Non controllability of first order quasilinear equations (Q1175616)

The author considers the first order quasilinear equation \(y_ t+\sum^ N_{i=1} (a_ j(y))x_ i=u\), in \(\mathbb{R}^ n\times \mathbb{R}^ +\), \(y(x,0)=y_ 0(x)\in\mathbb{R}^ n\) with control constraint \(\| u(t)\|_{L^ 1(\mathbb{R}^ n)}\leq\rho\). He proves that if \(a\) is continuous and satisfies \(\lim_{|\gamma|\to 0}\| a(\gamma)\|_{|\gamma|}<\infty\) then the feedback control \(u(t)=-\rho \text{sqn }y(t) t\geq 0\) steers every initial data \(y_ 0\in\overline{D(A)}\cap L^ \infty(\mathbb{R}^ n)\) into the origin in finite time \(T\leq \rho^{-1}\| y_ 0\|_{L^ 1(\mathbb{R}^ n)}\). Furthermore if \(\| a(\gamma)\|\leq \gamma|\gamma| \forall\gamma\in \mathbb{R}\), for every \(y_ 0\in\overline{D(A)}\) the system (1.1) is null controllable by the feedback control \(u=-By\) in time \(T\leq \rho^{-1}\| y_ 0\|_{L^ 1(\mathbb{R}^ n)}\). The proof relies on the property of certain nonlinear evolution equations to have compact support in \(t\).