Solving a Local Boundary Value Problem for a Nonlinear Nonstationary System in the Class of Feedback Controls

On a method for solving a local boundary problem for a nonlinear stationary system with perturbations in the class of piecewise constant controls

“…The authors prove a theorem that proves the validity (correctness) of the algorithm for solving the problem. The technique of its proof is much more complicated than a similar theorem in the work of Kvitko On the other hand, the condition of checking the local controllability of the Kalman type obtained in this paper is much easier to check than the similar condition in the work of Kvitko …”

“…The proposed study of the authors is most closely related to the works of Coron and Kvitko . In the book of Coron, the question about local controllability of nonlinear systems is considered.…”

“…In the work of Kvitko the algorithm of solving local BVP for nonlinear nonstationary controllable system of ODEs in the class of synthesizing controls without taking into account perturbations is presented. The main idea of solving the problem posed is that, by replacing the independent variable, the solving the original problem is reduced to solving the problem of constructing the synthesizing control that provides the exponential decrease in the functions of phase coordinate on an infinite time interval.…”

“…The general idea of constructing a solution to this problem coincides with the idea of solving the problem discussed in the work of Kvitko . However, its implementation is significantly different from the implementation in the aforementioned work, namely, The auxiliary control is introduced after replacing the independent and phase variables similar to the ones in the work of Kvitko . This control is associated with the original system of differential equations. A new auxiliary system is considered.…”

“…In this system, the origin control is considered as an phase coordinate. Therefore, the size of this new system in comparison with the similar system in the work of Kvitko is increased by the size of the origin control vector. The auxiliary control in the form of feedback is found, which provides an exponential decrease of the fundamental matrix of the linear part of the auxiliary system considered in Section 2. The solution of the Cauchy problem for an auxiliary system closed by auxiliary control feedback will give a differentiable control function in a new independent variable. Using this function, we find piecewise constant control. The transition in the resulting piecewise constant control to the original independent variable gives the desired control function. As a result, we get the desired programmed control in contrast to the synthesizing control in the work of Kvitko The authors prove a theorem that proves the validity (correctness) of the algorithm for solving the problem.…”