Convergence and proposed optimality of adaptive finite element methods for nonlinear optimal control problems

This paper investigates the adaptive finite element method for nonlinear optimal control problem, and the research content of reference (^[21] H. Leng and Y. Chen, 2017) is extended accordingly. Linear discretisation of the equation of state and the equation of common state is performed using continuous segmentation functions. At the same time, we use the bubble function technique to prove that the posterior error estimates are obtained from the upper and lower bounds. What is more, for the adaptive finite element method, we also consider convergence and quasi-optimality, where we find that the demand $h_0\ll 1$ on the initial grid is unconstrained for the convergence analysis of the proposed adaptive algorithm for the nonlinear optimal control problem. Simultaneously, some numerical simulation is used to verify our theoretical analysis.