On the Attainable Set for Scalar Nonlinear Conservation Laws with Boundary Control
On the Attainable Set for Scalar Nonlinear Conservation Laws with Boundary Control
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DOI：
10.1137/s0363012996304407
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发表时间：
1998
影响因子：
2.2
通讯作者：
F. Ancona;A. Marson
中科院分区：
文献类型：
--
作者：
F. Ancona;A. Marson
We consider the initial value problem with boundary control for a scalar nonlinear conservation law \begin{equation*} u_t+[f(u)]_x=0,\qquad\qquad u(0,x)=0,\qquad u(\cdot,0)= \tilde u\in{\cal U},\tag{$\ast$} \end{equation*} on the domain $\Omega=\{(t,x)\in\real^2: t\geq 0, x\geq 0\}$. Here $u=u(t,x)$ is the state variable, ${\cal U}$ is a set of bounded boundary data regarded as controls, and $f$ is assumed to be strictly convex. We give a characterization of the set of attainable profiles at a fixed time $T>0$ and at a fixed point $\bar x>0$: \begin{equation*} \begin{aligned} \rag &=\{u(T,\cdot): u \hbox{is a solution of} (\ast)\},\\ \ragx &=\{u(\cdot,\bar x): u \hbox{is a solution of} (\ast)\}, \end{aligned} \qquad\quad{\cal U}= \ellein({\Bbb R}^+). \end{equation*} Moreover we prove that $\rag$ and $\ragx$ are compact subsets of $\elleuno$ and $\elleuno_{loc}$, respectively, whenever ${\cal U}$ is a set of controls which pointwise satisfy closed convex constraints, together with some additional integral inequalities.
We consider the initial value problem with boundary control for a scalar nonlinear conservation law \begin{equation*} u_t+[f(u)]_x=0,\qquad\qquad u(0,x)=0,\qquad u(\cdot,0)= \tilde u\in{\cal U},\tag{$\ast$} \end{equation*} on the domain $\Omega=\{(t,x)\in\real^2: t\geq 0, x\geq 0\}$. Here $u=u(t,x)$ is the state variable, ${\cal U}$ is a set of bounded boundary data regarded as controls, and $f$ is assumed to be strictly convex. We give a characterization of the set of attainable profiles at a fixed time $T>0$ and at a fixed point $\bar x>0$: \begin{equation*} \begin{aligned} \rag &=\{u(T,\cdot): u \hbox{is a solution of} (\ast)\},\\ \ragx &=\{u(\cdot,\bar x): u \hbox{is a solution of} (\ast)\}, \end{aligned} \qquad\quad{\cal U}= \ellein({\Bbb R}^+). \end{equation*} Moreover we prove that $\rag$ and $\ragx$ are compact subsets of $\elleuno$ and $\elleuno_{loc}$, respectively, whenever ${\cal U}$ is a set of controls which pointwise satisfy closed convex constraints, together with some additional integral inequalities.