Modern Methods for Solving Nonconvex Optimal Control Problems

The paper presents a few remarks on the evolution of Irkutsk's school of O. V. Vasiliev on optimal control methods based on Pontryagin principle. Besides, one reviews some features of Pontryagin principle, in particular, its sufficiency and constructive property for linear (on the state) control systems and convex cost functionals. Further, some historical notes on the development of optimal control methods based on Pontryagin principle are considered. In particular, a separated attention has been paid to the impact of Irkutsk school of O. V. Vasiliev in the theory and method of optimal control, and the achievements of the former postgraduate student of O. V. Vasiliev professor V. A. Srochko. The mathematical presentation is concentrated on the story of the invention and investigations of the convergence and substantiation of the consecutive approximate's method based on Pontryagin principle. In addition, one considers new Global Optimality Conditions in a general nonconvex optimal control problem with Bolza goal functionals. Moreover, together with the necessity proof of global optimality conditions we investigate its relations to Pontryagin principle. Besides, the constructive (algorithmic) property of new optimality conditions is also demonstrated, and an example of nonconvex optimal control problems has been solved by means of global optimality conditions. In this example, we performed an improvement of a feasible control satisfying Pontryagin principle with a corresponding improvement of the cost functional. Finally, employing Pontryagin principle and new Global Optimality Conditions we give a demonstration of construction of a optimal control method and provide for new result on its convergence.