A Theory of Differential Games

Abstract

The differential games that we shall consider can be formulated intuitively as follows. The state of the game at time t is given by a vector x(t) in R ⁿ and is determined by a system of differential equations

$$ {{dx} \over {dt}} = f\left( {t,x,u(t),v(t)} \right)\,x({t_0}) = {x_0}, $$

(1.1)

where u(t) is chosen by Player I at each time t and v(t) is chosen by Player II at each time t. The choices are constrained by the conditions u(t) ∊ Y and v(t)∊ Z, where Y and Z are preassigned sets in euclidean spaces. The choice of u(t) is governed by a set of rules or “strategy” U selected by Player I prior to the start of play and the choice of v(t) is governed by a “strategy” V selected by Player II prior to the start of play. Play proceeds from the initial point (t₀,x₀) until the point (t, ϕ(t)), where ϕ is the solution of (1.1), reaches some preassigned terminal set T. The point at which (t, ϕ(t))reaches T is called the terminal point and is denoted by (t _f ,ϕ(t _f )),or (t _f ‚x _f ). The payoff is

$$ P({t_0},{x_0};U,V) = g({t_f},{x_f}) + \int_{{t_o}}^{{t_f}} {{f^o}(s,\phi (s),u(s),v(s))ds,} $$

(1.2)

where g and f ⁰ are preassigned functions. Player I wishes to choose U so as to maximize P while Player II wishes to choose V so as to minimize P.