Algorithm for constructing a polynomial solution of a program control problem for a dynamic system in partial derivatives
Authors: Zubova Svetlana 
Published: October 2023

The completely controlled dynamic system in partial derivatives is considered. The problem of constructing state and control functions in an analytical form is solved. The basic method is the cascade decomposition method, which is algorithmically implemented in three stages: forward cascade decomposition, central stage and reverse. The method is based on the properties of the matrix coefficient at the derivative of the control function. Decomposition means a p-step transition from the original system to a reduced system that is quite similar in form to the original one, but with respect to functions from subspaces. The given conditions are reduced in the process of decomposition. When passing to the p-th step system, additional conditions appear on the partial derivatives of the components of the state function. The number of extra conditions at each point is equal to the number of decomposition steps. The matrix coefficient at the derivative of the control function of the reduced system of the last step is surjective. It is this property that determines the presence of the property of complete controllability of the system under consideration. The first stage of decomposition - the stage of the direct move ends with the detection of the number of decomposition steps and the identification of the property of complete controllability. The task of the central stage of decomposition is to construct the state function of the reduced system of the last step in an analytical form. The state function of the reduced system is the basis function that determines the form of the state function of the original system. Necessary and sufficient conditions for the existence of a basis function in polynomial form are established. The minimum degree of the polynomial is also set, which is determined by the number of decomposition steps. Formulas for constructing vector functions - coefficients of the basis function polynomial are given. Formulas for constructing the control function of the reduced system are given in polynomial form. During the last stage of decomposition, the state function of the original system is successively restored in polynomial form. This polynomial function satisfies the given conditions at the start and end points. The final stage is the construction of the control function of the original system in polynomial form as well. While the last stage of decomposition the state function of the original system is successively restored in polynomial form. This polynomial function satisfies the given conditions at the start and end points. The final stage is the construction of the control function of the original system also in polynomial form. A step-by-step algorithm for solving the program control problem for a dynamic system in partial derivatives has been developed. Formulas for constructing state and control functions in polynomial form are given. An example of a three-dimensional dynamical partial differential system with a surjective matrix coefficient in the first-step splitting system is given. The implementation of the proposed algorithm is demonstrated. The state and control functions are constructed in the form of a polynomial of minimum degree.