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A posteriori error estimates of spectral method for nonlinear parabolic optimal control problem
Lin Li1, Zuliang Lu1,2,3, Wei Zhang4
11Key Laboratory for Nonlinear Science and System Structure, Chongqing Three Gorges University, Chongqing, P.R. China.
Journal of Inequalities and Applications
|August 24, 2018
Summary
This study presents a spectral approximation for optimal control problems in nonlinear parabolic equations. It provides a posteriori error estimates for state and control approximations using a backward Euler scheme and auxiliary equations.
Area of Science:
- Numerical Analysis
- Partial Differential Equations
- Optimal Control Theory
Background:
- Optimal control problems governed by nonlinear parabolic equations are crucial in various scientific and engineering fields.
- Efficient and accurate numerical methods are needed for solving these complex problems.
Purpose of the Study:
- To develop and analyze a spectral approximation scheme for nonlinear parabolic optimal control problems.
- To derive a posteriori error estimates for the approximated state and control.
Main Methods:
- A fully discrete spectral approximation scheme is constructed using the backward Euler method for time discretization.
- Orthogonal projection operators and auxiliary equations are employed to derive a posteriori error estimates.
Main Results:
- The paper presents a novel spectral approximation scheme for the addressed optimal control problem.
- Rigorous a posteriori error estimates are established for both the state and control approximations.
Conclusions:
- The proposed spectral approximation scheme is effective for nonlinear parabolic optimal control problems.
- The derived a posteriori error estimates provide valuable information for assessing the accuracy of the numerical solutions.
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