Optimization of body movement with variable structure in a viscous medium with non-constant density

Abstract

In the class of problems in the optimal control theory for objects motion in a viscous medium, new formulations with advanced mathematical models, including additional parameters of the medium, are investigated. Such problems are degenerate and their solution requires the development of a special mathematical apparatus. In particular, the formulation of the problem of optimal energy consumption for overcoming the resistance of a viscous medium of variable density, the translational displacement of a variable form solid from one phase state to another (the displacement time is specified) is considered. An analysis of the nonlinear relationships of the physical characteristics of a viscous medium has been carried out and a mathematical model has been constructed taking into account these relationships. The features of the of optimal control problem of the movement of bodies with variable geometry in a viscous medium of variable density from the point of view of the theory of optimal control are revealed. The necessary optimality conditions are obtained. The construction of such a displacement is associated with the solution of some two-point boundary value problem for a system of Navier-Stokes equations and having a similar structure of the conjugate system. © 2019 Author(s).