Finite element approximations of parabolic optimal control problem with measure data in time

ABSTRACT

The purpose of this paper is to study the a priori error analysis of finite element method for parabolic optimal control problem with measure data in a bounded convex domain. The solution of the state equation of this kind of problem exhibits low regularity which introduces some difficulties for both theory and numerics of finite element method. We first prove the existence, uniqueness and regularity results for the solutions of control problem under low regularity assumption on the state variable. For numerical approximations, we use continuous piecewise linear functions for the state and co-state variables, and piecewise constant functions for the control variable. Both spatially discrete and fully discrete finite element approximations of the control problem are analyzed. We derive a priori error estimates of order $\mathcal{O}\left(h\right)$ for the state, co-state and control variables for the spatially discrete problem. A time discretization scheme based on implicit backward-Euler method is applied to obtain error estimates of order $\mathcal{O}(h+k^{1/2})$ for the state, co-state and control variables. Numerical results are presented to illustrate our theoretical findings.

KEYWORDS:

• Parabolic optimal control problem
• measure data in time
• finite element method
• spatially discrete problem
• fully discrete problem
• a priori error estimates

AMS SUBJECT CLASSIFICATIONS:

• 49J20
• 49K20
• 65N15
• 65N30

Acknowledgements

The authors wish to thank the anonymous referee for his valuable suggestions and helpful comments which greatly improve the contents of this manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.