In this paper, we consider the equations for viscoelastic rate type fluids, which is viewed as the Navier–Stokes equation coupled with a rate type viscoelastic fluid model. The fluid velocity is assumed to satisfy a no-slip boundary condition, while the extra stress tensor is subject to a time-dependent Dirichlet boundary condition. First we prove the existence of weak solutions to (1.2)–(1.4). If d = 2, we further derive the existence and uniqueness of weak and strong solution. Next, based on the existence of global strong solution, we show that the control to state operator is Fréchet differentiable between appropriate Banach spaces. Finally, by using the properties of the control to state operator and the Lagrange principle, we prove the existence and necessary optimality condition of an optimal boundary control problem.
Yin et al. (2026) studied this question.