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Rufus Isaacs (whose work was published in 1965) formulated missile versus enemy aircraft pursuit schemes in terms of descriptive and navigation variables (state and control), and established a fundamental principle: the tenet of transition. The seminal contributions of Isaacs together with the classic research of Bellman on dynamic programming and Pontryagin et al. on optimal control laid the foundations of deterministic differential games. Early research in differential games centers on the extension of control theory problems.

N∗ (s) , x[2] (t0 ) = x0 , .. .. .. .. ∗ x˙ [n] (s) = f s, x[n] (s) , υ1∗ (s) , υ2∗ (s) , . . , υn−1 (s) , υn (s) , x[n] (t0 ) = x0 . The set of strategies {υ1∗ (s) , υ2∗ (s) , . . , υn∗ (s)} is known as a Nash equilibrium of the game. 1 Open-loop Nash Equilibria If the players choose to commit their strategies from the outset, the players’ information structure can be seen as an open-loop pattern in which η i (s) = {x0 }, s ∈ [t0 , T ] . Their strategies become functions of the initial state x0 and time s, and can be expressed as {ui (s) = ϑi (s, x0 ) , for i ∈ N }.

67) in which the payoff of Player 1 is the negative of that of Player 2. Under either MPS or CLPS information pattern, a Nash equilibrium solution can be characterized as follows. 2. 67) if there exists a function V : [t0 , T ] × Rm → R satisfying the partial differential equation: −Vt − 1 2 n Ω hζ (t, x) Vxh xζ h,ζ=1 = min1 max2 {g [t, x, u1 , u2 ] + Vx f [t, x, u1 , u2 ]} u1 ∈S u2 ∈S = max2 min1 {g [t, x, u1 , u2 ] + Vx f [t, x, u1 , u2 ]} = u2 ∈S u1 ∈S {g [t, x, φ∗1 (t, x) , φ∗2 (t, x)] + Vx f [t, x (t) φ∗1 (t, x) , φ∗2 (t, x)]} , V (T, x) = q (x) .

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Functional integral in statistic mechanics by Jensen

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