Boundary Control Problem Research Articles

Adaptive vibration control for stabilisation of the Euler–Bernoulli beam with an unknown payload

In this paper, an adaptive boundary controller is designed to solve the boundary control problem of Euler–Bernoulli beams with an unknown payload. Therefore, a boundary controller with an adaptive law is designed to compensate for the parameter uncertainty of the system. Partial Differential Equations (PDEs) are used to describe the Euler–Bernoulli beam system. The well-posedness of the system under the action of an adaptive boundary controller is proved by using the linear operator semigroup method. Meanwhile, the asymptotic stability of the closed-loop system is derived from the extended Krasovskii–LaSalle invariance principle. The effectiveness and superiority of the proposed method are illustrated by simulation in comparison with the existing results.

Boundary Value and Control Problems for the Stationary Magnetic Hydrodynamic Equations of Heat Conducting Fluid with Variable Coefficients

Boundary Value and Control Problems for the Stationary Magnetic Hydrodynamic Equations of Heat Conducting Fluid with Variable Coefficients

Neural Operator Approximations for Boundary Stabilization of Cascaded Parabolic PDEs

ABSTRACTThis article proposes a novel method to accelerate the boundary feedback control design of cascaded parabolic difference equations (PDEs) through DeepONet. The backstepping method has been widely used in boundary control problems of PDE systems, but solving the backstepping kernel function can be time‐consuming. To address this, a neural operator (NO) learning scheme is leveraged for accelerating the control design of cascaded parabolic PDEs. DeepONet, a class of deep neural networks designed for approximating nonlinear operators, has shown potential for approximating PDE backstepping designs in recent studies. Specifically, we focus on approximating gain kernel PDEs for two cascaded parabolic PDEs. We utilize neural operators to map only two kernel functions, while the other two are computed using the analytical solution, thus simplifying the training process. We establish the continuity and boundedness of the kernels, and demonstrate the existence of arbitrarily close DeepONet approximations to the kernel PDEs. Furthermore, we demonstrate that the DeepONet approximation gain kernels ensure stability when replacing the exact backstepping gain kernels. Notably, DeepONet operator exhibits computation speeds two orders of magnitude faster than PDE solvers for such gain functions, and their theoretically proven stabilizing capability is validated through simulations.

Integral BLF-Based Adaptive Dynamic Event-Triggered Boundary Control for a Flexible Riser System.

For the flexible riser systems modeled with partial differential equations (PDEs), this article explores the boundary control problem in depth for the first time using a dynamic event-triggered mechanism (DETM). Given the intrinsic time-space coupling characteristic inherent in PDE computations, implementing a state-dependent DETM for PDE-based flexible risers presents a significant challenge. To overcome this difficulty, a novel dynamic event-triggered control method is introduced for flexible riser systems, focusing on optimizing available control inputs. In order to save computational costs from the controller to the actuator, a dynamic event-triggered adaptive boundary controller is designed to effectively reduce boundary position vibrations. Additionally, considering external disturbances, an adaptive bounded compensation term is incorporated to counteract the influence of external disturbances on the system. Addressing boundary position constraints, a new integral barrier Lyapunov function (iBLF) tailored specifically for flexible riser systems is introduced, thereby alleviating conservatism in the controller design of flexible risers modeled by PDEs. At last, the validity of the proposed method is demonstrated through a simulation example.

Boundary control problem for a parabolic equation with involution

In this paper, we consider a boundary control problem for a parabolic equation with involution in a bounded one-dimensional domain. On the part of the border of the considered domain, the value of the solution with control function is given. Restrictions on the control are given in such a way that the average value of the solution in the considered domain gets a given value. The problem given by the method of separation of variables is reduced to the Volterra integral equation of the first kind. The existence of the control function was proved by the Laplace transform method.

