Tracking controllability for finite-dimensional linear systems

We develop a functional-analytic characterization of output tracking controllability for finite-dimensional linear systems. By formulating tracking as the surjectivity of the control-to-output map on suitable trajectory spaces, we show that exact tracking is equivalent to a nonstandard observability inequality for the adjoint dynamics. This characterization enables a Hilbert Uniqueness Method (HUM) type variational construction of minimum-norm tracking controls and makes explicit the intrinsic regularity requirements on reference trajectories induced by the system dynamics and the output operator. The same framework also yields a natural notion of approximate tracking when exact tracking fails. We provide explicit formulas in the scalar case and report numerical experiments for ODEs and semi-discretized PDEs, demonstrating the method for both smooth and non-smooth targets.