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On Control in the Presence of Small Random Perturbations by a Single Switching Operation
A. P. Čerenkov Moscow
Abstract:
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.
Received: 05.02.1963
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