Tracking controllability for finite-dimensional linear systems

TL;DR

This paper characterizes output tracking controllability for finite-dimensional linear systems using a functional-analytic approach, linking exact tracking to a nonstandard observability inequality and providing a variational method for control construction.

Contribution

It introduces a novel functional-analytic framework for output tracking, including a variational method for minimum-norm controls and explicit formulas in the scalar case.

Findings

Explicit formulas for scalar systems

Numerical experiments on ODEs and PDEs

Demonstration of both exact and approximate tracking

Abstract

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.