Inviscid Burgers

We finally consider non viscous Burgers' equation, with periodic boundary conditions, and for a time

such that there is no shock in the interval

Theorem 3.6 We consider one step of the BFN algorithm applied to the non viscous Burgers' equation (3.39), with observations $u_{obs}$ satisfying (3.39-F) with . We denote

$\begin{displaymath}\begin{array}{rcl} w(t) &=& u(t) - u_{obs}(t), \\ \widetilde{w}(t) &=& \widetilde{u}(t)-u_{obs}(t). \end{array}\end{displaymath}$

(3.40)

We assume that $u_{obs}\in W^{1,\infty}([0,T]\times\Omega)$ , i.e. there exists such that

$\displaystyle \vert\partial_x u_{obs}(t,x)\vert\le M, \quad \forall t\in [0,T], \forall x\in \Omega.$

(3.41)

Then:

If , then we have, for all $t\in[0,T]$ ,

$\displaystyle \Vert\widetilde{w}(t)\Vert \leq e^{(-K-K^\prime+M)(T-t)} \Vert w(t)\Vert.$ (3.42)
If $K(t,x)=K \mathbbm{1}_{[t_{1},t_{2}]}(t)$ with $0\leq t_{1} < t_{2}\leq T$ , then we have

$\displaystyle \Vert\widetilde{w}(0)\Vert \leq e^{(-K-K^\prime)(t_{2}-t_{1})+MT} \Vert w(0)\Vert.$ (3.43)

Proposition 3.2 We consider one forward (resp. backward) BFN step of the non viscous Burgers' equation (3.39-F) (resp. (3.39-B)). With the notations of theorem 3.6, if , then we have

$\displaystyle w(T,\psi(T,x)) = w(0,x) \exp \left( -\displaystyle\int_0^T K(\psi... ...\displaystyle\int_0^T \partial_x u_{obs}(\sigma,\psi(\sigma,x))d\sigma \right).$

(3.44)