This paper mainly investigates linear elastic optimal control problems with two constraints: distributed load control and boundary load control. The gradient of the objective functional is derived via an adjoint problem. We obtain the \(\varvec{H}^{\varvec{1}}\) -regularity for the distributed control solution. Moreover, we establish a suitable finite element interpolation to deal with the non-optimal regularity of the boundary control solution. The error estimates for the fully discretized problems is given. Then, together with the interior point method, a new numerical method for the optimal control of linear elasticity is exhibited. Finally, the effectiveness of presented scheme is demonstrated by various numerical simulations in two and three-space dimensions.
