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Scalar nonlinear conservation laws with integrable boundary data. (English) Zbl 0919.35081
The authors study the initial-boundary value problem for a scalar nonlinear conservation law $u_t+ f(u)_x=0, \;u=u(t,x), \quad t,x\geq 0; \qquad u(0,x)=\overline u(x), \;u(t,0)=\widetilde u(x)$ with integrable (possibly unbounded) initial and boundary data $$\overline u$$, $$\tilde u(x)$$. The flux function $$f(u)$$ is assumed to be a superlinear strictly convex function.
The authors generalize Le Floch’s explicit formula in the case $$\overline u \in L^1$$, $$f(\widetilde u)\in L^1_{\text{loc}}$$. A comparison principle for integrals of solutions is also derived. Then the problem under consideration is investigated for fixed $$\overline u=0$$ as a control problem governed by the boundary data. The corresponding attainable profiles at fixed time and at a fixed space variable are described. Finally the authors apply their results to some optimization problem of traffic flow.

MSC:
 35L65 Hyperbolic conservation laws 93C20 Control/observation systems governed by partial differential equations
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References:
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