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

TL;DR

This paper extends the Yakubovich Frequency Theorem to less restrictive conditions, characterizing when the infinite horizon quadratic minimization problem is solvable and identifying the set of initial data for which solutions exist.

Contribution

It establishes partial solvability conditions, characterizes initial data sets, and derives the minimum value and solutions under broader circumstances than previous results.

Findings

Identifies less restrictive conditions for problem solvability

Characterizes initial data sets with existing minima

Derives minimum value and solutions for broader cases

Abstract

The Yakubovich Frequency Theorem, in its periodic version and in its general nonautonomous extension, establishes conditions which are equivalent to the global solvability of a minimization problem of infinite horizon type, given by the integral in the positive half-line of a quadratic functional subject to a control system. It also provides the unique minimizing pair "solution, control" and the value of the minimum. In this paper we establish less restrictive conditions under which the problem is partially solvable, characterize the set of initial data for which the minimum exists, and obtain its value as well a minimizing pair. The occurrence of exponential dichotomy and the null character of the rotation number for a nonautonomous linear Hamiltonian system defined from the minimization problem are fundamental in the analysis.