Variational Approach for Finding the Cost-Optimal Trajectory

Different approaches are used to define the optimal path in terms of construction costs. Such problems in practice are usually solved by various heuristic procedures. To obtain a theoretically justified result, one can derive an integral cost functional under certain assumptions and use variational principles. Thus, the classical problem of the calculus of variations is obtained. The necessary condition for the minimum of such a functional has the form of an integrodifferential equation. This paper describes a numerical algorithm for solving this equation, which is based on the prominent shooting method, which has been studied in detail in the literature. Under additional assumptions, the existence of a solution is proved using Schauder’s fixed point principle. The problem of the uniqueness of the solution is studied. A numerical example is provided.