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We study the simultaneously reachable subspace for two strings controlled from a common endpoint. We give necessary and sufficient conditions for simultaneous spectral and approximate controllability. Moreover we prove the lack of simultaneous exact controllability and we study the space of simultaneously reachable states as a function of the position of the joint. For each type of controllability result we give the sharp controllability time.
We study the simultaneously reachable subspace for two strings
controlled from a common endpoint. We give necessary
and sufficient conditions for simultaneous spectral and approximate
controllability. Moreover we prove the lack of simultaneous exact
controllability
and we study the space of simultaneously reachable states
as a function of the position of the joint. For each type of controllability
result we give the sharp controllability time.
In this work we investigate a mathematical model for small vertical vibrations of a stretched string when the ends vary with the time t and the cross sections of the string is variable and the density of the material is also variable, that is, p=p(x). It contains Kirchhoff model for fixed ends. We obtain solutions by Galerkin method and estimates in Sobolev spaces.
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