Return condition for oscillating systems

Abstract

This paper is devoted to the problem of null-controllability for the oscillating linear system $\dot{x}_{2i-1} = x_{2i}, \dot{x}_{2i} = - x_{2i-1} + u$, $i = \overline{1,n}$ with constraints on the control $u \in [c, 1]$ and $u \in \{c, 1\}$ in the case when the origin is not an equilibrium point. Null-controllability means that there exists a moment of time $T_0$ such that for any time $T > T_0$ we are able to reach the origin in precisely this time. The criterion of controllability into a non-equilibrium point was obtained by V. I. Korobov and a new condition called the return condition on the interval was introduced, which must be satisfied, together with the classical conditions for controllability into an equilibrium point. This condition means that there exists a time interval $I = [T, T + \alpha]$, $\alpha > 0$ so that a trajectory starting from the origin may return there at any time $T\in I$.
The aim of this paper is to show that the return conditions are satisfied for the considered oscillatory system, and to obtain the analytical solution for the control that solves the return condition problem. The considered approach involves constructing a piecewise-constant control using values $u = c$ and $u = 1$ and transforming the problem into a trigonometric moment problem on an interval. This problem has a non-unique solution, and in our paper we present one involving $2n$ switching points and another with only $2$ in the case when $c \le \frac{1}{2}$. The solution with $2$ switching moments is especially interesting since it does not depend on the dimensionality of the system. We also generalize the problem to the case where the eigenvalues are of the form $\lambda_{2j}, \lambda_{2j-1} = \pm \nu_k i$, where $\nu_k$ are rational numbers. Additionally, we discuss some partial cases where $c > \frac{1}{2}$ and where the eigenvalues are irrational.