On an identification problem (Q1285474)

The identification theory of control systems is an important part of modern control theory. The problem of finding constant coefficients for the linear controlled object described by second-order ordinary differential equations \[ \ddot x + \alpha \dot x + \beta x = \rho u, \qquad x(0) = x_0, \dot x(0) = \dot x_0, \tag{1} \] where \(\alpha , \beta , \rho \in \mathbb{R}\) are unknown constants, \(u = u(t) \) is a control function, \(| u| \leq 1\), is considered. The whole identification process is going on in the fixed time interval \([0, T]\) by the construction of a piecewise constant control function \(u(t)\).