Boundary Control Research Articles

Boundary and distributed optimal control for a population dynamics PDE model with discontinuous in time Galerkin FEM schemes

We consider fully discrete finite element approximations for a semilinear optimal control system of partial differential equations in two cases: for distributed and Robin boundary control. The ecological predator-prey optimal control model is approximated by conforming finite element methods mimicking the spatial part, while a discontinuous Galerkin method is used for the time discretization. We investigate the sensitivity of the solution distance from the target function, in cases with smooth and rough initial data. We employ low, and higher-order polynomials in time and space whenever proper regularity is present. The approximation schemes considered are with and without control constraints, driving efficiently the system to desired states realized using non-linear gradient methods.

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.

Examining the training design and training transfer of a boundary management training: A randomized controlled intervention study

AbstractIn our mobile working world, boundaries between work and non‐work domains are more and more blurred, which can impair professionals' recovery and well‐being. Consequently, managing work–non‐work boundaries represents an important challenge for professionals. Research suggests that boundary work tactics conveyed in boundary management interventions may promote recovery and well‐being. However, the efficacy of boundary work tactics is largely unknown, as well as theoretical mechanisms that may explain the effectiveness of boundary management interventions in regard of both training design and training transfer. Building on the social cognitive theory of self‐regulation, we develop a web‐based boundary management training. Based on the integrated training transfer and effectiveness model, we evaluate its effects on the three levels of training effectiveness: (1) perceived learning, (2) cognitions and behaviours, with boundary control and boundary creation as indicators, and (3) recovery and well‐being. Results of our randomized controlled intervention study show several expected changes in boundary creation, suggesting that drawing on the social cognitive theory of self‐regulation for training design can result in effective behaviour change. Intervention effects on recovery and well‐being are more ambiguous, hinting at the power but likewise potential limitations of boundary creation.

Passivity-based boundary control for Korteweg-de Vries-Burgers equations

This article considers the passivity-based boundary control for Korteweg-de Vries-Burgers (KdVB) equations. Both the input strict passivity (ISP) and the output strict passivity (OSP) are studied. By the Lyapunov functional method and Wirtinger's inequality, sufficient criteria are derived to establish ISP and OSP for KdVB equations with boundary disturbances. Moreover, when parameter uncertainties arise in KdVB equations, the robust ISP and OSP are also investigated and sufficient criteria are proposed to achieve the robust ISP and OSP. Simulation examples are also given to verify the effectiveness of theoretical results.

Reduced-order modeling of conjugate heat transfer in lithium-ion batteries

Reduced-order modeling (ROM) techniques involving physics are gaining importance due to their ability to generate physically meaningful results swiftly and accurately. They are finding applications in design and optimization and in digital twins where real-time results are required. The proper orthogonal decomposition (POD)-Galerkin ROM based on finite volume discretization methodology is utilized here for solving a high-dimensional system in a reduced dimensional space in which the reduced basis is spanned by the POD modes. The reduced-order equations are obtained by projecting the discretized form of the full-order model (FOM) equations onto the reduced basis which eliminates the need for boundary control techniques. The proposed methodology is demonstrated for transient conjugate heat transfer (CHT) in an electric vehicle battery. The POD modes are computed using simulation data for three Reynolds numbers and the ROM is evaluated for transient analysis of temperature from initial state to steady state for seven Reynolds numbers. The ROM predictions are fairly accurate both in the interpolated and the extrapolated regions with significant computational speed-up. Five modes are sufficient for constructing the fluid and battery temperature ROM solution. The ROM computational speed-up obtained with 5 modes is approximately 4–7.4 times compared to the FOM. The proposed methodology is generic and can be readily applied for any engineering system involving CHT.

Boundary controllability for a 1D degenerate parabolic equation with a Robin boundary condition

In this paper, we prove the null controllability of a one-dimensional degenerate parabolic equation with a weighted Robin boundary condition at the left endpoint, where the potential has a singularity. We use some results from the singular Sturm–Liouville theory to show the well-posedness of our system. We obtain a spectral decomposition of a degenerate parabolic operator with Robin conditions at the endpoints, we use Fourier–Dini expansions and the moment method introduced by Fattorini and Russell to prove the null controllability and to obtain an upper estimate of the cost of controllability. We also get a lower estimate of the cost of controllability by using a representation theorem for analytic functions of exponential type.

Fixed-time synchronization for two-dimensional coupled reaction-diffusion complex networks: Boundary conditions analysis.

This paper examines fixed-time synchronization (FxTS) for two-dimensional coupled reaction-diffusion complex networks (CRDCNs) with impulses and delay. Utilizing the Lyapunov method, a FxTS criterion is established for impulsive delayed CRDCNs. Herein, impulses encompass both synchronizing and desynchronizing variants. Subsequently, by employing a Lyapunov-Krasovskii functional, two FxTS boundary controllers are formulated for CRDCNs with Neumann and mixed boundary condition, respectively. It is observed that vanishing Dirichlet boundary contributes to the synchronization of the CRDCNs. Furthermore, this study calculates the optimal constant for the Poincaré inequality in the square domain, which is instrumental in analyzing FxTS conditions for boundary controllers. Conclusive numerical examples underscore the efficacy of the proposed theoretical findings.

