What would be a good use for a memristor?
Memristors have a memory like characteristics, non-voltaile memory. So even when they're powered off they can hold on to the value. This sort of technology can replace how we design memory for computers such as RAM, SSD and Flash.

Resistive Random Access Memory (RRAM):


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Many applications can take advantage of it such as neural networks, robotics to mimic biological entities in a new way. DARPA is attempting to build a brain using Memristor chips, designed by HP Labs, for neuromorphic computing.


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They can also replace transistor-based crossbar latches since they take up less space.
Logic design is a big area these components can contribute. From crossbar latches to memory, gates and more

There already exist memristor-based control system patents out there (here's one example)