Limit domain of attainability for a linear oscillating third-order system of a special type

The problem under consideration is to find periodic trajectories lying on the boundary of the limit region of reachability of a linear time-invariant third order system with one controlling action bounded in absolute value. It is assumed that the characteristic equation of a homogeneous system has one negative real root and two complex conjugates roots, the real parts of all three roots are the same. The results obtained make it possible to construct the boundary of the limit reachability region (for an infinitely long control time) in the form of analytical expressions on the system parameters.