Frequency domain approach for stochastic systems

The classical approach of frequency domain is employed to tackle the stochastic linear quadratic (LQ) problems in infinite time horizon with control-dependent diffusions and indefinite cost weighting matrices, and the underlying linear matrix inequality, starting with the introduction of a frequency characteristic. The equivalence is established among the unique solvability of the LQ problem, the solvability of the associated stochastic Riccati equation, the coercivity of some bilinear form in the Hilbert space of the admissible controls, and the uniformly positive definiteness of the frequency characteristic matrix. Moreover, under the assumption that the drift part of the system is completely controllable, a stochastic analogue of the Kalman-Yakubovich-Popov theorem, also known as the positive real lemma, is established