Spectral mixed Galerkin method for state constrained optimal control problem governed by the first bi-harmonic equation

Abstract

An optimal control problem governed by the first bi-harmonic equation with the integral constraint for the state and its spectral approximations based on a mixed formulation are investigated. The optimality conditions of the exact and the discrete optimal control systems are derived. The a priori error estimates of high order spectral accuracy are obtained. Furthermore, a simple and efficient iterative algorithm is proposed to solve mixed discrete system. Some numerical examples are performed to verify the theoretical results.

Keywords:

• optimal control
• state constraint
• bi-harmonic equation
• mixed formulation
• gradient projection algorithm
• a priori error estimate

2000 AMS Subject Classifications :

• 49J20
• 49K20
• 65N30
• 65N35

Acknowledgements

We wish to express our thanks to referees for their useful suggestion and comments. Research supported partially by National Natural Science Foundation of China under the grant 11071080, Program of Shanghai Subject Chief Scientist, No. 09XD1401600, Fundamental Research Funds for the Central Universities of China and Shanghai Leading Academic Discipline Project: B407.

Additional information

Notes on contributors

Jianwei Zhou

Present address: Department of Mathematics, Linyi University, Linyi, Shandong, China.