Some algorithms of optimal control

In the first part of the paper, we describe the method of continuation with respect to a parameter in solution algorithms for nonlinear boundary value problems in ordinary differential equations. We present results of numerical experiments solving boundary value problems, including boundary value problems arising in optimal control theory. The parameter variation scheme (the continuation method) can be considered as a special development and modification of the classical Newton method. The basic idea of this approach can be shortly formulated as reducing a boundary value problem to a Cauchy problem. Regarding a Cauchy problem as an elementary operation, we arrive at a compact description of the algorithm of solving a boundary value problem by means of the method of continuation with respect to a parameter. The interest in this research area is related to studying numerical algorithms of solving the linear time-optimal control problem and is aimed at boundary problems of the maximum principle. We have developed a program BVP, which solves in the Maple environment regular boundary value problems for ordinary differential equations, some boundary value problems of the maximum principle arising in optimal control, problems of finding periodic solutions and limit cycles, and so on. In the second part of the paper, we describe a simple algorithm of constructing attainability (controllability) sets in plane linear controlled systems and give some examples of using it. The algorithm is based on parametric equations of the boundary of a plane strictly convex compact set given by its support function. This approach allows one to construct two-dimensional projections of attainability sets for multidimensional linear controlled systems. In the third part of the paper, we present sufficient optimality conditions for nonlinear controlled systems in terms of constructions of the Pontryagin maximum principle.