Abstract

We consider a linear timevarying control system with locally integrable and integrally bounded coefficients
$$\dot x =A(t)x+ B(t)u, \quad x\in\mathbb{R}^n,\quad u\in\mathbb{R}^m,\quad t\geqslant 0. \qquad\qquad (1)
$$
We construct control of the system $(1)$ as a linear feedback $u=U(t)x$ with measurable and bounded function $U(t),$ $t\geqslant 0.$ For the closedloop system
$$\dot x =(A(t)+B(t)U(t))x, \quad x\in\mathbb{R}^n, \quad t\geqslant 0, \qquad \qquad (2)$$
we study a question about the conditions for its uniform global attainability. The last property of the system $(2)$ means existence of a matrix $U(t),$ $t\geqslant 0,$ that ensure equalities $X_U((k+1)T,kT)=H_k$ for the statetransition matrix $X_U(t,s)$ of the system $(2)$ with fixed $T>0$ and arbitrary $k\in\mathbb N,$ $\det H_k>0.$ The problem is solved under the assumption of uniform complete controllability of the system $(1),$ corresponding to the closedloop system $(2),$ i.e. assuming the existence of such $\sigma>0$ and $\gamma>0,$ that for any initial time $t_0\geqslant 0$ and initial condition $x(t_0)=x_0\in \mathbb{R}^n$ of the system $(1)$ on the segment $[t_0,t_0+\sigma]$ there exists a measurable and bounded vector control $u=u(t),$ $\u(t)\\leqslant\gamma\x_0\,$ $t\in[t_0,t_0+\sigma],$ that transforms a vector of the initial state of the system into zero on that segment. It is proved that in twodimensional case, i.e. when $n=2,$ the property of uniform complete controllability of the system $(1)$ is a sufficient condition of uniform global attainability of the corresponding system $(2).$

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