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

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. ,( 1.14) and H m 0 (0, T ), which is the adherence of C ∞ c (0, T ) for the topology of . H m (0,T ) . For a ∈ [0, ∞) the (positive) fractional Sobolev space H a (0, T ) defined by interpolation and equipped with the norm . H a (0,T ) . 3 Controllability stemming from the linear orderWe start by studying the linearized system of (1.1) around the null equilibrium (z, u) = (0, 0):( 1.15) The controllability of systems such as (1.15) has been extensively studied. We refer in particular to [21] and [22] for the introduction of the moment method. The assumption that µ, ϕ k = 0 for all k ∈ N is obviously necessary for the linearized system to be null controllable (otherwise the component z(t), ϕ k of the state would evolve freely). Moreover, in order for the linearized system to be small-time null controllable, one must add the assumption that the sequence µ, ϕ k does not decay too fast (see Section 2.3).Theorem 1. Let µ ∈ H −1 N (0, π) such that µ, ϕ k = 0 for all k ∈ N and satisfying the decay assumption (2.25). The linear system (1.15) is small-time null controllable with L 2 controls.When dealing with nonlinear behaviors, especially in infinite dimension, the regularity of the controls plays a crucial role. In fact, and quite surprisingly, the regularity of the controls already plays an important role for control systems in finite dimension (see [7]). We define the following more precise notions, stressing the regularity imposed on the controls. Definition 1.3 (Small-time local null controllability). Let Γ be such that (1.6) holds. For m ∈ N * , we say that system (1.1) is H m -STLNC (respectively H m 0 -STLNC) when, for every T, η > 0, there exists δ > 0 such that, for every zDefinition 1.4 (Smooth small-time local null controllability). Let Γ be such that (1.6) holds. WeUnder appropriate assumptions, the smooth smal...

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