Conditions of Transversality as an Effective Instrument in the Mathematical Constructions of the Optimal Processes

Abstract

The topic of the article concerns the mathematical theory of the optimal processes and its methods. In particular, the extremely topical aspects of the methodology of the maximum principle and the procedure of its practical use are discussed. The article presents in details the transversality conditions, their role, place and significance in the general procedure for solving of the problems of finding the optimal functions. This work eliminates the existing methodical defect connected with an incomplete use of the transversality conditions. On concrete examples, it shows that the transversality conditions (as one of the necessary conditions of optimality in the form of the maximum principle) are an extremely effective mathematical tool (and even the only one, in some cases) for determination of the characteristic properties, laws and key characteristics (parameters, constants, integrals of motion) of the optimal solutions for the systems of ordinary differential equations (of the dynamic systems). The importance of the topic is explained by the fact that in the overwhelming majority of cases the authors believe that the transversality conditions complicate the problem of search for the optimum functions, instead of simplifying it. However, it is far f/om being so, and it is a standard error, which has become almost typical. In the paper, the authors convincingly demonstrate the following conclusion: for a maximally correct and complete solution of the problems of the optimal control, the transversality conditions, being the necessary conditions for a optimality similarly as the condition of the Hamiltonian maximum and the adjoint system of the differential equations, should be taken into account since the beginning and be included in the system of the equations, which formalize the maximum principle, directly after defining the adjoint variables, formation of the Hamil-tonian and the adjoint system of equations, determination of the properties of the optimal adjoint functions and the control variables. The concrete examples presented in the article confirm the exclusively significant role (quite often critical role) of the transversality conditions in the general algorithm of the procedure for application of the maximum principle.