Pointwise Constraints Research Articles

Analysis of backward Euler projection FEM for the Landau–Lifshitz equation

Abstract The paper focuses on the analysis of the Euler projection Galerkin finite element method (FEM) for the dynamics of magnetization in ferromagnetic materials, described by the Landau–Lifshitz equation with the point-wise constraint $|{\textbf{m}}|=1$. The method is based on a simple sphere projection that projects the numerical solution onto a unit sphere at each time step, and the method has been used in many areas in the past several decades. However, error analysis for the commonly used method has not been done since the classical energy approach cannot be applied directly. In this paper we present an optimal $\textbf{L}^2$ error analysis of the backward Euler sphere projection method by using quadratic or higher order finite elements under a time step condition $\tau =O(\epsilon _0 h)$ with some small $\epsilon _0>0$. The analysis is based on more precise estimates of the extra error caused by the sphere projection in both $\textbf{L}^2$ and $\textbf{H}^1$ norms, and the classical estimate of dual norm. Numerical experiment is provided to confirm our theoretical analysis.

Minimum weight topology optimization subject to unsteady heat equation and space-time pointwise constraints—toward automatic optimal riser design in the shape casting process

Minimum weight topology optimization subject to unsteady heat equation and space-time pointwise constraints—toward automatic optimal riser design in the shape casting process

Distributed Optimal Control of the 2D Cahn–Hilliard–Oberbeck–Boussinesq System for Nonisothermal Viscous Two-Phase Flows

We analyze a distributed optimal control problem where the state equation is governed by the coupling of the two-dimensional Cahn–Hilliard and Oberbeck–Boussinesq systems modelling incompressible viscous two-phase flows with convective heat transfer. Pointwise constraints are imposed on the controls that act as external sources in the fluid and convection–diffusion equations. The objective functional is of tracking-type that consists of a weighted energy of the difference between the state and a desired target. We establish the existence of optimal controls, the differentiability of the control-to-state operator, and the necessary and sufficient optimality conditions. For initial and target data with finite energy norms, limited space–time regularity of the adjoint states arises due to convection and surface tension.

Regularity of multipliers in second-order optimality conditions for semilinear elliptic control problems

A class of semilinear elliptic optimal control problems with mixed pointwise constraints is considered. Some criteria under which Lagrange multipliers are functions in -spaces are given. Based on the criteria, we establish first- and second-order necessary optimality conditions of KKT-type as well as second-order sufficient optimality conditions for the problem.

Comfort improvement in railway vehicles via optimal control of adaptive pneumatic suspensions

ABSTRACT This paper presents an adaptive optimal control strategy focused on improving ride comfort by minimising the root mean square of accelerations measured at seven locations on the carbody floor. It requires preview knowledge of both the train speed and the standard track quality (not the deterministic track disturbances) and differs from other works in considering pointwise constraints (upper and lower secondary suspension travel limits). Adaptive control is achieved by varying the diameter of the restriction orifice which connects the pneumatic spring to the auxiliary air reservoir, which confers different dynamic stiffness and damping to the front and rear secondary suspensions. Decision maps with the front and rear optimal diameters for five track irregularity levels and six different speeds were obtained using a Monte Carlo strategy based on solving the optimal control problem for 240 stochastic realisations of time-varying track irregularities of 300 s. Results show that the adaptive optimal control provides a successful compromise between comfort and safety which overshadows the conventional passive approach: it improves comfort up to 50% with respect to stiff passive configurations and it guarantees constraint fulfilment unlike compliant passive configurations.

New regularity results and finite element error estimates for a class of parabolic optimal control problems with pointwise state constraints

We study first-order necessary optimality conditions and finite element error estimates for a class of distributed parabolic optimal control problems with pointwise state constraints. It is demonstrated that, if the bound in the state constraint and the differential operator in the governing PDE fulfil a certain compatibility assumption, then locally optimal controls satisfy a stationarity system that allows to significantly improve known regularity results for adjoint states and Lagrange multipliers in the parabolic setting. In contrast to classical approaches to first-order necessary optimality conditions for state-constrained problems, the main arguments of our analysis require neither a Slater point, nor uniform control constraints, nor differentiability of the objective function, nor a restriction of the spatial dimension. As an application of the established improved regularity properties, we derive new finite element error estimates for the dG(0) − cG(1)-discretization of a purely state-constrained linear-quadratic optimal control problem governed by the heat equation. The paper concludes with numerical experiments that confirm our theoretical findings.

