Quadratic obstructions to small-time local controllability for scalar-input systems

Abstract

We consider nonlinear scalar-input differential control systems in the vicinity of an equilibrium. When the linearized system at the equilibrium is controllable, the nonlinear system is smoothly small-time locally controllable, i.e., whatever m > 0 and T > 0, the state can reach a whole neighborhood of the equilibrium at time T with controls arbitrary small in C m -norm. When the linearized system is not controllable, we prove that small-time local controllability cannot be recovered from the quadratic expansion and that the following quadratic alternative holds.Either the state is constrained to live within a smooth strict invariant manifold, up to a cubic residual, or the quadratic order adds a signed drift in the evolution with respect to this manifold. In the second case, the quadratic drift holds along an explicit Lie bracket of length (2k + 1), it is quantified in terms of an H −k -norm of the control, it holds for controls small in W 2k,∞ -norm. These spaces are optimal for general nonlinear systems and are slightly improved in the particular case of control-affine systems.Unlike other works based on Lie-series formalism, our proof is based on an explicit computation of the quadratic terms by means of appropriate transformations. In particular, it does not require that the vector fields defining the dynamic are smooth. We prove that C 3 regularity is sufficient for our alternative to hold.This work underlines the importance of the norm used in the smallness assumption on the control: depending on this choice of functional setting, the same system may or may not be small-time locally controllable, even though the state lives within a finite dimensional space.MSC: 93B05, 93C15, 93B10

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

“…The goal of this work is to illustrate possible behaviors for parabolic scalar-input control systems, stemming from the analysis of their second-order expansions. Some of these quadratic behaviors are already present in finite dimension (see [7], where the authors classified the possible quadratic behaviors for scalar-input control systems in finite dimension, or [32] for a short survey in French by the second author). Others are new and specific to control systems in infinite dimension.…”

“…Similarly as in finite dimension (see [7,Theorems 2 and 3]), Lie bracket considerations lead to obstructions related to quadratic coercive drifts, quantified by integer-order negative Sobolev norms. This means that the component z(t), ϕ 0 ineluctably moves in one direction, for instance it increases and therefore cannot reach values z(0), ϕ 0 , which prevents controllability.…”

“…Taking into account that ∆ N ϕ 0 = 0, we obtain ad 2 −1 ∆ N (Γ (0)) µ, ϕ 0 = Γ (0)∆ (1.29) Remark 1.8. For control-affine systems in finite dimension, we proved in [7,Theorem 3] that the optimal norm for the smallness assumption on u is the W 2n−3,∞ one. This norm is optimal in the following sense:…”

“…In finite dimension, there is no system for which the linearized system misses a direction and for which small-time local controllability is recovered with quadratic cost (we refer to [7] for more precise statements).…”

“…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.…”