  In this paper we consider a constrained parabolic optimal control problem.
The cost functional is quadratic and it combines the distance of the trajectory
of the system from the desired evolution profile together with the cost of a
control. The constraint is given by a term measuring the distance between the
final state and the desired state towards which the solution should be steered.
The control enters the system through the initial condition. We present a
geometric analysis of this problem and provide a closed-form expression for the
solution. This approach allows us to present the sensitivity analysis of this
problem based on the resolvent estimates for the generator of the system. The
numerical implementation is performed by exploring efficient rational Krylov
approximation techniques that allow us to approximate a complex function of an
operator by a series of linear problems. Our method does not depend on the
actual choice of discretization. The main approximation task is to construct an
efficient rational approximation of a generalized exponential function. It is
well known that this class of functions allows exponentially convergent
rational approximations, which, combined with the sensitivity analysis of the
closed form solution, allows us to present a robust numerical method. Several
case studies are presented to illustrate our results.
