Pointwise Control Research Articles

Moreau-Yosida regularization to optimal control of the monodomain model with pointwise control and state constraints in cardiac electrophysiology

In this work, we study the optimal control problem of a coupled reaction-diffusion system, which is a monodomain model in cardiac electrophysiology with pointwise bilateral control and state constraints. We adopt the Moreau-Yosida regularization as a penalization technique to deal with the state constraints. The regularized problem’s first-order optimality condition is derived. In addition, sufficient second-order optimality condition is derived for the regularized problem using the virtual control concept by proving equivalence between Moreau-Yosida regularization and the virtual control concept. The convergence of optimal controls of the regularized problems to the optimal control of the original problem is proved. Moreover, the semi-smooth Newton method for numerically finding the optimal solution to the regularization problem is presented. Finally, numerical experiments are conducted, and the results allow us to understand the extinction of the wave excitation in cardiac defibrillation in the presence of both control and state constraints.

Observer-based quantized control for networked interconnected PDE systems with actuator failures

This paper proposes an observer-based quantized controller for parabolic partial differential equation systems interconnected by a nonlinear coupling protocol. First, a Markov jump model is introduced to describe various randomly occurring actuator failures, and an observer-based pointwise controller is designed under the averaged measurement scheme. Furthermore, taking into account the limitation of network communication resources, a quantization method is adopted to relieve bandwidth pressure. In addition, stability conditions of the closed-loop system with ℋ∞ disturbance attenuation performance are derived by utilizing appropriate Lyapunov functional and inequality techniques. Ultimately, the proposed method is applied to the Fisher equation to verify its feasibility and effectiveness.

Optimal control of sweeping processes in unmanned surface vehicle and nanoparticle modelling

ABSTRACT This paper addresses novel applications to practical modelling of the newly developed theory of necessary optimality conditions in controlled sweeping/Moreau processes with free time and pointwise control and state constraints. Problems of this type appear, in particular, in dynamical models dealing with unmanned surface vehicles (USVs) and nanoparticles. We formulate optimal control problems for a general class of such dynamical systems and show that the developed necessary optimality conditions for constrained free-time controlled sweeping processes lead us to designing efficient procedures to solve practical models of this class. Moreover, the paper contains numerical calculations of optimal solutions to marine USVs and nanoparticle models in specific situations. Overall, this study contributes to the advancement of optimal control theory for constrained sweeping processes and its practical applications in the fields of marine USVs and nanoparticle modelling.

Optimality Conditions for Optimal Control of the Monodomain Model with Pointwise Control and State Constraints

Optimality Conditions for Optimal Control of the Monodomain Model with Pointwise Control and State Constraints

Virtual element discretization method to optimal control problem governed by Stokes equations with pointwise control constraint on arbitrary polygonal meshes

In this paper we investigate virtual element discretization of optimal control problem governed by Stokes equations. Based on the strategy of first-discretize-then-optimize we build up the virtual element discrete scheme and derive the discrete first order optimality system. A priori error estimates for the state, adjoint state and control variables in L2 and H1 norms are derived. Numerical experiments are presented to verify the theoretical findings.

New second order sufficient optimality conditions for state constrained parabolic control problems

We study a control problem governed by a semilinear parabolic equation with pointwise control and state constraints imposed at every point of the space-time cylinder. We obtain second order sufficient optimality conditions for local optimality in the sense of L 2 ( Q ) . Our results are valid for spatial domains of dimension less than or equal to three.

Hysteretic quantised security control for PDE systems under replay attacks

ABSTRACT This article designs a quantised security controller for partial differential equation (PDE) systems under replay attacks. Initially, the Takagi – Sugeno fuzzy model is utilised to represent the considered nonlinear system and combines pointwise measurement and pointwise control methods to save the whole system’s costs. Moreover, to reduce network transmission pressure and simulate the network, hysteretic quantiser and replay attacks are considered. Furthermore, based on the Lyapunov direct method and the inequality technique, sufficient conditions to assure the stability of the closed-loop system are achieved. Ultimately, the effectiveness and superiority of the submitted control strategy are confirmed through numerical examples and comparative analysis.

On the сontrollability, observability, and optimization of discrete nonstationary linear Volterra systems

In this article, we study discrete nonstationary linear dynamic systems of Volterra type. An essential feature of such kind of systems is that their current states depend on the previous states of this system. The formula Cauchy, which gives us the solution of linear Volterra systems with the control inputs, is obtained. The necessary and sufficient conditions of the pointwise controllability, pointwise output controllability, and observability are proven. Also the linear-quadratic optimization problem for the nonstationary Volterra control systems is studied.

