Observability of dynamic control games

Consider a controlled evolutionary game (CEG), which is described as a machine-human game. Assume the machines strategy updating rules are known, which are described as the evolutionary equation, and their payoffs y_i, i = 1, ···, n are also known, which are described as the outputs. Then the dynamic model can be described as a standard logical control systems. First, we give a necessary and sufficient condition to judge whether a CEG is observable. That is, for any two initial states x₀ ≠ x₀ can we find a control sequence u₀, u₁, ···, u_s, s <; ∞, such that the outputs are distinguishable? Secondly, we prove that if a CEG is observable, then we can find a control sequence u₀, u₁, ···, u_s, s <; ∞, such that the initial state x₀ is identifiable. An algorithm is provided to calculate it. The importance of this work is: Though the machines´ strategies may completely unknown, it can be recognized via observed payoffs. And then the optimal control and all other control goals can be realized.