scholarly journals A problem in control of elastodynamics with piezoelectric effects

2019 ◽  
Vol 40 (4) ◽  
pp. 2839-2870
Author(s):  
Harbir Antil ◽  
Thomas S Brown ◽  
Francisco-Javier Sayas
Keyword(s):  
Control Problem ◽  
Control Variable ◽  
Sufficient Conditions ◽  
The State ◽  
Adjoint Equations ◽  
Elastic Displacement ◽  
Complete Analysis ◽  
Well Posedness ◽  
State Equations ◽  
Piezoelectric Effects

Abstract We consider an optimal control problem where the state equations are a coupled hyperbolic–elliptic system. This system arises in elastodynamics with piezoelectric effects—the elastic stress tensor is a function of elastic displacement and electric potential. The electric flux acts as the control variable and bound constraints on the control are considered. We develop a complete analysis for the state equations and the control problem. The requisite regularity on the control, to show the well-posedness of the state equations, is enforced using the cost functional. We rigorously derive the first-order necessary and sufficient conditions using adjoint equations and further study their well-posedness. For spatially discrete (time-continuous) problems, we show the convergence of our numerical scheme. Three-dimensional numerical experiments are provided showing convergence properties of a fully discrete method and the practical applicability of our approach.

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.

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 On controllability and observability of fractional continuous-time linear systems with regular pencils

2017 ◽  
Vol 65 (3) ◽  
pp. 297-304
Author(s):  
A. Younus ◽  
I. Javaid ◽  
A. Zehra
Keyword(s):  
Linear Systems ◽  
Continuous Time ◽  
Sufficient Conditions ◽  
The State ◽  
Necessary And Sufficient Conditions ◽  
State Equations ◽  
Necessary And Sufficient ◽  
Controllability And Observability ◽  
Time Linear

AbstractIn this paper necessary and sufficient conditions of controllability and observability for solutions of the state equations of fractional continuous time linear systems with regular pencils are proposed. The derivations of the conditions are based on the construction of Gramian matrices.

scholarly journals Optimal Control for the Thin Film Equation: Convergence of a Multi-Parameter Approach to Track State Constraints Avoiding Degeneracies

2016 ◽  
Vol 16 (4) ◽  
pp. 685-702
Author(s):  
Markus Klein ◽  
Andreas Prohl
Keyword(s):  
Thin Film ◽  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
State Constraints ◽  
Necessary Optimality Conditions ◽  
State Constraint ◽  
The State ◽  
Well Posedness ◽  
Thin Film Equation

AbstractWe consider an optimal control problem subject to the thin-film equation. The PDE constraint lacks well-posedness for general right-hand sides due to possible degeneracies; state constraints are used to circumvent this problematic issue and to ensure well-posedness. Necessary optimality conditions for the optimal control problem are then derived. A convergent multi-parameter regularization is considered which addresses both, the possibly degenerate term in the equation and the state constraint. Some computational studies are then reported which evidence the relevant role of the state constraint, and motivate proper scalings of involved regularization and numerical parameters.

scholarly journals Adaptive finite element methods for sparse PDE-constrained optimization

2019 ◽  
Vol 40 (3) ◽  
pp. 2106-2142 ◽  
Author(s):  
A Allendes ◽  
F Fuica ◽  
E Otárola
Keyword(s):  
Piecewise Linear ◽  
Control Variable ◽  
Posteriori Error ◽  
The State ◽  
Adjoint Equations ◽  
Error Estimator ◽  
A Posteriori Error ◽  
Error Estimators ◽  
Variational Discretization ◽  
Posteriori Error Estimators

Abstract We propose and analyse reliable and efficient a posteriori error estimators for an optimal control problem that involves a nondifferentiable cost functional, the Poisson problem as state equation and control constraints. To approximate the solution to the state and adjoint equations we consider a piecewise linear finite element method, whereas three different strategies are used to approximate the control variable: piecewise constant discretization, piecewise linear discretization and the so-called variational discretization approach. For the first two aforementioned solution techniques we devise an error estimator that can be decomposed as the sum of four contributions: two contributions that account for the discretization of the control variable and the associated subgradient and two contributions related to the discretization of the state and adjoint equations. The error estimator for the variational discretization approach is decomposed only in two contributions that are related to the discretization of the state and adjoint equations. On the basis of the devised a posteriori error estimators, we design simple adaptive strategies that yield optimal rates of convergence for the numerical examples that we perform.

scholarly journals The Continuous Classical Boundary Optimal Control of Couple Nonlinear Hyperbolic Boundary Value Problem with Equality and Inequality Constraints

2019 ◽  
Vol 16 (4(Suppl.)) ◽  
pp. 1064 ◽  
Author(s):  
Jamil A. Ali Al-Hawasy
Keyword(s):  
Existence Theorem ◽  
Unique Solution ◽  
State Vector ◽  
Hyperbolic Equations ◽  
Sufficient Conditions ◽  
Inequality Constraints ◽  
The State ◽  
Nonlinear Hyperbolic Equations ◽  
State Equations ◽  
Classical Boundary

The paper is concerned with the state and proof of the existence theorem of a unique solution (state vector) of couple nonlinear hyperbolic equations (CNLHEQS) via the Galerkin method (GM) with the Aubin theorem. When the continuous classical boundary control vector (CCBCV) is known, the theorem of existence a CCBOCV with equality and inequality state vector constraints (EIESVC) is stated and proved, the existence theorem of a unique solution of the adjoint couple equations (ADCEQS) associated with the state equations is studied. The Frcéhet derivative derivation of the "Hamiltonian" is obtained. Finally the necessary theorem (necessary conditions "NCs") and the sufficient theorem (sufficient conditions" SCs") for optimality of the state constrained problem are stated and proved.  

scholarly journals A Quantitative Analysis of Preventive Measures to Mitigate the Infectious Diseases Using SIR model by Optimal Control Technique

2018 ◽  
Vol 5 (1) ◽  
pp. 11-19
Author(s):  
Jakia Sultana ◽  
Samiha Islam Tanni ◽  
Shamima Islam
Keyword(s):  
Optimal Control ◽  
Infectious Diseases ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Transmission Rate ◽  
Control Strategies ◽  
Sir Model ◽  
The State ◽  
Control Technique ◽  
State Equations

Optimal Control Problem with the state equations which describes the standard SIR Model is studied here. We considered the SIR Model with vaccination and without vaccination. We formulated an optimal control problem and derived necessary conditions. Existence of the state and the objective functional are also verified. We also characterized the optimal control by Pontryagin’s maximum principle which minimizes the number of infected individuals and cost of vaccination over some finite period. Whenever the vaccination is carried out for a long period of time, the simulated result effectively works for disease with high transmission rate. Observations from the numerical simulation revels that the infectious diseases are most successfully controlled whenever control strategies were adopted at early stages. GUB JOURNAL OF SCIENCE AND ENGINEERING, Vol 5(1), Dec 2018 P 11-19

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