Discrete-delay systems with a single delay are generally expressed as
where $h > 0$ is the delay, $x(t)$ is the state at time $t$, and $\phi$ is the functional initial condition. They generalize the usual dynamical systems whose future depends only on the current state. Delays are generally detrimental to stability and tend to destabilize systems and degrade performance, although in some problems they can improve stability or induce desirable oscillations such as limit cycles in biological systems.
Why they are challenging
Analyzing delay systems is harder than ordinary dynamical systems because of their infinite-dimensional nature. Both frequency-domain and time-domain techniques exist. When delays are time-varying, time-domain approaches are preferable, relying on the Lyapunov-Krasovskii and Lyapunov-Razumikhin theorems and on robust analysis methods such as integral quadratic constraints and well-posedness. When the delay is state-dependent, as in communication networks, very few general results exist and the problem remains largely open.
Control with delays
Controlling delay systems is itself a rich problem. For input-delay systems, state predictors can be used to anticipate future states for use in the control law. When the delay acts on the state, controllers involving delay components can be considered, either by using delayed signals (controllers with memory) or by letting the delay schedule the controller in an LPV fashion. The knowledge of the delay is critical for implementation, and when it is time-varying or uncertain its exact value is rarely available in real time, which is what makes the problem interesting.
Related publications
For the full list, see the publications page (search for "delay" or "TDS").