The stable regulator problem and its inverse

This paper provides a detailed survey and extension of certain properties of the stable regulator problem: determine <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">mininf{u} ∫min{0}max{∞} x'Qx + u'u dt</tex> subject to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ẋ = Fx + Gu; x(0) = x₀; {t → ∞} x(t) = 0</tex> where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</tex> is not necessarily sign definite. First, equivalence conditions recently given by Willems for the existence of the minimum are extended to include statements in terms of the Hamiltonian matrix and spectral factorization. This provides a precise relation between the time-domain and frequency-domain solutions to the problems. Second, the inverse problem of whether a given feedback <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u = -Kx</tex> is optimal for some <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</tex> is easily resolved, as is the redundancy problem of distinct Q <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> and Q <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> , resulting in the same optimal control.