Development and application of a gradient method for solving differential games
A technique for solving n-dimensional games is developed and applied to two pursuit-evasion games. The first is a two-dimensional game similar to the homicidal chauffeur but modified to resemble an airplane-helicopter engagement. The second is a five-dimensional game of two airplanes at constant altitude and with thrust and turning controls. The performance function to be optimized by the pursuer and evader was the distance between the evader and a given target point in front of the pursuer. The analytic solution to the first game reveals that both unique and nonunique solutions exist. A comparison between the gradient results and the analytic solution shows a dependence on the nominal controls in regions where nonunique solutions exist. In the unique solution region, the results from the two methods agree closely. The results for the five-dimensional two-airplane game are also shown to be dependent on the nominal controls selected and indicate that initial conditions are in a region of nonunique solutions.
Pursuit-Evasion of Two Aircraft in a Horizontal Plane
Pursuit-evasion between two aircraft in a horizontal plane is analyzed as a differential game using point-mass aircraft models. A suitable choice of real-space coordinates confers open-loop optimality on the game. The solution in the small is described in terms of the individual aircraft's extremal trajectory maps (ETM). Each is independent of role, adversary, and capture radius. An ETM depicts the actual trajectories flown by the aircraft in real space. A template method of generating constant control arcs is described. This is used to investigate bank saturation and throttle switching behavior exhaustively. Sections of the barrier are obtained by iteratively choosing pairs of extremals from the two ETM's. NALYTICAL studies of aircraft pursuit-evasion techniques seek to determine the influence of aircraft and weapon system performance upon combat outcome and to derive idealized tactics. Aircraft pursuit-evasion is considerably more difficult to analyze than optimal maneuvers because a closed-loop solution is a must in a differential game situation and exceptional and discontinuous surfaces are the rule rather than an exception. Analyses of aircraft pursuit-evasion have simplified the situation along three lines. First, pursuit-evasion has been studied as variants of the Game of Two Cars. 1~7 In these, the aircraft are assumed to fly at constant altitude and speeds; their optimal paths are composed of circular arcs and straight line segments. Second, in the energy approach,8'11 the aircraft are modeled realistically in terms of their energies and relative heading. However, the relative positions of the aircraft are ignored. The differential game studied is again trimensional. Third, in the dynamic modeling12 and numerical approaches, 13'16 combat is formulated as a game of prescribed duration thus avoiding discontinuous and exceptional surfaces. Even then, near-optimal closed-loop solutions have only been obtained for simple examples.16 Thus an analysis of constant-altitude pursuit-evasio n with varying aircraft speeds portrays the situation realistically. It also takes on, for the first time, a free-time combat game with five state variables. The game is formulated in a real space whose origin and axes change from party to party. The equations decompose into two sets, one for each aircraft. These are coupled in terms of the terminal quantities alone. This simplifies the solution in the small and effects significant computational economy. Moreover, each aircraft's optimal motion in real space can be described in terms of an extremal trajectory map (ETM) which is independent of the adversary and of role. The ETM thus serves as a measure of an individual aircraft's combat performance.
Globalization and the Internationalization of Legal Education in the United States: An Annotated Bibliography
Globalization and its influence on legal education is a topic of growing interest to the legal profession and to legal educators alike. The first articles on the subject appeared in the mid-1990s, and the frequency of publications has been increasing ever since. This bibliography provides annotations to English-language law journal articles published between 1995 and 2008 that focus on the impact of globalization on legal education in the United States.