The Game with a “Life-Line†for Simple Harmonic Motions of Objects

The purpose of this work is to study the pursuit-evasion problem and the “Life-line†game for two objects (called Pursuer and Evader) with simple harmonic motion dynamics of the same type in the Euclidean space. In this case, the objects move by controlled acceleration vectors. The controls of the objects are subject to geometrical constraints. In the pursuit problem, the strategy of parallel pursuit (in brief, the Π-strategy) is suggested for the Pursuer, and by this strategy a capture condition is achieved. In the evasion problem, a constant control function is offered for the Evader, and an evasion condition is derived. Employing the Π-strategy we generate an analytic formula for the attainability domain of the Evader (the set of all the meeting points of the objects), and we prove the Petrosjan type theorem describing that the attainability domain is monotonically decreasing with respect to the inclusion in time. In the “Life-line†problem, first, by virtue of the Π-strategy solvability conditions to the advantage of the Pursuer are achieved and next, in constructing a reachable domain of the Evader by a control function, solvability conditions to the advantage of the Evader are identified. Differential games under harmonic motions are more complex owing to some troubles in determining optimal strategies and in building the meeting domain of objects. Accordingly, such types of games have not been fairly investigated than the simple motion games. From this point of view, studying the pursuit, evasion, and “Life-line†problems for oscillated motions arouses a special interest.