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Given a family of curves constituting the general solution of a system of ordinary differential equations, the natural question occurs whether the family is identical with the totality of all extremals of an appropriate variational problem. Assuming the regularity of the latter problem, effective approaches are available but they fail in the non-regular case. However, a rather unusual variant of the calculus of variations based on infinitely prolonged differential equations and systematic use of...
This paper deals with some inverse and control problems for the Navier-Stokes and related systems. We will focus on some particular aspects that have recently led to interesting (theoretical and numerical) results: geometric inverse problems, Eulerian and Lagrangian controllability and vortex reduction oriented to shape optimization.
In dimension one it is proved that the solution to a total variation-regularized
least-squares problem is always a function which is "constant almost everywhere" ,
provided that the data are in a certain sense outside the range of the operator
to be inverted. A similar, but weaker result is derived in dimension two.
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