Quasilinear analysis of stochastic control systems with multiplicative nonlinearities
This article concerns the analysis of systems governed by nonlinear (systems of) differential equations \[ z'(t) = c(z,t) + d(z,t) V(t) \] such as might arise in feedback control systems, where \(c, d\) are nonrandom and known nonlinear functions but \(V(t)\) is a nonwhite noise. The idea of such an equation is that the real-time control errors embodied in \(d(z,t)\) interact multiplicatively with a random process \(V(t)\) whose first and second order moments can be assumed known. Then the approach of the paper toward analyzing the displayed equation is to derive differential equations for the mean and second moment of the solution \(z(t)\). The authors do this in some detail, including two examples, and they point out that the resulting solution process can differ from the modified `statistically linearized' equation for the mean of \(z(t)\) in which the left hand side of the equation is replaced by the derivative of the mean and the function \(z(t)\) on the right-hand side is replaced by the mean of \(z(t)\). The correct equations are briefly applied to study the so-called `p-stability' of the controlled stochastic differential systems.
scientific article; zbMATH DE number 919498