UNCONDITIONALLY OPTIMAL CONVERGENCE ANALYSIS OF SECOND-ORDER BDF SCHEME FOR LANDAU-LIFSHITZ EQUATION

<abstract> The Landau-Lifshitz equation is used to describe the evolution of spin fields in continuum ferromagnets and is a highly nonlinear parabolic problem with the constraint of unit length in the point-wise sense. This paper focuses on the unconditionally optimal error estimates of a linearized second-order BDF scheme for the numerical approximations of the solution to the Landau-Lifshitz equation. Since the point-wise constraint can be deduced from the partial differential equation, we do not take into account it in designing the numerical scheme. A rigorous error analysis is done and we derive the unconditionally optimal $ {\textbf{L}}^2 $ error estimate by using the error splitting technique. Numerical result is shown to check the theoretical analysis. </abstract>

Optimal Boundary Control of Hyperbolic Balance Laws with State Constraints

In this paper we analyze the optimal control of initial boundary value problems for entropy solutions of scalar hyperbolic balance laws with pointwise state constraints. Here, we suppose that the i...

On entropic solutions to conservation laws coupled with moving bottlenecks

Moving bottlenecks in road traffic represent an interesting mathematical problem, which can be modeled via coupled PDE-ODE systems. We consider the case of a scalar conservation law modeling the evolution of vehicular traffic and an ODE with discontinuous right-hand side for the bottleneck. The bottleneck usually corresponds to a slow moving vehicle influencing the bulk traffic flow via a moving flux pointwise constraint. The definition of solutions requires a special entropy condition selecting nonclassical shocks and we prove existence of such solutions for initial data with bounded variation. Approximate solutions are constructed via the wave-front tracking method and their limit are solutions of the Cauchy problem PDE-ODE.

Combining Model-Based Design and Model-Free Policy Optimization to Learn Safe, Stabilizing Controllers

Abstract This paper introduces a framework for learning a safe, stabilizing controller for a system with unknown dynamics using model-free policy optimization algorithms. Using a nominal dynamics model, the user specifies a candidate Control Lyapunov Function (CLF) around the desired operating point, and specifies the desired safe-set using a Control Barrier Function (CBF). Using penalty methods from the optimization literature, we then develop a family of policy optimization problems which attempt to minimize control effort while satisfying the pointwise constraints used to specify the CLF and CBF. We demonstrate that when the penalty terms are scaled correctly, the optimization prioritizes the maintenance of safety over stability, and stability over optimality. We discuss how standard reinforcement learning algorithms can be applied to the problem, and validate the approach through simulation. We then illustrate how the approach can be applied to a class of hybrid models commonly used in the dynamic walking literature, and use it to learn safe, stable walking behavior over a randomly spaced sequence of stepping stones.

Second-Order Lagrange Multiplier Rules in Multiobjective Optimal Control of Semilinear Parabolic Equations

We consider multiobjective optimal control problems for semilinear parabolic systems subject to pointwise state constraints, integral state-control constraints and pointwise state-control constraints. In addition, the data of the problems need not be twice Frechet differentiable. Employing the second-order directional derivative (in the sense of Demyanov-Pevnyi) for the involved functions, we establish necessary optimality conditions, via second-order Lagrange multiplier rules of Fritz-John type, for local weak Pareto solutions of the problems.

Second-order optimality conditions and regularity of Lagrange multipliers for mixed optimal control problems

This paper deals with second-order optimality conditions and regularity of Lagrange multipliers for a class of optimal control problems governed by semilinear elliptic equations with mixed pointwise constraints in which controls act both in the domain and on the boundary. We give some criteria under which the optimality conditions are of KKT type and the multipliers are of $$L^p$$ -spaces. Moreover, we show that the multipliers are Lipschitz continuous functions.

Regularity of solutions to a distributed and boundary optimal control problem governed by semilinear elliptic equations

This paper deals with the existence and regularity of solutions to an optimal control problem governed by semilinear elliptic equations with mixed pointwise constraints in which the controls act in the domain and on the boundary. We give some criteria under which the optimal solutions do exist and are Hölder continuous.

