| Abstract: |
| Given a continuous-time control system and $\varepsilon > 0$, we address the question of the existence of controls that maintain the corresponding state trajectories within the $\varepsilon$-neighborhood of any prescribed path in the state space. We first investigate this property, called approximate tracking controllability, in the linear finite-dimensional case, when our answers are negative: we demonstrate that approximate tracking controllability of the full state is impossible even in a certain weak sense, except for trivial situations. For finite-dimensional systems with quadratic nonlinearities, we prove approximate tracking controllability on any time horizon with respect to the relaxation metric. This approach is next generalized for a PDE system: the Euler equations with distributed control.
We underline the relevance of this weak setting by developing applications to coupled systems and by remarking obstructions that would arise for natural stronger norms. The main application concerns obtaining the enhanced dissipation of a passive scalar using a controlled fluid flow. |
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