Summary: The paper treats the initial boundary value problem for a scalar conservation law with strictly convex flux function. The boundary data is a Lebesgue-measurable and bounded function regarded as a control and constrained to remain in a prescribed set $U$ of admissible controls. A time $T>0$ being fixed, the authors characterize the set $A(T,U)$ consisting of the corresponding entropy solutions at the time $t=T$. Under natural assumptions on $U$, it is proven that $A(T,U)$ is a compact subset of $L^1$. Such a compactness property provides the key information in order to establish the existence of solutions for a class of optimisation problems. Finally the results are applied by the authors to an optimisation problem concerning a model of traffic flow on a highway.
On the attainable set for scalar nonlinear conservation laws with boundary control
ANCONA, FABIO;MARSON, ANDREA
1998
Abstract
Summary: The paper treats the initial boundary value problem for a scalar conservation law with strictly convex flux function. The boundary data is a Lebesgue-measurable and bounded function regarded as a control and constrained to remain in a prescribed set $U$ of admissible controls. A time $T>0$ being fixed, the authors characterize the set $A(T,U)$ consisting of the corresponding entropy solutions at the time $t=T$. Under natural assumptions on $U$, it is proven that $A(T,U)$ is a compact subset of $L^1$. Such a compactness property provides the key information in order to establish the existence of solutions for a class of optimisation problems. Finally the results are applied by the authors to an optimisation problem concerning a model of traffic flow on a highway.Pubblicazioni consigliate
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