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SUMMARYThis paper describes exact evaluations of various finite integrals whose integrands involve products of Daubechies' compactly supported wavelets and their derivatives and/or integrals. These finite integrals play an essential role in the wavelet-Galerkin approximation of differential or integral equations on a bounded interval.

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Collision avoidance remains the most important concern for ships at sea. Despite the electronic equipment now fitted on ships to support the mariner, expert experience is still essential when a ship is in danger of colliding with the others. To include these experts' experiences to resolve the problems of collision, we have designed a fuzzy collision-avoidance expert system that includes a knowledge base to store facts and rules, an inference engine to simulate experts' decisions and a fuzzy interface device. Either a quartermaster or an autopilot system can then implement the avoidance action proposed in the research. To perform the task of collision-avoidance effectively, a robust autopilot system using the state space H _ control methodology has been designed to steer a ship safely for various conditions at sea in performing course keeping, course-changing and route-tracking more robustly. The integration of fuzzy collision-avoidance and H _ autopilot systems is then proposed in this paper. KEY WORDS1. Collision avoidance.2. Autopilots. 3. Automation.. CHENG-NENG HWANG AND OTHERS VOL. 55