The objective of the delayed feedback control (DFC) is to stabilize an unstable periodic orbit embedded in a chaotic attractor by adding a delayed feedback term to the original differential system. The DFC is proposed by Pyragas (1992) as an alternative method, different from the one by Ott, Grebogi, and Yorke (1990).

For an ODE x' = f(x) and its orbitally unstable periodic solution γ(t)
with the period T > 0, we consider the following differential equation:

(*) x'(t) = f(x(t)) + K(x(t - τ) - x(t)).

Here K is an n × n real matrix, and τ = mT, where m is a positive integer. Then the problem of the DFC is to find K and τ such that γ(t) is orbitally stable as a solution of (*).

In this talk, as a special case of the above problem, we consider the stabilization problem of an unstable steady solution of an ODE by DFC.
In this case, we can take τ as an arbitrarily positive number. I will talk about the following:

1. Motivation:
(i) the OGY and the Pyragas method for controlling chaos,
(ii) the result of Fiedler et al. (2007) (the relationship between the stabilization of a periodic solution and a steady solution),

2. Background:
the dynamics of delay differential equations
((*) is not a usual ODE but a special "delay differentialequation."),

3. Main result:
"the stabilization of an unstable steady solution by DFC is possible on the assumption of the unstable eigenvalues of the corresponding steady state,"

4. Outline of proof:
the analysis of the characteristic equation by using the Lambert W function.