Identification problem for periodic coefficients of a linear controlled object (Q5947822)

Let the motion of a guided object be described by the linear differential equation with periodic coefficients of the form \[ x^{(n)}+ a_1(t)x^{(n-1)}+ \dots + a_n(t)x = \rho(t)u, \tag{1} \] where \(n\geq 1\) is an integer, \(a_i(t), \rho(t)\) are periodic functions of the period \(T>0\) continuous on \(\mathbb{R}^1\), \(u\) is the control parameter \((|u |\leq 1)\). The initial conditions for \(x(t)\) are given in the form \[ x(0)=\xi_1,\dots ,x^{(n-1)}(0)=\xi_n. \tag{2} \] The coefficients \(a_i(t), \rho(t)\) are considered to be unknown. It is required to find a piecewise constant control function \(u(t,x_t(.))\), where \(x_t(.)= \{x(s), 0 \leq s \leq t \}\) so that the unknown coefficients in (1) can be reconstructed for every vector \(\xi \in \mathbb{R}^n\) with coordinates \(\xi_i\) (from (2)).