  Active cyber defense is one important defensive method for combating cyber
attacks. Unlike traditional defensive methods such as firewall-based filtering
and anti-malware tools, active cyber defense is based on spreading "white" or
"benign" worms to combat against the attackers' malwares (i.e., malicious
worms) that also spread over the network. In this paper, we initiate the study
of {\em optimal} active cyber defense in the setting of strategic attackers
and/or strategic defenders. Specifically, we investigate infinite-time horizon
optimal control and fast optimal control for strategic defenders (who want to
minimize their cost) against non-strategic attackers (who do not consider the
issue of cost). We also investigate the Nash equilibria for strategic defenders
and attackers. We discuss the cyber security meanings/implications of the
theoretic results. Our study brings interesting open problems for future
research.
