Preprint 2002-027

On the Attainable set for Temple Class Systems with Boundary Controls

Fabio Ancona and Giuseppe Maria Coclite

Abstract: Consider the initial-boundary value problem for a strictly hyperbolic, genuinely nonlinear, Temple class system of conservation laws

ut+f(u)x=0,     u(0,x)= u0(x),     u(t,a)= ua(t),   u(t,b)= ub(t),         (1)
on the domain \Omega = {(t,x) in R2 : t >= 0, a <= x <= b}. We study the mixed problem (1) from the point of view of control theory, taking the initial data u0 fixed, and regarding the boundary data ua, ub as control functions that vary in prescribed sets Ua, Ub, of Linf boundary controls. In particular, we consider the family of configurations
A(T) = { u(T,.);    u is a solution to (1),    ua in Ua,  ub in Ub }
that can be attained by the system at a given time T>0, and we give a description of the attainable set A(T) in terms of suitable Oleinik-type conditions. We also establish closure and compactness of the set A(T) in the L1 topology.

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Fabio Ancona<, <>
Giuseppe Maria Coclite, <>
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<> May 15 2002.

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