Stability of two-dimensional SISO LTI system with bounded feedback gain that has bounded derivative

We consider a two-dimensional SISO LTI system closed by uncertain linear feedback. The feedback gain is time-varying, bounded, and has a bounded derivative (both bounds are known). We investigate the asymptotic stability of this system under all admissible behaviors of the gain. Note that the situation is similar to the classical absolute stability problem of Lurie--Aizerman with two differences: linearity and derivative constraint. Our method of analysis is therefore inspired by the variational ideas of Pyatnitskii, Barabanov, Margaliot, and others developed for the absolute stability problem. We derive the Hamilton--Jacobi--Bellman equation for a function describing the "most unstable" of the possible portraits of the closed-loop system. A numerical method is proposed for solving the equation. Based on the solution, sufficient conditions are formulated for the asymptotic stability and instability. The method is applied to an equation arising from the analysis of a power electronics synchronization circuit.