On the attainable set for scalar nonlinear conservation laws with boundary control. (English) Zbl 0919.35082

The authors consider the initial value problem with boundary control for a scalar nonlinear conservation law \(u_t+ f(u)_x= 0\), \(u(0, x)= 0\), \(u(.,0)= v\in U\) for \(x>0\), \(t>0\), where \(U\) is a set of bounded boundary data regarded as controls, and \(f\) is assumed to be strictly convex. They give a characterization of the set of attainable profiles at fixed time \(T\) {\(u(T,.)\); \(u\) is a solution} and at a fixed point \(x\) {\(u(.,x)\); \(u\) is a solution}. Moreover, they prove that these sets are compact subsets of \(L_1\) and \(L_{1,\text{loc}}\) respectively, whenever \(U\) is a set of controls which pointwise satisfy closed convex constraints, together with some additional integral inequalities. Finally, they apply obtained results to the model of traffic flow.
Reviewer: A.Doktor (Praha)


35L65 Hyperbolic conservation laws
35B37 PDE in connection with control problems (MSC2000)
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