Minimizing age of information with soft updates

We consider an information updating system where an information provider and an information receiver engage in an update process over time. Different from the existing literature where updates are countable (hard) and take effect either immediately or after a delay, but instantaneously in both cases, here updates start taking effect right away but gradually over time. We coin this setting soft updates. When the updating process starts, the age decreases until the soft update period ends. We constrain the number of times the information provider and the information receiver meet (number of update periods) and the total duration of the update periods. We consider two models for the decrease of age during an update period: In the first model, the rate of decrease of age is proportional to the current age, and in the second model, the rate of decrease of age is constant. The first model results in an exponentially decaying age, and the second model results in a linearly decaying age. In both cases, we determine the optimum updating schemes, by determining the optimum start times and optimum durations of the updates, subject to the constraints on the number of update periods and the total update duration.