Blending curves for landing problems by numerical differential equations, I. Mathematical modelling

In [1], the blending curves can be modelled as the solutions of a system of boundary value problems of Ordinary Differential Equations (ODEs), to resemble a flexible elastic beam. Such ODE blending approaches are different from the traditional interpolation methods. The advantages of the ODE approaches are optimal blending curves obtained in minimum energy, and flexibility to complicated blending problems. Following the ideas in [1], this paper intends to model the blending curves of airplane landing problems, also by means of the ODE solutions. Since the airstrip is a bounded straight line EE′, the landing point of an airplane (i.e., the blending curve) is just on line EE′, instead of the fixed point discussed in [1]. It is due to the rather complicated boundary conditions that the existence and uniqueness of the ODE solutions should first be studied. In this paper, we have concluded that the ODE solutions are existential and unique if the flying direction BB′ at the beginning point of the blending curve is not parallel to the airstrip EE′. Otherwise, the infinite solutions exist if the exterior force is also perpendicular to EE′. Moreover, the mathematical modelling for other landing problems, antimissile problems, and the ODE system with general linear boundary constraints are discussed.