On Control in the Presence of Small Random Perturbations by a Single Switching Operation

The motion of a system described by the vector differential equation $\dot x=f(t,x,\varepsilon)$ is consided, where $\varepsilon$ is the vector of random perturbations and
$$ f(t,x,\varepsilon)=\begin{cases} f^1(x,\varepsilon),&0\leqq\tau\leqq t, \\ f^2(x,\varepsilon),&t>\tau. \end{cases} $$
The aim of the control is to minimize $\lambda$, the principal part (with respect to the perturbations) of the variance of the functional $V[x(\tau+t,\varepsilon);t\geqq 0]$. Control is accomplished by regulating the moment of switching $\tau$. Let $u(t,\varepsilon)$ be a given set of functions. The decision when to switch is taken when a certain function $\varphi(u)$ (a characterizing function) becomes equal to a present value. The undisturbed motion and probability properties of the perturbations are known.The problem of finding the optimum characterizing function is studied. It is sufficient to consider only linear combinations $(\Phi,u)$ as $\varphi(u)$. An inhomogeneous system of linear algebraic equations is derived for determining the optimum coefficients $\Phi_1,\dots,\Phi_s$ and a suitable value for $\lambda$. The conditions for the existence of a solution, its uniqueness and of the equality $\lambda=0$ are investigated. An example is given.