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

In this work, we investigate the small-time global exact controllability of the Navier–Stokes equation, both towards the null equilibrium state and towards weak trajectories.We consider a viscous incompressible fluid evolving within a smooth bounded domain, either in 2D or in 3D. The controls are only located on a small part of the boundary, intersecting all its connected components. On the remaining parts of the boundary, the fluid obeys a Navier slip-with-friction boundary condition. Even though viscous boundary layers appear near these uncontrolled boundaries, we prove that small-time global exact controllability holds. Our analysis relies on the controllability of the Euler equation combined with asymptotic boundary layer expansions. Choosing the boundary controls with care enables us to guarantee good dissipation properties for the residual boundary layers, which can then be exactly canceled using local techniques.

The present paper is about a famous extension of the Prandtl equation, the so-called Interactive Boundary Layer model (IBL). This model has been used intensively in the numerics of steady boundary layer flows, and compares favorably to the Prandtl one, especially past separation. We consider here the unsteady version of the IBL, and study its linear well-posedness, namely the linear stability of shear flow solutions to high frequency perturbations. We show that the IBL model exhibits strong unrealistic instabilities, that are in particular distinct from the Tollmien-Schlichting waves. We also exhibit similar instabilities for a Prescribed Displacement Thickness model (PDT), which is one of the building blocks of numerical implementations of the IBL model.A similar expansion is assumed on the pressure. After plugging expansion (1.4) into the Navier-Stokes equation (1.1), one recovers the famous Prandtl system for the leading order profile (U 0 , V 0 )(t, x, y):with u e (t, x) := u 0 (t, x, 0). If we assume some fast enough convergence of U 0 − u e and of its y-derivatives to zero as y goes to infinity, system (1.5) further simplifies intostill with u e (t, x) = u 0 (t, x, 0).The Prandtl model (1.6) is the cornerstone of our understanding of the boundary layer behavior. One striking success of this model is the description of steady high Reynolds number flows along a thin plate. If we model the thin plate by the half line {x > 0, y = 0} and follow the Prandtl theory, we end up with a system of the type(1.7)In the steady case, a solution of this equation is given by the self-similar Blasius flowfor a profile f satisfying an integrodifferential equation, see [2]. Indeed, experiments and simulations show that this Blasius solution is an accurate approximation of the flow for Reynolds numbers up to 10 5 .Still, the range of validity of the Prandtl approximation is limited, due to hydrodynamic instabilities. Two destabilizing mechanisms are well known:

In this work, we are interested in the small time global null controllability for the viscous Burgers' equation yt − yxx + yyx = u(t) on the line segment [0, 1]. The second-hand side is a scalar control playing a role similar to that of a pressure. We set y(t, 1) = 0 and restrict ourselves to using only two controls (namely the interior one u(t) and the boundary one y(t, 0)). In this setting, we show that small time global null controllability still holds by taking advantage of both hyperbolic and parabolic behaviors of our system. We use the Cole-Hopf transform and Fourier series to derive precise estimates for the creation and the dissipation of a boundary layer.