## 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)

### MSC:

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