الفهرس 
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Abstract
This thesis handles the problem of intercepting a superior missile using a formation of lower-capabilities pursuers. The problem is formulated as a multi-player differential game and solved using both of dynamic programming and optimal control techniques. This thesis proposed two main approaches to make use of the Apollonius circle as a final path constraint of a formation of a set of players in order to achieve the capturing conditions. The first proposed approach considers finding one optimal trajectory, for each pursuer, from an initial state to the state of interception (called the controllability). In this approach, we formulated the dynamic game using Pontryagin Minimum Principle which converts the problem into two-points Boundary Value Problem then solved using the numerical analysis techniques. The second proposed approach considers solving the pursuit-evasion differential game using a well-known technique of optimal control, the principle of dynamic programming. This principle considers the viscosity solution of time-dependent Hamilton -Jacobi-Isaacs (HJI) Partial Differential Equations (PDEs), then taking advantage of an implicit surface function of the continuous Backward Reachable Set (BRS) (called the reachability) for the HJI viscosity solution. Although a considerable amount of literature had been published on reachability analysis of continuous-system for safety verification or obstacle-avoidance guarantee, we use it to investigate the interception of a superior evader/missile in a pursuit-evasion differential game. In both of the proposed approaches, we assume a constant speed ratio at engagement scenarios, a perfectly encircled formation of the pursuers around the missile is considered as the terminal path constraint (or the final value function of the HJI) which enforces the necessary condition to achieve the interception. The experimental results proved that the approach of trajectory optimization found an optimal trajectory of each pursuer until the interception with the missile. Also, the results of the reachability analysis of the pursuit-evasion game, based on Apollonius circle, helped in determining all reachable sets in which a group of slower pursuers could be in an interception course with a superior missile.