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We investigate a measurement-feedback process of repeated operations withtime delay. During a finite-time interval, measurement on the system isperformed and the feedback protocol derived from the measurement outcome isapplied with time delay. This protocol is maintained into the next intervaluntil a new protocol from the next measurement is applied. Unlike a feedbackprocess without delay, both memories associated with previous and presentmeasurement outcomes are involved in the system dynamics, which naturallybrings forth a joint system described by a system state and two memory states.The thermodynamic second law provides a lower bound for heat flow into athermal reservoir by the (3-state) Shannon entropy change of the joint system.However, as the feedback protocol depends on memory states sequentially, we candeduce a tighter bound for heat flow by integrating out irrelevant memorystates during dynamics. As a simple example, we consider the so-called colddamping feedback process where the velocity of a particle is measured and adissipative feedback protocol is applied to decelerate the particle. We confirmthat the heat flow is well above the tightest bound. We also examine thelong-time limit of this feedback process, which turns out to exhibit aninteresting instability transition as well as heating by controlling parameterssuch as measurement errors, time interval, protocol strength, and time delaylength. We discuss the underlying mechanism for instability and heating, whichmight be unavoidable in reality.

Information thermodynamics for a multi-feedback process with time delay