Solution of the state noise dependent optimal control problem in terms of Lyapunov iterations

1998 ◽  
Author(s):  
Z. Gajic ◽  
R. Losada
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
The State

scholarly journals Ana posteriorierror analysis for an optimal control problem with point sources

2018 ◽  
Vol 52 (5) ◽  
pp. 1617-1650 ◽  
Author(s):  
Alejandro Allendes ◽  
Enrique Otárola ◽  
Richard Rankin ◽  
Abner J. Salgado
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Weighted Sobolev Spaces ◽  
Control Variable ◽  
The State ◽  
Point Sources ◽  
Adjoint Equations ◽  
Error Estimator ◽  
A Posteriori

We propose and analyze a reliable and efficienta posteriorierror estimator for a control-constrained linear-quadratic optimal control problem involving Dirac measures; the control variable corresponds to the amplitude of forces modeled as point sources. The proposeda posteriorierror estimator is defined as the sum of two contributions, which are associated with the state and adjoint equations. The estimator associated with the state equation is based on Muckenhoupt weighted Sobolev spaces, while the one associated with the adjoint is in the maximum norm and allows for unbounded right hand sides. The analysis is valid for two and three-dimensional domains. On the basis of the deviseda posteriorierror estimator, we design a simple adaptive strategy that yields optimal rates of convergence for the numerical examples that we perform.

Optimal impulsive control for advertising strategy problems based on a gradient-based PSO algorithm

2018 ◽  
Vol 41 (8) ◽  
pp. 2280-2292 ◽  
Author(s):  
Xiang Wu ◽  
Jinxing Lin ◽  
Kanjian Zhang ◽  
Ming Cheng
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Inequality Constraints ◽  
Pso Algorithm ◽  
The State ◽  
Optimization Techniques ◽  
Advertising Strategy ◽  
Gradient Based ◽  
State Inequality Constraints

This paper considers an optimal advertising strategy problem. This is an important problem in marketing investment for new products in a free market. The main contributions of this paper are as follows. First, the problem is formulated as an optimal control problem of switched impulsive systems with the state inequality constraints, which is different from the existing nonlinear system models. As the complexity of such constraints and the switching instants are unknown, it is difficult to solve this problem by using conventional optimization techniques. To overcome this difficulty, by applying the penalty function, all the state inequality constraints are first written as non-differentiable penalty terms and imposed into the cost function. Then, the penalty terms are smoothed by using a novel smooth function, leading to a smooth optimal control problem with no state inequality constraints, and an improved gradient-based particle swarm optimization (PSO) algorithm is proposed for solving this problem. Error analysis results show that if the adjustable parameter is sufficiently small, the solution of the smooth optimal control problem is approximately equal to the original problem. Finally, a switched impulsive system for beer sales is established to illustrate the effectiveness of the developed algorithm.

scholarly journals Error estimates of finite volume method for Stokes optimal control problem

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Lin Lan ◽  
Ri-hui Chen ◽  
Xiao-dong Wang ◽  
Chen-xia Ma ◽  
Hao-nan Fu
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Error Estimates ◽  
Finite Volume ◽  
Stokes Equations ◽  
A Priori ◽  
The State ◽  
Element Approximation ◽  
Norm Error

AbstractIn this paper, we discuss a priori error estimates for the finite volume element approximation of optimal control problem governed by Stokes equations. Under some reasonable assumptions, we obtain optimal $L^{2}$ L 2 -norm error estimates. The approximate orders for the state, costate, and control variables are $O(h^{2})$ O ( h 2 ) in the sense of $L^{2}$ L 2 -norm. Furthermore, we derive $H^{1}$ H 1 -norm error estimates for the state and costate variables. Finally, we give some conclusions and future works.

Error estimates for the numerical approximation of a distributed optimal control problem governed by the von Kármán equations

2018 ◽  
Vol 52 (3) ◽  
pp. 1137-1172
Author(s):  
Gouranga Mallik ◽  
Neela Nataraj ◽  
Jean-Pierre Raymond
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Error Estimates ◽  
Numerical Approximation ◽  
The State ◽  
Distributed Optimal Control ◽  
Von Karman ◽  
Von Kármán Equations ◽  
Von Kármán

In this paper, we discuss the numerical approximation of a distributed optimal control problem governed by the von Kármán equations, defined in polygonal domains with point-wise control constraints. Conforming finite elements are employed to discretize the state and adjoint variables. The control is discretized using piece-wise constant approximations. A priori error estimates are derived for the state, adjoint and control variables. Numerical results that justify the theoretical results are presented.

