Small-time global stabilization of the viscous Burgers equation with three scalar controls

Abstract

We construct explicit time-varying feedback laws leading to the global (null) stabilization in small time of the viscous Burgers equation with three scalar controls. Our feedback laws use first the quadratic transport term to achieve the small-time global approximate stabilization and then the linear viscous term to get the small-time local stabilization.

Fredholm transformation on Laplacian and rapid stabilization for the heat equation

“…These PDEs usually do not enter in the classical Cauchy problem framework, but different techniques are now known to solve the kernel equation: successive approximations [33], explicit representations [33] or method of characteristics [19]. There exists now a vast literature on the backstepping method with Volterra transformations: let us cite for the heat/parabolic equation [3,9,22], for hyperbolic systems [6] and for the viscous Burgers equation [24]. We refer to [33] to a general overview of the backstepping method with Volterra transformations.…”

On the stability and null-controllability of an infinite system of linear differential equations

“…Obviously, Θ ∈ ℓ 2 and e tE Θ = e tθ Θ. Since −1 < λ ≤ 0 if we let θ = −λ + |1+λ| 2 ∈ (0, 1) then as t → +∞ one gets (10) e tA Θ 2 = e tλ e tE Θ 2 = e t|1+λ|/2 Θ 2 → +∞.…”

“…Control problems in Banach or Hilbert spaces arise naturally in processes described by partial differential equations (see for example [2,6,7,10,12,14,15,19,22] and references therein). Sometimes it is useful to reduce the control problem for partial differential equations to infinite systems of ODEs [3,4,8,9].…”

On the uniform controllability for a family of non-viscous and viscous Burgers-α systems

“…Remark 4.1 Proposition 8 is also true when y 0 ∈ L ∞ (0, L). Indeed, we can start using Proposition 3 that guarantees the existence and uniqueness of a solution (y α , z α ) to ( 42) satisfying (17) and (18). In particular, we have from ( 18) that…”

“…Global and local well-posedness are respectively established in [23] (when the boundary conditions are homogeneous) and [45,13]. Moreover, there are many important works dealing with the controllability properties of parabolic equations and systems (see [22,25,28,31]) and non-viscous and viscous Burgers equations (see [11,17,20,26,31,33,34,38,41,44]). In the case of the Burgers-α system, the local uniform null controllability of the viscous system (2) with distributed and boundary controls was studied in [1]; later, the results have been extended to any equation of the b-family in [27].…”