A linear-quadratic optimization problem and the frequency theorem for nonperiodic systems. I

Abstract

Here x ~ R ~, ~ R ~, the matrices A(t), b(t), G(t) = G(t)*, g(t), F(t) = F(t)* ~ ~01 m > 0 have appropriate orders and their elements are real, measurable, bounded, periodic functions with period T. (Equalities and inequalities hold almost everywhere. An asterisk denotes transposition, while in the case of complex vectors and matrices, it denotes Hermitian conjugation, i.e. transposition and complex conjugation.) Below we consider the following two problems. I. The optimization problem x (0) = a t $ = S $ (t, x (t), u (t)) dt--)- rain (0.3) 0 (a is a given vector), while those x(.), u(.) are admissible for which (0.1) and the stability condition Ix(.)l~L~(O,~), fu(.)l~L~(O, ~) hold. 2. The problem of the determination of the form V(t, x) = x*R0(t)x (the Lyapunov function) such that R0(t) = R0(t + T) = R0(t)*, R0(t) is an absolutely continuous matrix function and for the derivative we have, by virtue of Eq. (0.1), the equality