On the Attainable Set for Scalar Nonlinear Conservation Laws with Boundary Control

On the Uniform Controllability of the Burgers Equation

“…We consider the system constituted by (1), (2) and (3). Let us fix the initial condition u 0 in some Banach space X and a constant M = 0; then, our goal is to find two controls v 1 and v 2 such that the associated solution u satisfies u| t=T = M in (0, 1).…”

“…were considered in [1], for which the controllability problem is posed in the half line with null initial condition. The set of attainable states is completely described.…”

Hyperbolic equations and SBV functions

“…However, these theorems require some regularity assumptions on the initial data and it is known that they do not hold if we start from general Lipschitz initial data. Besides their intrinsic interest, the questions (Q1) and (Q2) arise naturally in some problems in the control theory for hyperbolic systems of conservation laws (see for instance [12], [6], [5], [7] and [13]). In this note we will survey some recent results that show that, while the answer to (Q1) is quite easily seen to be negative, an SBV -regularization effect holds for (0.1), (0.2) and (0.4) as soon as the equation is sufficiently nonlinear (note that such a hypothesis is needed since transport equations are special cases of all the equations considered so far).…”

A controllability result for the1-D isentropic Euler equation

“…• Ancona and Marson [4] (1998): for the scalar equation u t + (f (u)) x = 0 with f ≥ c > 0, they give a complete description of the attainable set starting from 0.…”