Indirect Boundary Controllability of Coupled Degenerate Wave Equations

Indirect Boundary Controllability of Coupled Degenerate Wave Equations

Boundary Control Strategies for Consensus of Fractional-Order Multi-Agent Systems Based on Coupling PDE-ODEs

Boundary Control Strategies for Consensus of Fractional-Order Multi-Agent Systems Based on Coupling PDE-ODEs

Quaternary Continuous Classical Boundary Optimal Control Problem Dominating by Parabolic System

In this paper, our purpose is to study the quaternary continuous classical boundary optimal control vector problem dominated by a quaternary linear parabolic boundary value problem. Under suitable assumptions and with a given quaternary continuous classical boundary control vector, the existence theorem for a unique quaternary state vector solution to the weak form is stated and demonstrated via the method of Galerkin. Furthermore, the continuity of the Lipschitz operator between the state vector solution to the weak form for the dominating equations and the corresponding are proved. The existence of a quaternary continuous classical boundary optimal control vector is stated and demonstrated under suitable Assumptions. The mathematical formulation for the quaternary adjoint boundary value problem associated with each considered boundary value problem is obtained and the Frѐchet derivative for the objective function is derived. Finally, the necessary conditions for the optimality theorem of the problem are stated and demonstrated.

Multi-strategy improved artificial rabbit optimization algorithm based on fusion centroid and elite guidance mechanisms

The Artificial Rabbit Optimization (ARO) algorithm has been proposed as an effective metaheuristic optimization approach in recent years. However, the ARO algorithm exhibits shortcomings in certain cases, including inefficient search, slow convergence, and vulnerability to local optima. To address these issues, this paper introduces a multi-strategy improved Artificial Rabbit Optimization (IARO) algorithm. Firstly, in the enhanced search strategy, we propose integrating the centroid guidance mechanism and elite guidance mechanism with the greedy strategy to update the position during the exploration phase. Additionally, the Levy flight strategy integrated with self-learning, is employed to update the position during the development phase to improve convergence speed and prevent falling into local optima. Secondly, the algorithm incorporates a per-dimension mirror boundary control strategy to map individuals exceeding the boundary back within the boundary back inside the boundary. This boundary control strategy ensures the algorithm operates within bounds and enhances convergence speed. Finally, within the survival of the fittest strategy, an adaptive factor is introduced to gradually enhance the population's overall adaptability. This factor regulates the balance between exploration and exploitation, allowing the algorithm to fully explore the search space and improve its robustness. To substantiate the effectiveness of the proposed IARO algorithm, a rigorous and systematic verification analysis was undertaken. Comparative experiments for qualitative and quantitative analysis were conducted on three benchmark test sets, namely CEC2017, CEC2020, and CEC2022. The analysis results, including the Wilcoxon rank-sum test, consistently demonstrates that this improved algorithm outperforms ARO and other state-of-the-art optimization algorithms comprehensively. Finally, the feasibility of the IARO algorithm has been verified in seven classical constrained engineering problems.

On optimal control in a nonlinear interface problem described by hemivariational inequalities

Abstract This article is devoted to the existence of optimal controls in various control problems associated with a novel nonlinear interface problem on an unbounded domain with non-monotone set-valued transmission conditions. This interface problem involves a nonlinear monotone partial differential equation in the interior domain and the Laplacian in the exterior domain. Such a scalar interface problem models non-monotone frictional contact of elastic infinite media. The variational formulation of the interface problem leads to a hemivariational inequality (HVI), which, however, lives on the unbounded domain, and thus cannot be analyzed in a reflexive Banach space setting. Boundary integral methods lead to another HVI that is amenable to functional analytic methods using standard Sobolev spaces on the interior domain and Sobolev spaces of fractional order on the coupling boundary. Broadening the scope of this article, we consider extended real-valued HVIs augmented by convex extended real-valued functions. Under a smallness hypothesis, we provide existence and uniqueness results; moreover, we establish a stability result with respect to the extended real-valued function as a parameter. Based on the latter stability result, we prove the existence of optimal controls for four kinds of optimal control problems: distributed control on the bounded domain, boundary control, simultaneous boundary-distributed control governed by the interface problem, and control of the obstacle driven by a related bilateral obstacle interface problem.

