Level Sweeping of the Value Function in Linear Differential Games

Abstract

In this chapter one considers a linear antagonistic differential game with fixed terminal time T, geometric constraints on the players’ controls, and continuous quasi-convex payoff function ϕ depending on two components x i, x j of the phase vector x. Let $$ \mathcal{M}_c = \{ x:\varphi (x_1 ,x_j ) \leqslant c\} $$ be a level set (a Lebesgue set) of the payoff function. One says that the function ϕ possesses the level sweeping property if for any pair of constants c 1 < c 2 the relation $$ \mathcal{M}_{c2} = \mathcal{M}_{c1} + (\mathcal{M}_{c2} - *\mathcal{M}_{c1} ) $$ holds. Here, the symbols + and $$ - $$ mean algebraic sum (Minkowski sum) and geometric difference (Minkowski difference). Let $$ \mathcal{W}_c $$ be a level set of the value function $$ (t,x) \mapsto \mathcal{V}(t,x) $$ . The main result of this work is the proof of the fact that if the payoff function ϕ possesses the level sweeping property, then for any t ∈ [t 0, T] the function $$ x \mapsto \mathcal{V}(t,x) $$ also has the property: $$ \mathcal{W}_{c2} (t) = \mathcal{W}_{c1} (t) + (\mathcal{W}_{c2} (t) - *\mathcal{W}_{c1} (t)) $$ . Such an inheritance of the level sweeping property by the value function is specific to the case where the payoff function depends on two components of the phase vector. If it depends on three or more components of the vector x, the statement, generally speaking, is wrong. This is shown by a counterexample.