scholarly journals Optimal Control of Plasticity with Inertia

2021 ◽  
Vol Volume 2 (Original research articles) ◽  
Author(s):  
Stephan Walther
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Monotone Operator ◽  
Maximal Monotone Operator ◽  
Kinematic Hardening ◽  
Maximal Monotone ◽  
Small Strain ◽  
State Equation ◽  
Elasto Plasticity

The paper is concerned with an optimal control problem governed by the equations of elasto plasticity with linear kinematic hardening and the inertia term at small strain. The objective is to optimize the displacement field and plastic strain by controlling volume forces. The idea given in [10] is used to transform the state equation into an evolution variational inequality (EVI) involving a certain maximal monotone operator. Results from [27] are then used to analyze the EVI. A regularization is obtained via the Yosida approximation of the maximal monotone operator, this approximation is smoothed further to derive optimality conditions for the smoothed optimal control problem.

scholarly journals High-order homogenization in optimal control by the Bloch wave method

2021 ◽  
Author(s):  
Agnes Lamacz-Keymling ◽  
Irwin Yousept
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Optimal Solution ◽  
Periodic Coefficient ◽  
State Equation ◽  
Bloch Wave ◽  
High Order ◽  
Effective Control ◽  
Linear Quadratic

This article examines a linear-quadratic elliptic optimal control problem in which the cost functional and the state equation involve a highly oscillatory periodic coefficient $A^\eps$. The small parameter $\eps>0$ denotes the periodicity length. We propose a high-order effective control problem with constant coefficients that provides an approximation of the original one with error $O(\eps^M)$, where $M\in\N$ is as large as one likes. Our analysis relies on a Bloch wave expansion of the optimal solution and is performed in two steps. In the first step, we expand the lowest Bloch eigenvalue in a Taylor series to obtain a high-order effective optimal control problem. In the second step, the original and the effective problem are rewritten in terms of the Bloch and the Fourier transform, respectively. This allows for a direct comparison of the optimal control problems via the corresponding variational inequalities, leading to our main theoretical result on the high-oder approximation.

A 𝑃1 Finite Element Method for a Distributed Elliptic Optimal Control Problem with a General State Equation and Pointwise State Constraints

2021 ◽  
Vol 21 (4) ◽  
pp. 777-790
Author(s):  
Susanne C. Brenner ◽  
Sijing Liu ◽  
Li-Yeng Sung
Keyword(s):  
Finite Element Method ◽  
Optimal Control ◽  
Finite Element ◽  
Optimal Control Problem ◽  
Control Problem ◽  
State Constraints ◽  
State Equation ◽  
Pointwise State Constraints ◽  
Primal Dual ◽  
Element Method

Abstract We investigate a P 1 P_{1} finite element method for an elliptic distributed optimal control problem with pointwise state constraints and a state equation that includes advective/convective and reactive terms. The convergence of this method can be established for general polygonal/polyhedral domains that are not necessarily convex. The discrete problem is a strictly convex quadratic program with box constraints that can be solved efficiently by a primal-dual active set algorithm.

scholarly journals Sparse Optimal Control for Fractional Diffusion

2018 ◽  
Vol 18 (1) ◽  
pp. 95-110 ◽  
Author(s):  
Enrique Otárola ◽  
Abner J. Salgado
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
A Priori ◽  
Control Variable ◽  
Fractional Diffusion ◽  
State Equation ◽  
Solution Technique ◽  
State Variables ◽  
Regularity Results

AbstractWe consider an optimal control problem that entails the minimization of a nondifferentiable cost functional, fractional diffusion as state equation and constraints on the control variable. We provide existence, uniqueness and regularity results together with first-order optimality conditions. In order to propose a solution technique, we realize fractional diffusion as the Dirichlet-to-Neumann map for a nonuniformly elliptic operator and consider an equivalent optimal control problem with a nonuniformly elliptic equation as state equation. The rapid decay of the solution to this problem suggests a truncation that is suitable for numerical approximation. We propose a fully discrete scheme: piecewise constant functions for the control variable and first-degree tensor product finite elements for the state variable. We derive a priori error estimates for the control and state variables.

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.

REDUCED MODEL OF PLASMA DISCHARGE CONTROL IN TOKAMAK WITH IRON CORE

2020 ◽  
Vol 7 (3) ◽  
pp. 11-22
Author(s):  
VALERY ANDREEV ◽  
◽  
ALEXANDER POPOV
Keyword(s):  
Optimal Control ◽  
Inverse Problem ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Nonlinear Behavior ◽  
Plasma Discharge ◽  
Reduced Model ◽  
Iron Core ◽  
Power Sources ◽  
Real Power

A reduced model has been developed to describe the time evolution of a discharge in an iron core tokamak, taking into account the nonlinear behavior of the ferromagnetic during the discharge. The calculation of the discharge scenario and program regime in the tokamak is formulated as an inverse problem - the optimal control problem. The methods for solving the problem are compared and the analysis of the correctness and stability of the control problem is carried out. A model of “quasi-optimal” control is proposed, which allows one to take into account real power sources. The discharge scenarios are calculated for the T-15 tokamak with an iron core.

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