Algorithm for constructing control of a dynamical system in partial derivatives

Abstract

The structural analysis of the dynamical system described by the differential equation in partial derivatives is carried out. The problem is to transfer the system from the initial state to the final state. The difference of this problem from the classical one is the presence of two required vector functions: state and control. The control problem is solved: the properties of matrix coefficients and functions that imply complete controllability (or uncontrollability of the system) are revealed; a step-by-step algorithm for constructing control functions and the corresponding state for a fully controlled system is presented. The study is carried out by the algorithmic method of cascade decomposition, which consists in a stage-by-stage (step-by-step) transition from the original system to systems of ever-decreasing sizes, and which allows optimizing the process of numerical implementation of the controlled process. The practical implementation of the method does not require exact formulas for constructing matrix coefficients, which makes it possible to avoid cumbersome matrix transformations and get by with the change of variables procedure. The implementation of the method is based on the properties of the matrix coefficient at the derivative of one of the desired functions and is algorithmically implemented in forward and backward steps. Each case of a (zero, reversible or irreversible) coefficient is considered in detail and, in the case of an irreversible coefficient, the system is split into a hierarchically structured set of subsystems of the first and second levels in the process of implementing the forward move. Further, in order to reveal the properties of matrix coefficients, the procedure of structural analysis of the subsystem of the second level is implemented, which is quite similar to the original system, but in a space of lower dimension. The finite-dimensionality of the original spaces implies the complete realization of the forward decomposition in a finite number of steps, which does not exceed the dimension of the original space. Uncontrollability or complete controllability of the system at the last decomposition step is revealed. In the case of revealing the property of complete controllability of the system of splitting the last step, the reverse course of the algorithm is implemented: obtaining formulas for constructing the control function and the corresponding state function. Introduced method allows one to vary when constructing the desired vector functions: functions that satisfy given boundary conditions can be constructed in exponential, fractional-rational, polynomial form, or in any other form that best suits the needs of the study.