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MATHEMATICS
The criterion of uniform global attainability of periodic systems
A. A. Kozlov Polotsk State University, ul. Blokhina, 29, Novopolotsk, 211440,
Belarus
Abstract:
We consider a linear time-varying control system
\begin{equation}
\dot x =A(t)x+ B(t)u, \quad x\in\mathbb{R}^n,\quad u\in\mathbb{R}^m,\quad t\in \mathbb{R}
\end{equation}
with piecewise continuous and bounded
$\omega$-periodic coefficient matrices
$A (\cdot)$ and
$B (\cdot).$ We construct control of the system
$(1)$ as a linear feedback
$u=U(t)x$ with piecewise continuous and bounded matrix function
$U(t)$,
$t\in \mathbb{R}$. For the closed-loop system
\begin{equation}
\dot x =(A(t)+B(t)U(t))x, \quad x\in\mathbb{R}^n, \quad t\in \mathbb{R},
\end{equation}
the conditions of its uniform global attainability are studied. The latest property of the system (2) means existence of matrix
$U(t)$,
$t\in \mathbb{R}$, ensuring equalities
$X_U((k+1)T,kT)=H_k$ for the state-transition matrix
$X_U(t,s)$ of the system (2) with fixed
$T>0$ and arbitrary
$k\in\mathbb{Z}$,
$\det H_k>0$. The problem is solved under the assumption of uniform complete controllability (by Kalman) of the system (1), corresponding to the closed-loop system (2), i.e. assuming the existence of such numbers
$\sigma>0$ and
$\alpha_i>0,$ $i=\overline{1,4}$, that for any number
$t_0\in\mathbb{R}$ and vector
$\xi\in \mathbb{R}^n$ the following inequalities hold:
$$\alpha_1\|\xi\|^2\leqslant \xi^*\int\nolimits_{t_0}^{t_0+\sigma}X(t_0,s)B(s)B^*(s)X^*(t_0,s)\,ds\,\xi\leqslant\alpha_2\|\xi\|^2,$$
$$\alpha_3\|\xi\|^2\leqslant\xi^*\int\nolimits_{t_0}^{t_0+\sigma}X(t_0+\sigma,s)B(s)B^*(s)X^*(t_0+\sigma,s)\,ds\,\xi\leqslant\alpha_4 \|\xi\|^2,$$
where
$X(t,s)$ is the state-transition matrix of linear system (1) with
$u(t)\equiv0.$ It is proved that the property of uniform complete controllability (by Kalman) of the periodic system (1) is a necessary and sufficient condition of uniform global attainability of the corresponding system (2).
Keywords:
linear control system with periodic coefficients, uniform complete controllability, uniform global attainability.
UDC:
517.926,
517.977
MSC: 34D08,
34H05,
93C15 Received: 30.08.2019
DOI:
10.35634/vm200206