scholarly journals An Approximation of Solutions for the Problem with Quasistatic Contact in the Case of Dry Friction

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 904
Author(s):  
Nicolae Pop ◽  
Miorita Ungureanu ◽  
Adrian I. Pop
Keyword(s):  
Boundary Condition ◽  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Displacement Field ◽  
Dry Friction ◽  
Control Variable ◽  
Traction Force ◽  
The State ◽  
State System

In this paper, we discuss the question of finding an optimal control for the solutions of the problem with dry friction quasistatic contact, in the case that the friction law is modeled by a nonlocal version of Coulomb’s law. In order to get the necessary optimality conditions, we use some regularization techniques, and this leads us to a problem of control for an inequality of the variational type. The optimal control problem consists, in our case, of minimizing a sequence of optimal control problems, where the control variable is given by a Neumann-type boundary condition. The state system is represented by a limit of a sequence, whose terms are obtained from the discretization, in time with finite difference and space with the finite element method of a regularized quasistatic contact problem with Coulomb friction. The purpose of this optimal control problem is that the traction force (the control variable) acting on one side of the boundary (the Neumann boundary condition) of the elastic body produces a displacement field (the state system solution) close enough to the imposed displacement field, and the traction force from the boundary remains small enough.

scholarly journals A Mixed Discontinuous Galerkin Approximation of Time Dependent Convection Diffusion Optimal Control Problem

2017 ◽  
Vol 2017 ◽  
pp. 1-14
Author(s):  
Qingjin Xu ◽  
Zhaojie Zhou
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Discontinuous Galerkin ◽  
Mixed Finite Element ◽  
Galerkin Approximation ◽  
Control Variable ◽  
The State ◽  
Time Dependent ◽  
Convection Diffusion

In this paper, we investigate a mixed discontinuous Galerkin approximation of time dependent convection diffusion optimal control problem with control constraints based on the combination of a mixed finite element method for the elliptic part and a discontinuous Galerkin method for the hyperbolic part of the state equation. The control variable is approximated by variational discretization approach. A priori error estimates of the state, adjoint state, and control are derived for both semidiscrete scheme and fully discrete scheme. Numerical example is given to show the effectiveness of the numerical scheme.

Morley FEM for a Distributed Optimal Control Problem Governed by the von Kármán Equations

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Sudipto Chowdhury ◽  
Neela Nataraj ◽  
Devika Shylaja
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Error Estimates ◽  
Lower Order ◽  
Control Variable ◽  
The State ◽  
Distributed Optimal Control ◽  
Von Kármán Equations ◽  
Adjoint Variables

AbstractConsider the distributed optimal control problem governed by the von Kármán equations defined on a polygonal domain of {\mathbb{R}^{2}} that describe the deflection of very thin plates with box constraints on the control variable. This article discusses a numerical approximation of the problem that employs the Morley nonconforming finite element method (FEM) to discretize the state and adjoint variables. The control is discretized using piecewise constants. A priori error estimates are derived for the state, adjoint and control variables under minimal regularity assumptions on the exact solution. Error estimates in lower-order norms for the state and adjoint variables are derived. The lower-order estimates for the adjoint variable and a post-processing of control leads to an improved error estimate for the control variable. Numerical results confirm the theoretical results obtained.

scholarly journals SUPG-Stabilized Virtual Element Method for Optimal Control Problem Governed by a Convection Dominated Diffusion Equation

Entropy ◽  
2021 ◽  
Vol 23 (6) ◽  
pp. 723
Author(s):  
Qiming Wang ◽  
Zhaojie Zhou
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Diffusion Equation ◽  
Control Problem ◽  
State Equation ◽  
The State ◽  
Virtual Element Method ◽  
Adjoint State ◽  
Virtual Element ◽  
Convection Dominated

In this paper, the streamline upwind/Petrov Galerkin (SUPG) stabilized virtual element method (VEM) for optimal control problem governed by a convection dominated diffusion equation is investigated. The virtual element discrete scheme is constructed based on the first-optimize-then-discretize strategy and SUPG stabilized virtual element approximation of the state equation and adjoint state equation. An a priori error estimate is derived for both the state, adjoint state, and the control. Numerical experiments are carried out to illustrate the theoretical findings.

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