Arbitrary aperture synthesis with nonlocal leaky-wave metasurface antennas

The emergence of new technological needs in 5 G/6 G networking and broadband satellite internet access amplifies the demand for innovative wireless communication hardware, including high-performance low-profile transceivers. In this context, antennas based on metasurfaces – artificial surfaces engineered to manipulate electromagnetic waves at will – represent highly promising solutions. In this article, we introduce leaky-wave metasurface antennas operating at micro/millimeter-wave frequencies that are designed using the principles of quasi-bound states in the continuum, exploiting judiciously tailored spatial symmetries that enable fully customized radiation. Specifically, we unveil additional degrees of control over leaky-wave radiation by demonstrating pointwise control of the amplitude, phase and polarization state of the metasurface aperture fields by carefully breaking relevant symmetries with tailored perturbations. We design and experimentally demonstrate metasurface antenna prototypes showcasing a variety of functionalities advancing capabilities in wireless communications, including single-input multi-output and multi-input multi-output near-field focusing, as well as far-field beam shaping.

Hysteresis quantified control for switched reaction–diffusion systems and its application

This article addresses the exponential input-to-state stabilization problem for switched reaction–diffusion systems, in which the systems’ mode jumping complies with the persistent dwell-time switching mechanism. On the basis of point measurement, a novel pointwise controller is designed to reduce the amount of sensors and actuators. Then, a hysteresis quantizer with an adjustable parameter is employed to balance the quantitative effect and system’s performance, which can improve the bandwidth utilization of the network, simultaneously. Finally, the effectiveness of the proposed approach is demonstrated by an application of the temperature control of power semiconductor chips.

Semismoothness for Solution Operators of Obstacle-Type Variational Inequalities with Applications in Optimal Control

We prove that solution operators of elliptic obstacle-type variational inequalities (or, more generally, locally Lipschitz continuous functions possessing certain pointwise-a.e. convexity properties) are Newton differentiable when considered as maps between suitable Lebesgue spaces and equipped with the strong-weak Bouligand differential as a generalized set-valued derivative. It is shown that this Newton differentiability allows to solve optimal control problems with -cost terms and one-sided pointwise control constraints by means of a semismooth Newton method. The superlinear convergence of the resulting algorithm is proved in the infinite-dimensional setting, and its mesh independence is demonstrated in numerical experiments. We expect that the findings of this paper are also helpful for the design of numerical solution procedures for quasi-variational inequalities and the optimal control of obstacle-type variational problems.

On the Strong Subregularity of the Optimality Mapping in an Optimal Control Problem with Pointwise Inequality Control Constraints

This paper presents sufficient conditions for strong metric subregularity (SMsR) of the optimality mapping associated with the local Pontryagin maximum principle for Mayer-type optimal control problems with pointwise control constraints given by a finite number of inequalities G_j(u)le 0. It is assumed that all data are twice smooth, and that at each feasible point the gradients G_j'(u) of the active constraints are linearly independent. The main result is that the second-order sufficient optimality condition for a weak local minimum is also sufficient for a version of the SMSR property, which involves two norms in the control space in order to deal with the so-called two-norm-discrepancy.

Energy Estimates for the Tracefree Curvature of Willmore Surfaces and Applications

We prove an $$\varepsilon $$ -regularity result for the tracefree curvature of a Willmore surface with bounded second fundamental form. For such a surface, we obtain a pointwise control of the tracefree second fundamental form from a small control of its $$L^2$$ -norm. Several applications are investigated. Notably, we derive a gap statement for surfaces of the aforementioned type. We further apply our results to deduce regularity results for conformal minimal spacelike immersions into the de Sitter space $${\mathbb {S}}^{4,1}$$ .

Distributed robust output feedback control of uncertain fractional-order reaction-diffusion systems

This article focuses on the problem of distributed robust stabilization for a class of uncertain fractional-order reaction–diffusion systems (FRDS) by using two distributed output feedback control strategies: pointwise control based on collocated pointwise measurements and piecewise control based on collocated piecewise measurements. First, the corresponding distributed robust feedback controllers are designed based on the two measurement methods. Then, based on the Lyapunov direct method, Mittag-Leffler (M-L) function, and linear matrix inequality technique (LMI), sufficient conditions for the Mittag-Leffler stability of the closed-loop system are obtained in terms of linear matrix inequalities, respectively. Finally, numerical studies are given to verify the effectiveness of the robust distributed controllers designed in this article.