Boundary control problem for the reaction– advection– diffusion equation with a modulus discontinuity of advection

Boundary control problem for the reaction– advection– diffusion equation with a modulus discontinuity of advection

Boundary control problem for the reaction-advection-diffusion equation with a modulus discontinuity of advection

Рассмотрена периодическая задача для сингулярно возмущенного параболического уравнения реакция-диффузия-адвекция типа Бюргерса с модульной адвекцией, имеющая решение вида движущегося фронта. Сформулированы условия существования такого решения, построено его асимптотическое приближение. Поставлена задача управления, в которой требуемый закон движения фронта достигается за счет специальным образом подобранного краевого условия. Построено асимптотическое решение задачи граничного управления. При помощи асимптотического метода дифференциальных неравенств получены оценки точности решения задачи управления. Предложен оригинальный численный алгоритм решения сингулярно возмущенных задач с модульной адвекцией.

Boundary Value and Control Problems for the Stationary Heat Transfer Model with Variable Coefficients

Boundary Value and Control Problems for the Stationary Heat Transfer Model with Variable Coefficients

Exponential Stability of the Numerical Solution of a Hyperbolic System with Nonlocal Characteristic Velocities

In this paper, we investigate the problem of the exponential stability of a stationary solution for a hyperbolic system with nonlocal characteristic velocities and measurement error. The formulation of the initial boundary value problem of boundary control for the specified hyperbolic system is given. A difference scheme is constructed for the numerical solution of the considered initial boundary value problem. The definition of the exponential stability of the numerical solution in ℓ2-norm with respect to a discrete perturbation of the equilibrium state of the initial boundary value difference problem is given. A discrete Lyapunov function for a numerical solution is constructed, and a theorem on the exponential stability of a stationary solution of the initial boundary value difference problem in ℓ2-norm with respect to a discrete perturbation is proved.

Optimal boundary control for a competitive population system with size structure and time delay in a polluted environment

In this paper, we consider the optimal boundary control problem of a two-species competitive system with time delay and size structure in a polluted environment. First, the well posedness of the system is studied by using the characteristic line method and the fixed point principle. Second, the necessary conditions for optimal boundary control are obtained by conjugate system and normal cone property. Finally, the existence and uniqueness of the optimal strategy are proved by Ekeland variational principle.

Application of coastal acoustic tomography: calibration of open boundary conditions on a numerical ocean model for tidal currents

Coastal acoustic tomography (CAT), which measures path-averaged currents from reciprocal acoustic transmission experiments and reconstructs velocity fields from the multiple path-averaged current data, is useful for monitoring tidal currents in coastal shallow water, especially if data assimilation is employed. Previous CAT data assimilation studies have focused on state estimation problems, i.e., the reconstruction of tidal currents and following dynamical discussion. In this study, we investigate the use of path-averaged currents in a boundary control problem. Specifically, we aim to use the observed path-averaged currents to determine the parameters of a numerical ocean model, which were tidal amplitudes and phases as the open boundary conditions in this study. We investigate two methods: using the ensemble Kalman filter (EnKF) results and a linearization approach called model Green’s function method. Both calibration methods decreased the amplitudes of tidal constituents at the open boundaries. We compare the model performance between the model predictions with and without the calibration of the open boundary conditions. The model predictions with the calibrated open boundary conditions improved the agreement with the observed path-averaged current. We also implemented the sequential updates of EnKF with the two calibrated open boundary conditions. The EnKF results with the independently calibrated two open boundary conditions improved the agreement with the comparison data obtained by acoustic Doppler current profiler measurement compared with the original EnKF result with the initial open boundary conditions.

Extremal Problems for Hyperbolic Systems with Boundary Conditions Involving Integral Time Lags

Extremal problems for integral time lag hyperbolic systems are presented. The optimal boundary control problems for hyperbolic systems in which integral time lags appear in the Neumann boundary conditions are solved. Such systems constitute, in a linear approximation, a universal mathematical model for many processes in which transmission signals at a certain distance with electric, hydraulic and other long lines take place. The time horizon is fixed. Making use of Dubovicki-Milyutin scheme, necessary and sufficient conditions of optimality for the Neumann problem with the quadratic performance functionals and constrained control are derived.

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.

