the covering method for the solution of terminal control problem

A new method for solving the terminal control problem for dynamical systems is formulated. This problem is to determine a program trajectory and a program control that takes the system from a given initial state to a given final state. The method is based on the addition of equations with control derivative to the source system and reformulation of the problem in the boundary value problem for the augmented system E. Additional equations must be chosen so as to satisfy the following conditions. There is a surjective map (covering) from the phase space E to the phase space of some dynamical system Y. The covering takes solutions of E to solutions of Y. Boundary conditions in the final moment are mapped to the boundary conditions on the solutions of Y. Any solution of Y satisfies the boundary conditions in the initial moment. Then the solution of the terminal control problem is as the solution of two Cauchy problems for dynamical systems E and Y. Augmented system E satisfying mentioned properties is called r-closure of the terminal control problem. It is shown that this approach generalizes the well-known method for solving the terminal control problem for flat systems. A flat system is a system whose solutions are uniquely determined by a certain set of functions of time (flat output). The mentioned well-known method is based on polynomial dependence of flat output of time and do not take into account constraints on the system.

It is proved that for an arbitrary flat system r-closure can be chosen any determined system of ordinary differential equations of the corresponding order. It is showed how to construct a covering with the above-mentioned properties using the general solution of this system. The properties of the covering are proved only locally, i.e. when the initial time is close to the final time, and the initial conditions are close the final conditions. But this covering may be applicable to other terminal problems with the same final conditions. This result can be used to solve the terminal control problem for flat systems with constraints. In addition, an example demonstrates the possibility of applying this method to non-flat systems.