## 1 Introduction  
This paper proposes a chasing controller so that a pursuer can chase and capture a maneuvering evader which moves at high speed. This problem is related to the challenging missile guidance problem of intercepting a high-speed missile [1; 2; 3; 4; 5; 6]. The pursuer must move at high speed to capture a high-speed evader. Consider the terminal phase when the pursuer is sufficiently close to the high-speed evader. The accurate control of the pursuer in the terminal phase is crucial, since it is hard to capture a high-speed evader if the pursuer misses the evader in the terminal phase. Our strategy in the terminal phase is to slow down the pursuer's speed. As we slow down the relative speed of the pursuer with respect to the evader, we can decrease the probability of missing the high-speed evader. (Consider the case where two spaceships dock each other. It is desirable to slow down the relative speed for safe and accurate docking). Thus, our strategy is to make the pursuer decrease its speed in the terminal phase, while ensuring that the distance between the evader and the pursuer monotonically decreases as time passes. As far as we know, no paper in the literature on chasing targets considered changing the pursuer's speed so as to capture a maneuvering evader in a provably complete manner. Our controller is designed to assure that the angular rate of Line-of-sight joining the pair (the pursuer and the evader) is exactly zero at all time indexes. This type of movement is called the _motion camouflage with respect to a fixed point at infinity_[7; 8; 9]. This motion camouflage is employed by various visual insects and animals to achieve prey capture, mating, or territorial combat [8; 10; 11; 12]. This movement is a time-optimal solution to capture a pursuee moving with a constant velocity (speed and heading) [10]. In addition [10], argued that this motion minimizes time-to-capture of an unpredictably moving pursuee. Assuming that the pursueecan readily observe optical flow, but only poorly detect looming, this pursuer's movement is hardly detected by the prey [8]. This paper introduces a chasing controller to enable a pursuer to chase a maneuvering evader while not rotating the LOS joining the pair (the evader and the pursuer). We make the pursuer decrease its speed in the terminal phase, while ensuring that the distance between the evader and the pursuer monotonically decreases as time passes. The literature is abundant with papers on planning path of robots [8; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27]. [PERSON] [28] developed a path planning algorithm for an underwater robot approaching a static target while not being detected by the target. The path was planned to reduce both the time required to meet the target and the robot's sound measured by the static target. The authors of [29; 30; 31] presented the autonomous tracking and following of a marine vessel by an Unmanned Surface Vehicle (USV) in the presence of dynamic obstacles. In [29; 30], the path planning for the USV with International Regulations for Preventing Collisions at Sea (COLREGS) rules was achieved. The authors of [31] presented a trajectory planning and tracking approach for following a differentially constrained target vehicle operating in an obstacle field. [PERSON] [31] predicted the target state several time steps forward in time and generated a collision-free trajectory to allow the USV to safely reach the predicted target state. As far as we know, no paper on chasing targets handled changing the pursuer's speed so as to capture a maneuvering evader in a provably complete manner. Many controllers have been developed to mimic motion camouflage in nature. To capture an evader [7; 8; 9; 32] presented a chasing controller based on biologically plausible sensing. [PERSON] and [PERSON] [9; 32] developed a motion camouflage controller in noisy environments. Note that [7; 8; 9] only considered a pursuer which moves with a constant speed.