Boundary Control Research Articles

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>

On Boundary Control of the Transport Equation. Assigning Real Spectra and Exponential Decay

On Boundary Control of the Transport Equation. Assigning Real Spectra and Exponential Decay

Adaptive boundary control of an axially moving system with large acceleration/deceleration under the input saturation.

We present the dynamical equation model of the axially moving system, which is expressed through one partial differential equation (PDE) and two ordinary differential equations (ODEs) obtained using the extended Hamilton's principle. In the case of large acceleration/deceleration axially moving system with system parameters uncertainty and input saturation limitation, the combination of Lyapunov theory, S-curve acceleration and deceleration (Sc A/D) and adaptive control techniques adopts auxiliary systems to overcome the saturation limitations of the actuator, thus achieving the purpose of vibration suppression and improving the quality of vibration control. Sc A/D has better flexibility than that of constant speed to ensure the operator performance and diminish the force of impact by tempering the initial acceleration. The designed adaptive control law can avoid the control spillover effect and compensate the system parameters uncertainty. In practice, time-varying boundary interference and distributed disturbance exist in the system. The interference observer is used to track and eliminate the unknown disturbance of the system. The control strategy guarantees the stability of the closed-loop system and the uniform boundedness of all closed-loop states. The numerical simulation results test the effectiveness of the proposed control strategy.

A two level finite element method for Stokes constrained Dirichlet boundary control problem

In this paper we present a finite element analysis for a Dirichlet boundary control problem governed by the Stokes equation. The Dirichlet control is considered in a convex closed subset of the energy space $\mathbf{H}^1(\Omega).$ Most of the previous works on the Stokes Dirichlet boundary control problem deals with either tangential control or the case where the flux of the control is zero. This choice of the control is very particular and their choice of the formulation leads to the control with limited regularity. To overcome this difficulty, we introduce the Stokes problem with outflow condition and the control acts on the Dirichlet boundary only hence our control is more general and it has both the tangential and normal components. We prove well-posedness and discuss on the regularity of the control problem. The first-order optimality condition for the control leads to a Signorini problem. We develop a two-level finite element discretization by using $\mathbf{P}_1$ elements(on the fine mesh) for the velocity and the control variable and $P_0$ elements (on the coarse mesh) for the pressure variable. The standard energy error analysis gives $\frac{1}{2}+\frac{\delta}{2}$ order of convergence when the control is in $\mathbf{H}^{\frac{3}{2}+\delta}(\Omega)$ space. Here we have improved it to $\frac{1}{2}+\delta,$ which is optimal. Also, when the control lies in less regular space we derived optimal order of convergence up to the regularity. The theoretical results are corroborated by a variety of numerical tests.

Null boundary controllability for some biological and chemical diffusion problems

We examine the null boundary controllability of diffusion problems (with non-local boundary conditions) arising from the morphogen concentration process and the electrochemistry. Two different types of semi-periodic conditions are addressed. Also, towards further applications to biological and chemical diffusion problems, we comment on anti-semi-periodic boundary conditions. Such boundary conditions make the auxiliary spectral problems non-self-adjoint and, therefore the classical eigenfunctions expansion method does not work. The systems of eigenfunctions form a Riesz basis in the $ L_2 $ space by adding associated eigenfunctions. For the controllability, we determine the admissible classes of initial data in terms of their Fourier coefficients. Finally, we present the null boundary controllability of these problems by the reduction to the moment problem.

Feedback stabilization of the linearized Viscous Saint-Venant system by constrained Dirichlet boundary control

Feedback stabilization of the linearized Viscous Saint-Venant system by constrained Dirichlet boundary control

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.

Boundary control of a flexible beam with output constraint under saturation input

Boundary control of a flexible beam with output constraint under saturation input

Boundary control of a flexible beam with output constraint under saturation input

Boundary control of a flexible beam with output constraint under saturation input

Boundary control for prescribed-time stabilization of delayed parabolic reaction-diffusion systems

Boundary control for prescribed-time stabilization of delayed parabolic reaction-diffusion systems

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.

Turnpike properties of optimal boundary control problems with random linear hyperbolic systems

Turnpike properties of optimal boundary control problems with random linear hyperbolic systems

Approximate controllability of neutral fractional stochastic differential systems with control on the boundary

In this paper, the approximate boundary controllability results for neutral fractional stochastic differential system with fractional Brownian motion in Hilbert space is discussed. The existence mild solution of neutral fractional stochastic differential system with fractional Brownian motion is proved by using fractional calculus, compact semigroup, Schauder fixed-point theorem and stochastic analysis. The sufficient conditions for approximate boundary controllability of this system is established under the corresponding linear system is approximately controllable. Finally, an example is given to show the application of our results.

