Convergence analysis of finite element approximations for a nonlinear second order hyperbolic optimal control problems

Huanhuan Li; Meiling Ding; Xianbing Luo; Shuwen Xiang; Huanhuan Li; Meiling Ding; Xianbing Luo; Shuwen Xiang

doi:10.3934/nhm.2024038

Networks and Heterogeneous Media

2024, Volume 19, Issue 2: 842-866. doi: 10.3934/nhm.2024038

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Convergence analysis of finite element approximations for a nonlinear second order hyperbolic optimal control problems

School of Mathematics and Statistics, Guizhou University, Guiyang 550025, China

Received: 12 July 2024 Revised: 19 August 2024 Accepted: 22 August 2024 Published: 28 August 2024

This paper focused on approximating a second-order nonlinear hyperbolic optimal control problem. By introducing a new variable, the hyperbolic equation was converted into two parabolic equations. A second-order fully discrete scheme was obtained by combining the Crank-Nicolson formula with the finite element method. The error estimation for this scheme was derived utilizing the second-order sufficient optimality condition and auxiliary problems. To validate the effectiveness of the fully discrete scheme, a numerical example was presented.
- second order hyperbolic equation,
- optimal control,
- finite element method,
- a priori error estimates
Citation: Huanhuan Li, Meiling Ding, Xianbing Luo, Shuwen Xiang. Convergence analysis of finite element approximations for a nonlinear second order hyperbolic optimal control problems[J]. Networks and Heterogeneous Media, 2024, 19(2): 842-866. doi: 10.3934/nhm.2024038

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Abstract

This paper focused on approximating a second-order nonlinear hyperbolic optimal control problem. By introducing a new variable, the hyperbolic equation was converted into two parabolic equations. A second-order fully discrete scheme was obtained by combining the Crank-Nicolson formula with the finite element method. The error estimation for this scheme was derived utilizing the second-order sufficient optimality condition and auxiliary problems. To validate the effectiveness of the fully discrete scheme, a numerical example was presented.

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