This work aims to understand the velocity tracking control problem for a class of non-Newtonian fluids. We introduce the third-grade-Voigt equations in the two-dimensional torus \(\mathbb {T}^2\) and prove the existence and uniqueness of the solution. Then, we show the existence and uniqueness of solution to the corresponding linearized state equation and adjoint equation. Additionally, we provide a suitable stability result for the state equation and demonstrate that the Gateaux derivative of the control-to-state mapping agrees with the solution of the linearized state equation. Next, we establish the first order optimality conditions and show the existence of an optimal solution. Ultimately, we are able to provide a uniqueness result for the coupled system consisting of the adjoint equation, the state equation, and the first order optimality condition. Therefore, under appropriate conditions on the data, the uniqueness of the optimal solution holds.