Robust semi-supervised nonnegative matrix factorization for image clustering

Nonnegative matrix factorization (NMF) is a powerful dimension reduction method, and has received increasing attention in various practical applications. However, most traditional NMF based algorithms are sensitive to noisy data, or fail to fully utilize the limited supervised information. In this paper, a novel robust semi-supervised NMF method, namely correntropy based semi-supervised NMF (CSNMF), is proposed to solve these issues. Specifically, CSNMF adopts a correntropy based loss function instead of the squared Euclidean distance (SED) in constrained NMF to suppress the influence of non-Gaussian noise or outliers contaminated in real world data, and simultaneously uses two types of supervised information, i.e., the pointwise and pairwise constraints, to obtain the discriminative data representation. The proposed method is analyzed in terms of convergence, robustness and computational complexity. The relationships between CSNMF and several previous NMF based methods are also discussed. Extensive experimental results show the effectiveness and robustness of CSNMF in image clustering tasks, compared with several state-of-the-art methods.

A Lagrange multiplier method for semilinear elliptic state constrained optimal control problems

In this paper we apply an augmented Lagrange method to a class of semilinear elliptic optimal control problems with pointwise state constraints. We show strong convergence of subsequences of the primal variables to a local solution of the original problem as well as weak convergence of the adjoint states and weak-* convergence of the multipliers associated to the state constraint. Moreover, we show existence of stationary points in arbitrary small neighborhoods of local solutions of the original problem. Additionally, various numerical results are presented.

Pointwise‐constrained optimal control of a semiactive vehicle suspension

SummaryThis article presents an optimal control strategy (OCS) for semiactive vehicle suspensions with road profile sensors. The suspension is modeled as a quarter‐car model with a magnetorheological (MR) damper. The OCS main objective is to minimize the fourth‐power acceleration of the sprung mass. In addition, three pointwise constraints of the model are taken into account when the optimal control problem is solved: suspension travel limits (upper and lower) and tyre vertical force. In order to deal with a large number of constraints, we implement the gradient optimization method based on the method of moving asymptotes routine, which shows very good performance reaching optimal controls while satisfying the constrains. The solution has been compared with two passive MR damper configurations (low and high damping) as well as Skyhook and Balance control strategies for three different road inputs. Results show that OCS fulfills the constraints and reduces the sprung mass acceleration peak and the root‐mean‐quad acceleration up to 59%, in comparison to passive strategies.

A frequency‐constrained geometric Pontryagin maximum principle on matrix Lie groups

AbstractWe present a geometric discrete‐time Pontryagin maximum principle (PMP) on matrix Lie groups that incorporates frequency constraints on the control trajectories in addition to pointwise constraints on the states and control actions directly at the stage of the problem formulation. This PMP gives first‐order necessary conditions for optimality and leads to two‐point boundary value problems that may be solved by numerical techniques to arrive at optimal trajectories. We demonstrate our theoretical results with numerical simulations on the optimal trajectory generation of a wheeled inverted pendulum and an attitude control problem of a spacecraft on the Lie group SO(3).

A One Dimensional Elliptic Distributed Optimal Control Problem with Pointwise Derivative Constraints

We consider a one dimensional elliptic distributed optimal control problem with pointwise constraints on the derivative of the state. By exploiting the variational inequality satisfied by the derivative of the optimal state, we obtain higher regularity for the optimal state under appropriate assumptions on the data. We also solve the optimal control problem as a fourth order variational inequality by a C 1 finite element method, and present the error analysis together with numerical results.

Error estimates for the FEM approximation of optimal sparse control of elliptic equations with pointwise state constraints and finite‐dimensional control space

SummaryIn this work, we derive an a priori error estimate of order for the finite element approximation of a sparse optimal control problem governed by an elliptic equation, which is controlled in a finite dimensional space. Furthermore, box‐constrains on the control are considered and finitely many pointwise state‐constrains are imposed on specific points in the domain. With this choice for the control space, the achieved order of approximation for the optimal control is optimal, in the sense that the order of the error for the optimal control is of the same order of the approximation for the state equation.

$$L^\infty $$-Stability of a Parametric Optimal Control Problem Governed by Semilinear Elliptic Equations

This paper studies local stability of a parametric optimal control problem governed by semilinear elliptic equations with mixed pointwise constraints. We show that if the unperturbed problem satisfies the strictly nonnegative second-order optimality conditions, then the solution map is upper Holder continuous in $$L^\infty $$-norm of control variable.