Are viscoelastic flows under control or out of control?

The question of controllability is whether it is possible to steer a system to a given state with a given class of control inputs. In the context of partial differential equations, control inputs are typically restricted to the boundary or to a given part of the domain. In this paper, we consider the controllability of flows of linear viscoelastic fluids. An important issue is whether the stresses, in addition to the motion, can be controlled. For the controls, we allow a body force in the equation of motion. We show that no control is possible, unless the initial conditions for the stresses satisfy constraints. We then consider the special situation of parallel shear flow, under the assumption that these constraints are satisfied. If inertia is negligible, we show there is no controllability unless the control is on the entire interval. With inertia, we prove that exact controllability holds in the case of a single relaxation mode, and in the case of several relaxation modes if the control is on the entire interval. If several relaxation modes are present, and the control is restricted to a subinterval, we can show only approximate controllability.