On the Mathematical Theory of a Time-varying Bilinear Stochastic Differential System and its Application to Two Dynamic Circuits

抄録

Bilinear systems are ubiquitous in dynamics and control literature. The concept of bilinear systems is attributed to input-output coupling terms. Stochastically influenced bilinear systems are described via bilinear stochastic differential equations. Bilinear systems are attractive and popular in dynamical systems and control literature for two reasons: (i) first, they offer closed-form solutions for time-varying as well as time-invariant settings (ii) they preserve some of the qualitative characteristics of non-linear stochastic systems. This paper chiefly intends to construct a mathematical theory of a scalar time-varying bilinear ‘Stratonovich’ stochastic differential equation with a vector random input by deriving its closed-form solution and related results. Secondly, the analytic results of the paper are applied to a series RL electrical circuit and sampling mixer circuit Stochastic Differential Equations (SDE). The theory of this paper hinges on the ‘Stratonovich calculus’. This paper will be of interest to dynamists, stochasticians looking for advances in bilinear systems and their control. More specifically, this paper opens up research directions in stability and control of bilinear stochastic systems by exploiting the analytic results of this paper.