Optimal evasion with a path-angle constraint and against two pursuers

We consider the problem of optimal evasion when the pursuer is known to employ fixed-gain proportional naviga- tion. The performance index is a measure of closest approach. The analysis is done for planar motions at constant speed, and the kinematics are first linearized around a nominal collision course. Two cases are studied. The first involves optimal evasion with a terminal path-angle constraint for the evader, and the second, optimal evasion against more than one pursuer. The latter is studied as an optimal evasion against one pursuer with state constraints imposed by the others. The optimal controls are shown to be "bang-bang," with the number of switches depending on the pursuer's navigation gain and the particular constraints of each case. URSUIT-EVASION problems have been traditionally classified among the classical examples of differential game theory. In the last 15 years, a different approach has been applied to these problems,1"4 namely, to fix the pursuer's strat- egy and form a one-sided optimal control problem for the evader. This approach, being conceptually simpler than the former, enables more realistic models to be applied for the dynamics of the opponents. In general, the fixed pursuer's strategy has been taken as constant gain, proportional naviga- tion that, under some formulations, is an optimal strategy for the pursuer. We shall consider two related problems in which the optimal evasive maneuvers are subjected to some state constraints. The first problem to be considered in this work is the optimal eva- sion with a terminal path-angle constraint. The motivation for this problem is derived from cases in which the evading vehicle is a missile, guiding toward a fixed target, and the pursuer is an interceptor trying to protect this target. Thus, the evasive maneuver is constrained by a terminal path- (or heading) angle constraint to guarantee capture of the fixed target. The inter- ceptor is successful if the evader is destroyed, or if it is made to deviate significantly from its course. It will be shown that this problem has some interesting features whose importance may exceed the bounds of the pursuit-evasion conflict. The motivation for the second problem is self-evident. We shall consider the problem of optimal evasion against more than one, starting with two pursuers. This problem will be studied as an optimal evasion against one pursuer subject to a minimum required miss distance of the others. We shall allow the pursuers to select their launch time so as to minimize the closest approach. The models to be used are all linear, due to the relative complexity of the problems. Consequently, the results will be valid in the vicinity of the nominal collision courses.