Galerkin spectral approximation for optimal control problem of a fourth-order equation with L²-norm control constraint

Abstract

We investigate the Galerkin spectral approximation of an optimal control problem governed by a fourth-order partial differential equation (PDE), in which an $L^2$-norm constraint on control variable is equipped. First, the optimality conditions for both the original control problem and its spectral approximation problem are, respectively, obtained. Then, a priori error estimates of the spectral approximation problem are established in detail. Next, a posteriori error estimates for the approximation problem are also investigated, which include not only $H^2$-error estimate for the state and co-state but also $L^2$-error estimate for the control, state and co-state. Finally, three numerical examples are executed to validate the theoretical analysis.

Keywords:

• Optimal control
• fourth-order equation
• spectral approximation
• control constraint
• a priori error
• a posteriori error

AMS subject classifications:

• 49M25
• 49M41
• 65M60
• 65N35

Acknowledgements

The authors would like to thank the editor and the anonymous referees for their very careful reading and constructive suggestions that improve the manuscript substantially.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported in part by the National Natural Science Foundation of China under Grant No. 11471036.