Controlling the controller: hysteresis-delay differential equations

Abstract:

Our talk revolves around differential equations with hysteresis and delay terms. We focus on the problem of stability analysis of periodic solutions of such equations. This problem is infinite-dimensional and discontinuous due to the delay and hysteresis. We present a technique to reduce it, in certain cases, to the spectral problem for a linear finite-dimensional operator.

Our main application is a thermal control model. It consists of a parabolic equation with hysteresis on the boundary. Gurevich and Tikhomirov recently showed the existence of both stable and unstable periodic solutions for such a model. Their result naturally raises the question of whether it is possible to change the stability properties of such solutions.

We use the well-known Pyragas control to change the stability of periodic solutions of the thermal control model. Using this method, one adds an additional delay term to the boundary without destroying the known periodic solution. This results in a parabolic equation with both hysteresis and delay terms on the boundary. Using Fourier decomposition, this equation is reduced to a system of ODE. Then we can apply our finite dimensional reduction technique and show that Pyragas control can change stability of periodic solutions.