Searching for the cost-optimal road trajectory on the relief of the terrain

The article analyzes the problem of obtaining the cost-optimal trajectory for building a road. Using the apparatus of mathematical modelling, the authors derive the cost functional, the argument of which is the function that describes the path trajectory. The resulting functional after some additional transformations is written in a simpler form. For the problem of the calculus of variations obtained in this manner, an optimality condition is derived. This condition takes into account the specifics of the constructed functional. Unlike the classical Euler—Lagrange condition, it leads not to a differential, but to an integro-differential equation. An illustrative example of the numerical solution of the obtained equation using the methods of computational mathematics is provided.