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.

我们考虑标量输入控制系统在平衡附近，在该平衡处线性化系统是不可控制的。对于有限维控制系统，作者最近对可能的二次行为进行了分类。二次项会在动力学中引入强制性漂移，并通过整数负Sobolev范数来量化，该规范与Lie括号相关联，并且阻碍了整个非线性系统的平稳局部局部可控性。

在非线性抛物方程的上下文中，我们证明了相同的障碍仍然存在。更重要的是，我们证明了出现了两个新的行为，这在有限维中是不可能的。首先，存在一个连续的二次障碍族，它由分数负Sobolev范数或它们的加权变化来量化。其次，更令人惊讶的是，有时可以从二次扩展中恢复少量的局部零可控性。我们还构建了一个系统，该系统使用二次展开来恢复无数个方向。

与有限维情况一样，控件的规则性与可能的二次障碍的强度之间的关系在我们的分析中起着关键作用。