Representation of Contractive Solutions of a Class of Algebraic Riccati Equations as Characteristic Functions of Maximal Dissipative Operators

Abstract.

Let

$$ j_{mm} = \left(\matrix{{I_m}&{0}\cr 0&{-I_m}\cr}\right), \quad {\mathcal{J}}_{m} = \left(\matrix{{0}&{-iI_m}\cr {iI_m}&{0}\cr}\right), $$

I _m is the identity matrix of order m. Let W(λ) be an entire matrix valued function of order 2m, W(0) = I _2m , the values of W(λ) are j _mm -unitary at the imaginary axis and strictly j _mm -expansive in the open right half-plane. The blocks of order m of the matrix W(λ) with appropriate signs are treated as coefficients of algebraic Riccati equation. It is proved that for any λ with positive real part this equation has a unique contractive solution θ(λ). The matrix valued function θ(λ) can be represented in a form θ(λ) = θ _A (iλ) where θ _A (μ) is the characteristic function of some maximal dissipative operator A. This operator is in a natural way constructed starting from the Hamiltonian system of the form

$$ \frac{dx(\tau)}{d \tau} = i \mathcal{J}_{m}K(\tau)x(\tau), \quad \tau \in [0;+\infty) $$

with periodic coefficients.