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The linear Boltzmann equation describing electron flow in a semiconductor is considered. The Cauchy problem for space-independent solutions is investigated, and without requiring a bounded collision frequency the existence of integrable solutions is established. Mass conservation, an H-theorem, and moment estimates also are obtained, assuming weak conditions. Finally, the uniqueness of the solution is demonstrated under a suitable hypothesis on the collision frequency.

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The purpose of this paper is to present the basic mathematical modeling of microcopters, which could be used to develop proper methods for stabilization and trajectory control. The microcopter taken into account consists of six rotors, with three pairs of counter-rotating fixedpitch blades. The microcopter is controlled by adjusting the angular velocities of the rotors which are spun by electric motors. It is assumed as a rigid body, so the differential equations of the microcopter dynamics can be derived from both the Newton-Euler and Euler-Lagrange equations. Euler-angle parametrization of three-dimensional rotations contains singular points in the coordinate space that can cause failure of both dynamical model and control. In order to avoid singularities, the rotations of the microcopter are parametrized in terms of quaternions. This choice has been made taking into consideration the linearity of quaternion formulation, their stability and efficiency.