all approximated by polynomials of degree k. As for the control variable, it is discretized by the variational discretization concept. Then, an efficient and reliable a posteriori error estimator is introduced, and we prove that the error estimator provides an upper bound and a lower bound for the errors. Finally, numerical results are presented to illustrate the performance of the obtained a posteriori error estimator.

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.

Singular asymptotic expansion of the exact control for the perturbed wave equation

The Petrowsky type equation y t t ε + ε y x x x x ε − y x x ε = 0, ε > 0 encountered in linear beams theory is null controllable through Neumann boundary controls. Due to the boundary layer of size of order ε occurring at the extremities, these boundary controls get singular as ε goes to 0. Using the matched asymptotic method, we describe the boundary layer of the solution y ε and derive a rigorous second order asymptotic expansion of the control of minimal weighted L 2 -norm, with respect to the parameter ε. The weight in the norm is chosen to guarantee the smoothness of the control. In particular, we recover and enrich earlier results due to J.-L. Lions in the eighties showing that the leading term of the expansion is a null Dirichlet control for the limit hyperbolic wave equation. The asymptotic analysis also provides a robust discrete approximation of the control for any ε small enough. Numerical experiments support our study.

Input-to-state stabilization of an ODE-wave system with disturbances

In this paper, we consider the input-to-state stabilization of an ODE-wave feedback-connection system with Neumann boundary control, where the left end displacement of the wave equation enters the ODE, while the output of the ODE is fluxed into boundary of the wave equation. The disturbance is appeared as a nonhomogeneous term in the ODE. Based on the backstepping approach, a state feedback control law is designed to guarantee the exponential input-to-state stability of the closed-loop system. The resulting closed-loop system has been shown to be well-posed by the semigroup approach. Moreover, we construct an exponentially convergent state observer based on which an output feedback control law is obtained, and the closed-loop system is proved to be input-to-state stable.

Observers and Disturbance Rejection Control for a Heat Equation

The article is concerned with active disturbance rejection control of a heat equation. The considered heat equation satisfies the Dirichlet boundary condition on one part of the boundary. On the other part of the boundary is located a Neumann boundary control. The heat equation system suffers from both a model uncertainty in the heat flow modeling and an unknown external disturbance. Our control approach is based on the design of an exponentially converging observer to estimates both the state and the unknown uncertainty. The estimated state and the estimated uncertainty are used to build a stabilizing feedback control law such that the closed-loop system is exponentially stabilized, and the external disturbance is rejected.