Tracking controllability for finite-dimensional linear systems

This paper develops 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 trajectory-space observability inequality associated with the dual input-output structure.

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 illustrative numerical examples for ODEs and semi-discretized PDEs, demonstrating the method for both smooth and nonsmooth targets.