From classical to quantum quadratic cost control

Controlling the size of the solution of a (deterministic, stochastic or quantum stochastic) differential equation, by minimizing an appropriate cost functional, is very important in classical and quantum engineering. Of particular importance is the case of linear differential equations and quadratic cost functionals, since in that case the control processes can be explicitly calculated. In this paper we review some basic aspects of the classical theory and we present our results in the quantum case, obtained over the past few years.