In this paper, we first investigate optimal control problem for semilinear parabolic and introduce the standard $L^2(\mathrm{\Omega })$-orthogonal projection and the elliptic projection. Then we present some necessary intermediate variables and their error estimates. At last, we derive the error estimates between the finite element solutions and $L^2$-orthogonal projection or the elliptic projection of the exact solutions.

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.