Pointwise Control Research Articles

Solving the optimal control problem of the parabolic PDEs in exploitation of oil

In this paper, the optimal control problem is governed by weak coupled parabolic PDEs and involves pointwise state and control constraints. We use measure theory method for solving this problem. In order to use the weak solution of problem, first problem has been transformed into measure form. This problem is reduced to a linear programming problem. Then we obtain an optimal measure which is approximated by a finite combination of atomic measures. We find piecewise-constant optimal control functions which are an approximate control for the original optimal control problem.

Semideterministic Global Optimization Method: Application to a Control Problem of the Burgers Equation

This paper has two objectives. We introduce a new global optimization algorithm reformulating optimization problems in terms of boundary-value problems. Then, we apply this algorithm to a pointwise control problem of the viscous Burgers equation, where the control weight coefficient is progressively decreased. The results are compared with those obtained with a genetic algorithm and an LM-BFGS algorithm in order to check the efficiency of our method and the necessity of using global optimization techniques.

Superconvergence Properties for Optimal Control Problems Discretized by Piecewise Linear and Discontinuous Functions

An optimal control problem for an elliptic equation is investigated with pointwise control constraints. This paper is concerned with discretization of the control by piecewise linear but discontinuous functions. The state and the adjoint state are discretized by linear finite elements. Approximations of the optimal solution of the continuous optimal control problem will be constructed by a projection of the discrete adjoint state. It is proved that these approximations have convergence order h 2.

Optimal control of the convection-diffusion equation using stabilized finite element methods

In this paper we analyze the discretization of optimal control problems governed by convection-diffusion equations which are subject to pointwise control constraints. We present a stabilization scheme which leads to improved approximate solutions even on corse meshes in the convection dominated case. Moreover, the in general different approaches “optimize-then- discretize” and “discretize-then-optimize” coincide for the proposed discretization scheme. This allows for a symmetric optimality system at the discrete level and optimal order of convergence.

On Nash Equilibria for Noncooperative Games Governed by the Burgers Equation

Existence of a Nash equilibrium in a noncooperative game governed by the one-dimensional Burgers equation, proposed in the case of pointwise controls in Ref. 1, is proved under data qualifications that guarantee the diffusion term in the Burgers’ equation to be dominant enough with respect to the uniform convexity of the payoffs.

Error Estimates for the Numerical Approximation of a Distributed Control Problem for the Steady-State Navier–Stokes Equations

We obtain error estimates for the numerical approximation of a distributed control problem governed by the stationary Navier-Stokes equations, with pointwise control constraints. We show that the $L^2$-norm of the error for the control is of order $h^2$ if the control set is not discretized, while it is of order $h$ if it is discretized by piecewise constant functions. These error estimates are obtained for local solutions of the control problem, which are nonsingular in the sense that the linearized Navier-Stokes equations around these solutions define some isomorphisms, and which satisfy a second order sufficient optimality condition. We establish a second order necessary optimality condition. The gap between the necessary and sufficient second order optimality conditions is the usual gap known for finite dimensional optimization problems.

Primal–dual active set method for control constrained optimal control of the Stokes equations

In this paper, we study the distributed and boundary optimal control of the Stokes equations in the presence of pointwise control constraints. The analysis of the problems involve existence results, as well as necessary conditions and optimality systems. A primal–dual active set strategy is studied for the numerical solution of the problems. For the distributed case, global and local convergence results for the infinite dimensional method are presented. For the boundary control case, due to the lack of regularity of the pointwise constraint multiplier, the method is applied to a discretized version of the original problem. Finally, some numerical experiments, which illustrate the performance of the method for both cases, are commented.

Lyapunov controller design for transverse vibration of a cable-linked transporter system

Modeling and Lyapunov controller design for transverse vibration of a flexible cable transporter system with arbitrarily varying lengths are presented using Hamilton's principle and Lyapunov theory. The axial velocity of the system is assumed to be unknown in the model. This is different from existing literature where the axial velocity is assumed either to be constant or prescribed. The governing equations include two coupled non-linear Partial Differential Equations (PDEs) and boundary conditions. The interactions between the cables and the slider, pulleys, and motors are included in the model. Numerical solution of the derived governing equations is obtained using Galerkin's method. Based on the Lyapunov theory, we propose two boundary controllers and one domain point-wise controller, which suppress the transverse vibration of the cables effectively while assuring the attainment of the slider goal. The proposed controllers dissipate the vibration energy and guarantee the stability of the closed-loop system in Lyapunov sense. Simulation results demonstrate the effectiveness of the proposed controllers.

An efficient computational method for the optimal control problem for the Burgers equation

Pointwise control of the viscous Burgers equation in one spatial dimension is studied with the objective of minimizing the distance between the final state function and target profile along with the energy of the control. An efficient computational method is proposed for solving such problems, which is based on special orthonormal functions that satisfy the associated boundary conditions. Employing these orthonormal functions as a basis of a modal expansion method, the solution space is limited to the smallest lower subspace that is sufficient to describe the original problem. Consequently, the Burgers equation is reduced to a set of a minimal number of ordinary nonlinear differential equations. Thus, by the modal expansion method, the optimal control of a distributed parameter system described by the Burgers equation is converted to the optimal control of lumped parameter dynamical systems in finite dimension. The time-variant control is approximated by a finite term of the Fourier series whose unknown coefficients and frequencies giving an optimal solution are sought, thereby converting the optimal control problem into a mathematical programming problem. The solution space obtained is based on control parameterization by using the Runge–Kutta method. The efficiency of the proposed method is examined using a numerical example for various target functions.