Fuzzy Event-Triggered Control for PDE Systems With Pointwise Measurements Based on Relaxed Lyapunov–Krasovskii Functionals

In this article, an event-triggered control problem for partial differential equation systems with pointwise measurements is investigated via relaxed Lyapunov–Krasovskii functionals. First, the Takagi–Sugeno fuzzy model is introduced to describe the nonlinear systems and a fuzzy event-triggered pointwise controller is proposed with pointwise measurements, which can make a tradeoff between the system’s performance and implementation complexity subject to limited transmission bandwidth. Second, some relaxed conditions are established to ensure the closed-loop system’s stability by using the Lyapunov method and inequality techniques. Finally, two simulation examples are provided to demonstrate the effectiveness and practicability of the designed controller.

Convergence of adaptive nonconforming finite element method for Stokes optimal control problems

This paper aims at proving the convergence and quasi-optimality of an adaptive nonconforming finite element method for Stokes distributed control problems with pointwise control constraints. Nonconforming P1/P0 pair (Crouzeix–Raviart elements) and variational discretization are used to approximate the state equation and the control variable, respectively. A posteriori error estimates with upper and lower bounds are first derived for the state and adjoint variables. Then we prove the contraction property for the sum of the energy error of the state and adjoint state and the scaled error estimator on two consecutive adaptive meshes. The resulting linear convergence is finally used to show the quasi-optimal convergence rate of the adaptive algorithm. Additionally, some numerical results are provided to support our theoretical analysis.

State estimation-based robust optimal control of influenza epidemics in an interactive human society

This paper presents a state estimation-based robust optimal control strategy for influenza epidemics in an interactive human society in the presence of modeling uncertainties. Interactive society is influenced by random entrance of individuals from other human societies whose effects can be modeled as a non-Gaussian noise. Since only the number of exposed and infected humans can be measured, the states of the influenza epidemics are first estimated by an extended maximum correntropy Kalman filter (EMCKF) to provide a robust state estimation in the presence of the non-Gaussian noise. An online quadratic program (QP) optimization is then synthesized subject to a robust control Lyapunov function (RCLF) to minimize susceptible and infected humans, while minimizing and bounding the rates of vaccination and antiviral treatment. The main contribution of this work is twofold. First, the joint QP-RCLF-EMCKF strategy meets multiple design specifications such as state estimation, tracking, pointwise control optimality, and robustness to parameter uncertainty and state estimation errors that have not been achieved simultaneously in previous studies. Second, the uniform ultimate boundedness (UUB)/convergence of all error trajectories is guaranteed by using a Lyapunov stability argument. Simulation results show that the proposed approach achieves appropriate tracking and state estimation performance with good robustness.

Controllability Problems for the Heat Equation in a Half-Plane Controlled by the Dirichlet Boundary Condition with a Point-Wise Control

Controllability Problems for the Heat Equation in a Half-Plane Controlled by the Dirichlet Boundary Condition with a Point-Wise Control

Iterative learning control for linear parabolic distributed parameter systems with multiple point sensors

AbstractThis article presents an iterative learning control (ILC) approach for linear parabolic distributed parameter systems with multiple actuators and multiple sensors. The distribution functions of actuators and sensors are chosen as delta function to produce pointwise control and pointwise measurement. A P‐type ILC law is proposed based on the iterative inputs and outputs to ensure the iterative process of the system is convergent under the ILC law. By utilizing integration by parts, triangle inequality, property of delta function, and Gronwall lemma, a sufficient condition based on an inequality constraint for the convergence analysis of the track error system is presented. Finally, the effectiveness of the proposed design method is verified by numerical simulation results.

Locally positive stabilization of infinite-dimensional linear systems by state feedback

An overview is given of the locally positive stabilization problem for positive infinite-dimensional linear systems with a bounded control operator. The impossibility of solving this problem by using a nonnegative input is established. Two methods for solving the problem by means of state feedback, namely spectral decomposition and control invariance, are described. The results are illustrated by means of a perturbed diffusion equation with Dirichlet boundary conditions and a diffusion equation with Neumann boundary conditions and pointwise control.