Control over some asymptotic invariants of two-dimensional linear control systems with an observer

We consider a linear time-varying control system with an observer with locally integrable and integrally bounded coefficients
\begin{gather} \dot x =A(t)x+ B(t)u, \quad x\in\mathbb{R}^n,\quad u\in\mathbb{R}^m,\quad t\geqslant 0, \\ y =C^*(t)x, \quad y\in\mathbb{R}^p. \end{gather}
We study a problem of control over asymptotic invariants for the system closed by linear dynamic output feedback with time-varying coefficients. The research method presented in the paper is based on the construction of a system of asymptotic estimation for the state of the system (1), (2), introduced by R. Kalman. For solving the problem, we use the extension of the notion of uniform complete controllability (in the sense of Kalman) proposed by E.L. Tonkov for systems with coefficients from wider functional classes. The notion of uniform complete observability (in the sense of Tonkov) is given for the system (1), (2). For $n=2$, it is proved that uniform complete controllability and uniform complete observability (in the sense of Tonkov) of the system (1), (2) with locally integrable and integrally bounded coefficients are sufficient for arbitrary assignability of the upper Bohl exponent and of the complete spectrum of the Lyapunov exponents for the system closed-loop by linear dynamic output feedback. For the proof, we use the previously established results on uniform global attainability of a two-dimensional system (1), closed by linear time-varying static state feedback, under the condition of uniform complete controllability (in the sense of Tonkov) of the open-loop system (1).