On the solvability of the Yakubovich linear-quadratic infinite horizon minimization problem

The Yakubovich Frequency Theorem, in its periodic and in its general nonautonomous extension, establishes conditions which result to be equivalent to the global solvability of a minimization problem of infinite horizon type, given by an integral of a quadratic functional subject to a control system. The solvability of the minimization problem is formulated in terms of the property of a correspondinglinear Hamiltonian systems associated to the problem. In the paper, some conditions under which the problem is partially solvable are presented ; moreover the set of initial data for whichthe minimum exists is characterized and the values of the minimum aswell of the minimizing pair are provided. In the analysis result tobe fundamental the occurence of an exponential dichotomy and the null character of the rotation number for the corresponding linear nonautonomous Hamiltonian system.