Error estimates for linear-quadratic control problems with control constraints

An abstract linear-quadratic optimal control problem with pointwise control constraints is investigated. This article is concerned with the discretization of the control by piecewise linear functions. Under the assumption that the optimal control and the optimal adjoint state are Lipschitz continuous and piecewise of class C 2, an approximation of order h 3/2, is proved for the solution of the control discretized problem with respect to the solution of the continuous one. The theoretical results are illustrated by numerical tests.

Optimal Control of the Stokes Equations: A Priori Error Analysis for Finite Element Discretization with Postprocessing

An optimal control problem for 2d and 3d Stokes equations is investigated with pointwise control constraints. This paper is concerned with the discretization of the control by piecewise constant functions. The state and the adjoint state are discretized by finite element schemes. In the paper a postprocessing strategy is suggested, which allows for significant improvement of the accuracy.

Differential Stability of Control-Constrained Optimal Control Problems for the Navier-Stokes Equations

ABSTRACT Distributed optimal control problems for the time-dependent and the stationary Navier-Stokes equations subjected to pointwise control constraints are considered. Under a coercivity condition on the Hessian of the Lagrange function, optimal solutions are shown to be directionally differentiable functions of perturbation parameters such as the Reynolds number, the desired trajectory, or the initial conditions. The derivative is characterized as the solution of an auxiliary linear-quadratic optimal control problem. Thus, it can be computed at relatively low cost. Taylor expansions of the minimum value function are provided as well.

Linear and Discontinuous Approximations for Optimal Control Problems

An optimal control problem for 2D and 3D elliptic equations is investigated with pointwise control constraints. This paper is concerned with the discretization of the control by piecewise linear but discontinuous functions. The state and the adjoint state are discretized by linear finite elements. The paper is focused on similarities and differences to piecewise constant and piecewise linear (continuous) approximation of the controls. Approximation of order h in the L ∞-norm is proved in the main result.

A Primal-Dual Active Set Method for Bilaterally Control Constrained Optimal Control of the Navier–Stokes Equations

This paper is devoted to the analysis and numerical solution of distributed optimal control of the Navier–Stokes equations in presence of bilateral pointwise control constraints. The analysis of the problem involves the proof of existence of an optimal solution, as well as the presentation of necessary and sufficient conditions for optimality. For the numerical solution of the problem we apply a primal-dual active set strategy and show global and local convergence properties of the method. Finally, some numerical experiments, which illustrate the performance of the method, are discussed.

$L^\infty$-Estimates for Approximated Optimal Control Problems

An optimal control problem for a two-dimensional elliptic equation is investigated with pointwise control constraints. This paper is concerned with discretization of the control by piecewise linear functions. The state and the adjoint state are discretized by linear finite elements. Approximation of order h in the $L^\infty$-norm is proved in the main result.

Pointwise stabilization of strings with non‐linear feedback

AbstractWe consider an initial and boundary value problem for a homogenous string subject to an internal pointwise control. The solution resulting from a non‐linear feedback is studied. We give various explicit decay estimates depending on the control position and the feedback non‐linearity. Copyright © 2004 John Wiley & Sons, Ltd.

Error Estimates for Parabolic Optimal Control Problems with Control Constraints

An optimal control problem for the 1-d heat equation is investigated with pointwise control constraints. This paper is concerned with the discretization of the control by piecewise linear functions. The connection between the solutions of the discretized problems and the continuous one is investigated. Under an additional assumption on the adjoint state an approximation order σ^3/2 is proved for uniform discretizations. In the general case it is shown that a non-uniform control discretization ensures an approximation of order σ^3/2. Numerical tests confirm the theoretical part.

Basis property and stabilization of a translating tensioned beam through a pointwise control force

Introducing a recently developed powerful method, Riesz basis method as it is called, and applying it to the stabilization of a translating tensioned beam through a pointwise control force are the two purposes of the paper. An asymptotic analysis, which is different from the one in treating the same problem in literature, is used to get a complete asymptotic expansion of the eigenvalues and eigenfunctions for the beam. The controlled system is written as an evolution equation in a state Hilbert space whose energy norm (induced by inner product) is the total energy of the beam. The spectrum-determined growth condition is obtained by showing that there is a set of generalized eigenfunctions of the system, which forms a Riesz basis for the state space. As a consequence, the stability, particularly the exponential stability of the system, is developed. The analysis used here can be extended to other distributed systems and the results significantly improve that in [1].

Superconvergence Properties of Optimal Control Problems

An optimal control problem for a two-dimensional (2-d) elliptic equation is investigated with pointwise control constraints. This paper is concerned with discretization of the control by piecewise constant functions. The state and the adjoint state are discretized by linear finite elements. Approximations of the optimal solution of the continuous optimal control problem will be constructed by a projection of the discrete adjoint state. It is proved that these approximations have convergence order h2 .

Optimal control of systems of parabolic PDEs in exploitation of oil

Optimal control problem for the exploitation of oil is investigated. The optimal control problem under consideration in this paper is governed by weak coupled parabolic PDEs and involves with pointwise state and control constraints. The properties of solution of the state equations and the continuous dependence of state functions on control functions are investigated in a suitable function space; existence of optimal solution of the optimal control problem is also proved.