Boundary Value and Control Problems for Mass Transfer Equations with Variable Coefficients

Boundary Value and Control Problems for Mass Transfer Equations with Variable Coefficients

Prescribing transport equation solution's decay via multiplicity manifold and autoregressive boundary control

AbstractThis article addresses the boundary control problem of the transport equation. Namely, we propose a control method, which is merely a delayed output feedback relying on a partial pole placement idea, that consists in assigning an appropriate exponential decay rate to the closed‐loop system's solution. The proposed control structure appearing in the transport boundary, which has proven its effectiveness in controlling finite dimensional systems, consists of an autoregressive relation linking the transport equation's input and output. The obtained result provides an analytical lower bound for the solution's exponential decay. The contribution is concluded with a numeric discussion on the concrete implementation of the method with a special emphasis on the delay seen as a control parameter.

Error analysis for the finite element approximations of Dirichlet parabolic boundary control problem with measure data

This article investigates the convergence analysis of finite element method for Dirichlet boundary control problem governed by parabolic equation with measure data. The presence of measure data in the source term and Dirichlet boundary control lead the solution of the state equation to have low regularity, which creates a challenging task for the error analysis. We prove the existence, uniqueness and regularity results of the solution to the control problem. For the discretization of the state and adjoint variables, we employ continuous piecewise linear polynomials while the control variable is approximated by piecewise constant functions. The backward Euler technique provides the foundation for the temporal discretization. For the completely discrete optimal control problem, a priori error bounds for the state variable in the L2(0,T;L2(Ω))-norm and for the control variable in the L2(0,T;L2(∂Ω))-norm are derived. Numerical experiment is performed to validate our theoretical analysis.

Numerical investigation for a boundary optimal control of reaction-advection-diffusion equation using penalization technique

<p>The objective of this paper is to describe the problem of boundary optimal control of the reaction-advection-diffusion equation for not very regular Dirichlet data and enumerate its qualitative properties. We check that the state equation is well-posed and we introduce the penalization technique to take into account the boundary condition of the Dirichlet type. Then, we consider the corresponding optimal boundary control problem and give the optimality conditions. Finally, we conducted a numerical investigation of the convergence of the solution of the penalized problem to the solution of the non-penalized one when the penalty parameter tends to zero in regular and non-regular domains.</p>

On the boundary control problem associated with a fourth order parabolic equation in a two-dimensional domain

On the boundary control problem associated with a fourth order parabolic equation in a two-dimensional domain

Null controllability of the advection-dispersion equation with respect to the dispersion parameter using the Fokas method

We investigate the null controllability of the linear advection-dispersion (LAD) equation posed on a finite interval. To this end, we employ the Fokas method to determine boundary controls and the controlled solutions concerning varying dispersion parameters and final times. By employing the Fokas method for the LAD equation, which provides an explicit integral representation of the solution, we enable a comprehensive analysis of null controllability. Consequently, we derive a boundary control input that effectively steers the solution from its initial state to the zero state within a given final time. The proposed approach offers a straightforward means to achieve the null controllability for the LAD equation, even as the dispersion parameter and the final time vary. Importantly, the effectiveness of this approach can be extended to various types of boundary control problems, thereby enhancing its broader applicability and potential impact.

Optimal Dirichlet boundary control by Fourier neural operators applied to nonlinear optics

We present an approach for solving optimal boundary control problems of nonlinear optics by using deep learning. For computing high resolution approximations of the solution to the nonlinear wave model, we propose higher order space-time finite element methods in combination with collocation techniques. Thereby, Cl-regularity in time of the global discrete solution is ensured. The simulation data is used to train solution operators that effectively leverage the higher regularity of the data. The solution operator is represented by Fourier Neural Operators and can be used as the forward solver in the optimal Dirichlet boundary control problem.The proposed algorithm is implemented and tested on high-performance computing platforms, with a focus on efficiency and scalability. The effectiveness of the approach is demonstrated on the problem of generating Terahertz radiation in periodically poled Lithium Niobate. The neural network is used as the solver in the optimal control setting to optimize the parametrization of the optical input pulse and maximize the yield of 0.3THz-frequency radiation.We exploit the periodic layering of the crystal to design the neural networks. The networks are trained to learn the propagation through one period of the layers. The recursive application of the network onto itself yields an approximation to the full problem. Our results indicate that the proposed method can achieve a speedup by a factor of 360 compared to classical methods. A comparison of our results to experimental data shows the potential to revolutionize the way we approach optimization problems in nonlinear optics.