On Level Sets With "Narrow Throats" in Linear Differential Games

Abstract

Examples with zero-sum linear differential games of fixed terminal time and a convex terminal payoff function depending on two components of the phase vector are considered. Such games can have an indifferent zone with constant value function. The level set of the value function associated with the indifferent zone is called the "critical" tube. In the selected examples, the critical tube and the neighboring level sets exhibit "narrow throats". Presence of such throats requires extremely precise computations for constructing the level sets. The paper presents different forms of critical tubes with narrow throats and indicates the combinations of problem parameters that can produce them.

Zero-Sum Pursuit-Evasion Differential Games with Many Objects: Survey of Publications

“…Its "peculiarity" is in the presence of two time periods with narrow "throats." Earlier, for three-dimensional solvability sets in model problems 1 × 1 of cosmic pursuit, examples with one narrow throat have been constructed [32]. For a problem 2 × 1 with dynamics of form (25) in work [29], an example is given where in some period of time, the solvability set disjoins into two parts.…”

Solvability Sets in Pursuit Problem with Two Pursuers and One Evader

“…Its "peculiarity" is in the presence of two time periods with narrow "throats". Earlier, for three-dimensional solvability sets in model problems 1 × 1 of cosmic pursuit, examples with one narrow throat have been constructed (Kumkov et al (2005)). For a problem 2 × 1 with dynamics of form (2) in work (Kumkov et al (2013)), an example is given, where in some period of time, the solvability set disjoins into two parts.…”

“…Efficient procedures (Isakova et al (1984); Patsko and Turova (2001); Kumkov et al (2005)) for backward constructing the level sets of the value function (the solvability sets) had been elaborated for linear differential games of general type with the geometric constraints u ∈ P, v ∈ Q on the players' controls, a fixed terminal instant T , and continuous payoff function ϕ that depends on two components of the phase vector at the terminal instant.…”

Extremal aiming in problems with an unknown level of dynamic disturbance

“…In the above mentioned papers, the structure of the value function level sets in a differential game with the dynamics (6.1) was investigated in detail for different versions of the elliptic constraints P and Q. Three-dimensional representations of the level sets have been presented. 29 In this paper, we abandon the a priori specification of a geometric constraint on the second player's control. We take the initial data for problem (6.1) in the form…”