Approximate Tracking Controllability of Systems with Quadratic Nonlinearities

TL;DR

This paper analyzes approximate tracking controllability for finite‑dimensional time‑continuous systems of the form $\dot{x}(t)+A x(t)+f(x(t))=B u(t)$. It shows that in the linear case ($f\equiv0$) weak approximate tracking is possible only when the map $B$ is onto; otherwise, duality arguments yield obstructions. For nonlinear systems with quadratic drift $f(x)=\Gamma(x,x)$, the authors prove weak approximate tracking controllability on any horizon under a saturation/convexification assumption (Assumption 2), using an enlarged system with two inputs and a relaxation norm to obtain proximity $|x(\tau)-\psi(\tau)|$ and $|||x-\psi|||_\tau$. They then apply the result to coupled systems and motion planning, obtaining global results even if uncontrolled dynamics may blow up in finite time, and discuss limitations under stronger norms. The work highlights how norm choice and nonlinearity structure enable constructive weak tracking in finite dimensions and provides a pathway to practical control strategies via smooth inputs.

Abstract

Given a finite-dimensional time continuous control system and $\varepsilon>0$, we address the question of the existence of controls that maintain the corresponding state trajectories in the $\varepsilon$-neighborhood of any prescribed path in the state space. We investigate this property, called approximate tracking controllability, for linear and quadratic time invariant systems. Concerning linear systems, our answers are negative: by developing a systematic approach, we demonstrate that approximate tracking controllability of the full state is impossible even in a certain weak sense, except for the trivial situation where the control space is isomorphic to the state space. Motivated by these negative findings for linear systems, we focus on nonlinear dynamics. In particular, we prove weak approximate tracking controllability on any time horizon for a general class of systems with arbitrary linear part and quadratic nonlinear terms. The considered weak notion of approximate tracking controllability involves the relaxation metric. We underline the relevance of this weak setting by developing applications to coupled systems (including motion planning problems) and by remarking obstructions that would arise for natural stronger norms. The exposed framework yields global results even if the uncontrolled dynamics might exhibit singularities in finite time.

Theorems & Definitions (37)

• Definition 1.1
• Remark 1.2
• Remark 1.3
• Definition 1.4
• Proposition 1.5
• Proposition 1.8
• Example 1.9
• Remark 1.10
• Definition 2.1
• Remark 2.2