Finite-time synchronization of delayed semi-Markov reaction–diffusion systems: An asynchronous boundary control scheme

This paper tries to study the problem of finite-time synchronization for delayed semi-Markov reaction–diffusion systems. Based on the spatial and parametric characteristics of the considered systems, a new asynchronous boundary control scheme is proposed to ensure the finite-time synchronization of the drive and response systems. In the asynchronous boundary control scheme, only an actuator should be placed at the spatial boundary, which is more easier to implement and economical than the other non-boundary control strategies. Besides, the system parameters and controller follow two asynchronous semi-Markov chains for jumping, which is more practical than obeying one semi-Markov chain. Moreover, for the considered systems, we proposes a new lemma of finite-time stability, and by employing the inequality methods and variable substitution, we derive the criterion of finite-time synchronization and a correlative corollary. Finally, a numerical example and an application example on secure communication are carried out to support the developed approach.

Boundary controllability for variable coefficients one-dimensional wave equation with interior degeneracy

Abstract In this paper, we study boundary controllability for the linear extension problem of a wave equation with space-dependent coefficients and having an internal degeneracy. For this purpose, we mainly focus on the well-posedness and the boundary null controllability of a relaxed version of the original problem, namely, to some degenerate transmission problem. The key ingredient is to derive direct and inverse inequalities for the associated homogeneous degenerate adjoint problem. By these inequalities, we deduce that the transmission problem has a unique solution by transposition and this solution is null controllable. Moreover, we give an explicit formula of the controllability time.

Event-triggered boundary control of an unstable reaction diffusion PDE with input delay

In chemical, biological, or population (epidemiological) processes the feedback action may be considerably delayed by time-consuming chemical measurements or biological tests. With such large delays on the control action in mind, and motivated by the fact that in some of these systems only piecewise-constant inputs can be applied between time instants at which measurements trigger changes in control, we consider the problem of event-triggered stabilization of 1-D reaction–diffusion PDE systems with input delay. The approach relies on reformulating the delay problem as an actuated transport PDE which cascades into the reaction–diffusion PDE, and on the emulation of backstepping control. The paper proposes a static (state-dependent) triggering condition which establishes the time instants at which the control value needs to be updated. It is shown that under the proposed event-triggered boundary control, there exists a minimal dwell-time (independent of the initial conditions) between two triggering times which allows to guarantee the well-posedness of the closed-loop system, and the exponential stability. The stability analysis is based on Input-to-State stability theory for PDEs and small-gain arguments. A simulation example is presented to validate the theoretical results.

Outlier-resistant observer-based fuzzy sampled-data boundary control for the hot strip mill cooling process

Outlier-resistant observer-based fuzzy sampled-data boundary control for the hot strip mill cooling process

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.

Real-time active constraint generation and enforcement for surgical tools using 3D detection and localisation network.

Introduction: Collaborative robots, designed to work alongside humans for manipulating end-effectors, greatly benefit from the implementation of active constraints. This process comprises the definition of a boundary, followed by the enforcement of some control algorithm when the robot tooltip interacts with the generated boundary. Contact with the constraint boundary is communicated to the human operator through various potential forms of feedback. In fields like surgical robotics, where patient safety is paramount, implementing active constraints can prevent the robot from interacting with portions of the patient anatomy that shouldn't be operated on. Despite improvements in orthopaedic surgical robots, however, there exists a gap between bulky systems with haptic feedback capabilities and miniaturised systems that only allow for boundary control, where interaction with the active constraint boundary interrupts robot functions. Generally, active constraint generation relies on optical tracking systems and preoperative imaging techniques. Methods: This paper presents a refined version of the Signature Robot, a three degrees-of-freedom, hands-on collaborative system for orthopaedic surgery. Additionally, it presents a method for generating and enforcing active constraints "on-the-fly" using our previously introduced monocular, RGB, camera-based network, SimPS-Net. The network was deployed in real-time for the purpose of boundary definition. This boundary was subsequently used for constraint enforcement testing. The robot was utilised to test two different active constraints: a safe region and a restricted region. Results: The network success rate, defined as the ratio of correct over total object localisation results, was calculated to be 54.7% ± 5.2%. In the safe region case, haptic feedback resisted tooltip manipulation beyond the active constraint boundary, with a mean distance from the boundary of 2.70 mm ± 0.37 mm and a mean exit duration of 0.76 s ± 0.11 s. For the restricted-zone constraint, the operator was successfully prevented from penetrating the boundary in 100% of attempts. Discussion: This paper showcases the viability of the proposed robotic platform and presents promising results of a versatile constraint generation and enforcement pipeline.

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.

Sliding mode control for the stabilization of fractional heat equations subject to boundary uncertainty

By adopting the sliding mode control (SMC) and the generalized Lyapunov method, the boundary feedback stabilization issue is studied for the fractional diffusion system subject to boundary control matched disturbance. The classical sliding surface and the fractional integral sliding function are constructed and sliding mode controllers are designed respectively to realize the Mittag-Leffler (M-L) stabilization of the considered system. The controller based on the newly-introduced fractional integral sliding function not only helps to relax the constraints on the disturbance but also realizes the same stabilization effect as that of the classical one. The well-posedness result of the solution is also obtained for discontinuous fractional heat equations. Besides, a numerical experiment validates the theoretical outcomes.