Unexpected quadratic behaviors for the small-time local null controllability of scalar-input parabolic equations

TL;DR

This paper investigates quadratic obstructions to small-time local null controllability in scalar-input parabolic equations, revealing new behaviors and conditions under which controllability can be recovered or obstructed.

Contribution

It extends finite-dimensional quadratic controllability obstructions to infinite-dimensional parabolic PDEs, discovering new phenomena unique to the infinite-dimensional setting.

Findings

Existence of a continuous family of quadratic obstructions quantified by fractional Sobolev norms.

Small-time local null controllability can sometimes be recovered from quadratic expansion.

An example system with infinitely many controllable directions via quadratic expansion.

Abstract

We consider scalar-input control systems in the vicinity of an equilibrium, at which the linearized systems are not controllable. For finite dimensional control systems, the authors recently classified the possible quadratic behaviors. Quadratic terms introduce coercive drifts in the dynamics, quantified by integer negative Sobolev norms, which are linked to Lie brackets and which prevent smooth small-time local controllability for the full nonlinear system. In the context of nonlinear parabolic equations, we prove that the same obstructions persist. More importantly, we prove that two new behaviors occur, which are impossible in finite dimension. First, there exists a continuous family of quadratic obstructions quantified by fractional negative Sobolev norms or by weighted variations of them. Second, and more strikingly, small-time local null controllability can sometimes be recovered from the quadratic expansion. We also construct a system for which an infinite number of directions are recovered using a quadratic expansion. As in the finite dimensional case, the relation between the regularity of the controls and the strength of the possible quadratic obstructions plays a key role in our analysis.