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Journal of Industrial and Management Optimization (JIMO)
 

Existence and stability analysis for nonlinear optimal control problems with $1$-mean equicontinuous controls

Pages: 1409 - 1422, Volume 11, Issue 4, October 2015      doi:10.3934/jimo.2015.11.1409

 
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Hongyong Deng - Department of Mathematics, Guizhou University, Guizhou, 550025, China (email)
Wei Wei - Department of Mathematics, Guizhou University, Guizhou, 550025, China (email)

Abstract: In this paper, the existence and stability of solutions of nonlinear optimal control problems with $1$-mean equicontinuous controls are discussed. In particular, a new existence theorem is obtained without convexity assumption. We investigate the stability of the optimal control problem with respect to the right-hand side functions, which is important in computational methods for optimal control problems when the function is approximated by a new function. Due to lack of uniqueness of solutions for an optimal control problem, the stability results for a class of optimal control problems with the measurable admissible control set is given based on the theory of set-valued mappings and the definition of essential solutions for optimal control problems. We show that the optimal control problems, whose solutions are all essential, form a dense residual set, and so every optimal control problem can be closely approximated arbitrarily by an essential optimal control problem.

Keywords:  Optimal control, existence, stability, set-valued mapping, essential solution.
Mathematics Subject Classification:  Primary: 49J15, 49J53; Secondary: 93D09.

Received: February 2014;      Revised: October 2014;      Available Online: March 2015.

 References