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Control of Diffeomorphisms and Densities Andrei A. Agrachev SISSA, Trieste, Italy and Steklov Mathematical Institute, Moscow, Russia e-mail: agrachev@sissa.it Consider a classical control system as it was defined by Pontryagin: x = f (x, u), x M , u U. (1)

Assume that the state space M is a smooth manifold, the set of control parameters U is a closed subset of another smooth manifold, the righthand side f is smooth, and a reasonable completeness assumption allows to extend solutions of ordinary differential equations to the whole time axis. We call controls the mappings u : (t, x) u(t, x) with values in U that are smooth with respect to x and measurable bounded with respect to t: a mixture of the program and feedback controls. Now plug-in a control in system (1) and obtain a time-varying ordinary differential equation x = f (x, u(t, x)), (2)

which generates a family of diffeomorphisms Pt : M M , where P0 (x) = x ands the curves t Pt (x) satisfy (2) for any x M . We say that t Pt is an admissible "tra jectory" in the group of diffeomorphisms associated to the control u. Given an integral cost functional
T

J (u(·)) =
0

(x(t), u(t)) dt

and a probability measure µ on M , we set
T

Jµ (u) =
0 M

(Pt (x), u(t, x)) dµdt

a functional on the space of controls u. In my talk, I am going to discuss the controllabilty and optimal control issues for the defined in this way systems on the group of diffeomorphisms.