Stabilization and destabilization of a circulatory system by small velocity-dependent forces

A linear autonomous mechanical system under non-conservative positional forces is considered. The influence of small forces proportional to generalized velocities on the stability of the system is studied. Necessary and sufficient conditions are obtained for the matrix of dissipative and gyroscopic forces to make the system asymptotically stable. A system with two degrees of freedom is studied in detail. Explicit formulae describing the structure of the stabilizing matrix and the stabilization domain in the space of the matrix elements are found and plotted. As a mechanical example, a problem of stability of the Ziegler–Herrmann–Jong pendulum is analyzed.