A Multi-Task Learning Method for Relative Geologic Time, Horizons, and Faults With Prior Information and Transformer

Horizon extraction and fault detection are essential in seismic interpretation and closely related to each other. Most existing methods tend to deal with these two tasks independently, and may not work well in interpreting seismic images with complex geologic structures. We propose a multi-task learning (MTL) network with two branches to extract all horizons and detect faults simultaneously by estimating a relative geologic time (RGT) map as well as computing a fault map. These two branches share training datasets, feature maps, and network parameters during the training. The RGT estimation branch, constructed with a transformer architecture, is more lightweight compared to previous CNN methods but provides a larger and structure-oriented receptive field to adaptively capture global structural information for estimating a globally optimal RGT map. The fault detection branch is a simple convolutional neural network (CNN) which merges feature maps shared by the transformer and the derivatives of the estimated RGT to compute a fault map. The fault detection branch provides boundary control for the RGT estimation branch while the later provides global constraints for the former to improve its robustness to noise. Note that our RGT estimation by globally fitting all structures in a seismic image is a volumetric horizon interpretation method with which we are able to obtain a whole volume of horizons, all at once, by simply extracting contours of the RGT map. In our method, we further enable convenient human interactions by integrating manually interpreted horizons (or horizon segments) into the network, which imposes expert knowledge on the network to estimate reasonable RGT results from seismic images with complex fault systems, unconformities, and poor data quality. Moreover, when using 3D horizons as constraints, we are able to decompose the computationaly expensive 3D RGT estimation from a seismic volume into independently parallel 2D estimations slice by slice and combine them to obtain a laterally consistent 3D result.

On the approximate boundary controllability of some partial functional integrodifferential equations with finite delay in Banach spaces

<p style='text-indent:20px;'>This work concerns the study of approximate boundary controllability for some nonlinear partial functional integrodifferential equations with finite delay arising in the modeling of materials with memory, in the framework of general Banach spaces. We give sufficient conditions that ensure the approximate controllability of the system by supposing that its linear undelayed part is approximately controllable, admits a resolvent operator in the sense of Grimmer, and by making use of the Banach fixed-point Theorem and the continuity of the resolvent operator in the uniform norm-topology. As a result, we obtain a generalization of several important results in the literature, without assuming the compactness of the resolvent operator and the uniform boundedness of the nonlinear term. An example of applications is given for illustration.</p>

Mittag-Leffler stability analysis for time-fractional hyperbolic systems with space-dependent reactivity using backstepping-based boundary control

Mittag-Leffler stability analysis for time-fractional hyperbolic systems with space-dependent reactivity using backstepping-based boundary control

A comparative study of two globally convergent numerical methods for acoustic tomography

The comparative study of two globally convergent numerical methods for acoustic tomography is carried out in two dimensions. These are the boundary control method and the quasi-reversibility method. The novelty is that in the latter a nonlinear inverse problem is reduced to a family of the linear integral equation of the first kind via the Lavrentiev approach and this reduction is used within the quasi-reversibility method. The analysis of its stability is carried out via Carleman estimates. The computational effectiveness of these methods is tested in the numerical experiments with the smooth and discontinuous coefficients to be recovered from the tomographic data.

Bipartite consensus for multi-agent networks of fractional diffusion PDEs via aperiodically intermittent boundary control.

In this paper, the exponential bipartite consensus issue is investigated for multi-agent networks, whose dynamic is characterized by fractional diffusion partial differential equations (PDEs). The main contribution is that a novel exponential convergence principle is proposed for networks of fractional PDEs via aperiodically intermittent control scheme. First, under the aperiodically intermittent control strategy, an exponential convergence principle is developed for continuously differentiable function. Second, on the basis of the proposed convergence principle and the designed intermittent boundary control protocol, the exponential bipartite consensus condition is addressed in the form of linear matrix inequalities (LMIs). Compared with the existing works, the result of the exponential intermittent consensus presented in this paper is applied to the networks of PDEs. Finally, the high-speed aerospace vehicle model is applied to verify the effectiveness of the control protocol.

Optimal Boundary Control of a Distributed Heterogeneous Vibrating System with Given States at Intermediate Times

The problem of optimal boundary control of a distributed heterogeneous vibrating system governed by the one-dimensional wave equation with piecewise constant characteristics is considered. It is assumed that each homogeneous segment is traveled by a wave over the same time. The control is performed via displacements of both ends. The cost functional is specified on the whole time interval. A constructive approach is proposed for finding an optimal control function that transfers the vibrations from an initial state through multipoint intermediate states to a terminal state over a given time interval. The results are illustrated by an example.

Global stabilization of a 1D semilinear heat equation via modal decomposition and direct Lyapunov approach

In this paper we consider state-feedback global stabilization of a semilinear 1D heat equation with a nonlinearity exhibiting a linear growth bound. We study both non-local and boundary control via a modal decomposition approach. For both cases, we suggest a direct Lyapunov method applied to the full-order closed-loop system. The nonlinear terms are compensated by using Parseval’s inequality, leading to efficient and constructive linear matrix inequality (LMI) conditions for obtaining the controller dimension and gain. For non-local control we provide sufficient conditions that guarantee global stabilization for any linear growth bound via either linear or nonlinear controller, provided the number of actuators is large enough. We prove that the nonlinear controller achieves at least the same performance as the linear one. For the case of boundary control, we employ a multi-dimensional dynamic extension, whereas in the numerical example we manage with a larger linear growth bound. The introduced direct Lyapunov approach gives tools for a variety of robust control problems for semilinear parabolic PDEs.