Optimal synthesis in an infinite-dimensional space

For a class of optimal control problems and Hamiltonian systems generated by these problems in the space $l_2$, we prove the existence of extremals with a countable number of switchings on a finite time interval. The optimal synthesis that we construct in the space $l_2$ forms a fiber bundle with piecewise smooth two-dimensional fibers consisting of extremals with a countable number of switchings over an infinite-dimensional basis of singular extremals.