Finite element method for parabolic optimal control problems with a bilinear state equation

Shakya, Pratibha; Sinha, Rajen Kumar

doi:10.1016/j.cam.2019.112431

Cited by 15 publications

(4 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Han and Zhang [6] studied neutral delay-reaction-diffusion equations with Galerkin finite element method and Crank-Nicolson method, they proved that fully discrete Galerkin finite element scheme was uniquely solvable, stable and convergent. Shakya and Sinha [16] considered optimize-then-discretize strategy and used continuous piecewise linear finite element to approximate spatial state variable and linearized backward Euler scheme was applied to the time discretization. For more information on the application of Galerkin finite element method to differential equations, interested readers can refer to the literatures [27][28][29][30].…”

Section: Introductionmentioning

confidence: 99%

Convergence and stability of Galerkin finite element method for hyperbolic partial differential equation with piecewise continuous arguments

Chen

Wang

2023

Numerical Methods Partial

View full text Add to dashboard Cite

In this paper, the convergence and stability of Galerkin finite element method for a hyperbolic partial differential equations with piecewise continuous arguments are investigated. Firstly, the variation formulation is derived by applying Green's formula and Galerkin finite element method to spatial direction of the original equation. Next, semidiscrete and fully discrete schemes are obtained and the convergence is analyzed in L 2 -norm rigorously. Moreover, the stability analysis shows that the semidiscrete scheme can achieve unconditionally stability. The sufficient conditions of stability for fully discrete scheme are also obtained under which the analytic solution is asymptotically stable. Finally, some numerical experiments are provided to demonstrate our theoretical results.

show abstract

Section: Introductionmentioning

confidence: 99%

Convergence and stability of Galerkin finite element method for hyperbolic partial differential equation with piecewise continuous arguments

Chen

Wang

2023

Numerical Methods Partial

View full text Add to dashboard Cite

show abstract

“…To solve this problem, scholars have carried out many research works and proposed various optimal control algorithms. 29–32 Gavrikov et al 33 study the optimal boundary control problem with linear quadratic cost function in the heat transfer process of a cylinder; Mechelli and Volkwein 34 study an optimal boundary and bilateral control problem of heat transfer equation with convection term; Dekhkonov 35 recently studies the boundary control problem, based on the heat transfer model, for the heat exchange process; Bollo et al 36 solve OCPs of Neumann boundary for a n -dimensional heat equation; Abbasi and Malek 37 solve pointwise optimal control problem on and inside a tissue subject to thermal wave model with Dirichlet and Rubin boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

Parameterization optimal control of an unsteady partial differential equations with convection term by an improved three-term spectrum conjugate gradient algorithm

Yu,

Wang,

Pang

et al. 2023

Measurement and Control

View full text Add to dashboard Cite

The unsteady partial differential equations (UPDE) with convection term gives a clear descriptions for the solidification process of a slab in dynamic production of continuous casting. To give a suitable setting value of secondary cooling water flow rate for the dynamic control system, this study investigates an optimal control problem (OCP) of UPDE with convection term. Firstly, control vector discretization of OCP and the solution of UPDE are given. Secondly, due to the rapidity for gradient, this paper analyzes the expression of the gradient calculation method based on Hamiltonian function costate system by approximate treatment, matrix calculation and composite trapezoidal integral method. Thirdly, an improved three-term spectrum conjugate gradient algorithm (ITSCGA) is proposed to solve the OCP of UPDE, and the global convergence of the ITSCGA is demonstrated. Lastly, the performance of ITSCGA is demonstrated by experimental simulations. The results demonstrate that the ITSCGA provides a smaller temperature fluctuations, and improves the quality of a slab.

show abstract

“…For linear and semi-linear parabolic OCPs, a priori error estimate of space-time finite element discretization was derived in [18,19], and the superconvergence properties of semi-discrete and fully discrete FEMs were obtained in [8,10], and [22,23], respectively. For bilinear parabolic OCPs, some convergence and superconvergence results of FEMs and mixed FEMs can be found in [6,16,20,24].…”

Section: Introductionmentioning

confidence: 99%

A new superconvergence of finite elements for bilinear parabolic optimal control problems

2022

J. Nonlinear Funct. Anal.

View full text Add to dashboard Cite

In this paper, we consider a fully discrete finite element approximation of bilinear parabolic optimal control problems with an integral constraint. First, we give an approximation scheme of the model problem, where triangular finite element and backward Euler methods are used. Second, we introduce some useful intermediate variables, interpolation operators and related error estimates. Third, we derive a new superconvergence between the numerical solutions and projection functions of exact solutions. Last, a numerical example is provided to verify our results.

show abstract

Finite element method for parabolic optimal control problems with a bilinear state equation

Cited by 15 publications

References 16 publications

Convergence and stability of Galerkin finite element method for hyperbolic partial differential equation with piecewise continuous arguments

Convergence and stability of Galerkin finite element method for hyperbolic partial differential equation with piecewise continuous arguments

Parameterization optimal control of an unsteady partial differential equations with convection term by an improved three-term spectrum conjugate gradient algorithm

A new superconvergence of finite elements for bilinear parabolic optimal control problems

Contact Info

Product

Resources

About