1. An object recognition apparatus comprising: a first detecting device configured to detect first position information indicating a position of a first peripheral object that exists within a first detection region established around periphery of a mobile object; a second detecting device configured to detect second position information indicating a position of a second peripheral object that exists within a second detection region, wherein the second detection region overlaps with all or a portion of the first detection region; a target object setting device configured to set a target object by matching the first peripheral object and the second peripheral object using the first position information and the second position information; and a behavior support control device configured to control a behavior supporting action of the mobile object in relation to the target object; wherein the behavior support control device is configured to: set a first target area which is used to judge whether or not the target object is to be treated as a target of the behavior supporting action, as a portion of the first detection region; and set a second target area which is used to judge whether or not the target object is to be treated as the target of the behavior supporting action, as a portion of the second detection region; wherein, if the position of the first peripheral object exists within the first target area and the position of the second peripheral object exists within the second target area, the target object is treated as the target of the behavior supporting action; and wherein, if the position of the first peripheral object does not exist within the first target area or the position of the second peripheral object does not exist within the second target area, the target object is removed from being the target of the behavior supporting action, or the behavior supporting action with respect to the target object is suppressed.