On the controllability of the Burger equation

Abstract

Abstract. We p r e s e n t here a return method to describe some attainable sets on an interval of the classical Burger equation by means of the variation of the domain. Statement of the main resultWe are here interested in the following problem of controllability: Let T c be an arbitrary real number, X a given normed space of real functions of the real variable de ned on 1 2], z 1 z 0 elements of X. Does there exist a weak (entropic) solution of: where " > 0 is a priori given, and if this holds for any ( z 0 z 1 ) 2 X X and any " > 0, one says that there is approximate controllability i n X.According to the settlement of the controllability problem, we take here X = BV ( 1 2 here Df is the derivative o f f in the sense of measures.The following conditions (ES) are the conditions necessarily satis ed by spatial values of weak entropic solutions of the Cauchy problem (1.2) with initial value in the class of functions with bounded variation: we s a y that a Universit e d e V ersailles, Analyse appliqu ee,

Asymptotic stabilization of entropy solutions to scalar conservation laws through a stationary feedback law

“…we can apply all the results of the preceding section to the entropy solution u of (14), (17) and (18). We begin the proof of Theorem 1 with the following geometric lemma.…”

“…The feedback laws (17) and (29) act in two steps. In the first step the control g uniformly increases the state u(t, .)…”

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Remarks on exact controllability for the Navier-Stokes equations

“…we obtain from the previous inequality 18) where the operators B * (v, ·), B * (·,v) are defined in (2.3). A short calculation shows, that L * is the adjoint of the differential operator which corresponds to the linearization of the Navier-Stokes equations at the pointv.…”

Remarks on global controllability for the Burgers equation with two control forces

“…In [10], the author describes the attainable set of the inviscid one-dimensional Burgers equation. In particular, he proves that by means of a boundary control, the Burgers equation can be driven from the null initial condition to a constant final state M in a time T 1/M.…”