Solution of the multipoint control problem for a dynamic system in partial derivatives

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.

Algorithm for constructing control of a dynamical system in partial derivatives

Zubova

2022

Structural analysis of the control function of a dynamic system in partial derivatives of different order

Zubova

2023

For a dynamic system described by a partial differential equation of different order, the problem of constructing software control in an analytical form is solved. The primary method of research is the cascade decomposition method, the improved algorithm of which includes three main stages: the forward move, the central stage and the reverse move. The method is based on the properties of the matrix coefficient with a second-order partial derivative of the control function. The noetherianness of the coefficient causes the splitting of the original space into direct sums of subspaces. A scheme for structuring subspaces in accordance with the properties of the matrix coefficient is given. The direct move of decomposition is realized, which consists in a step-by-step transition to equivalent hierarchically structured systems of two levels in subspaces. The step-by-step structuring of the components of the state function into functions from subspaces while using matrix coefficients - projectors is performed. The resulting functions from subspaces are called pseudo-state and pseudo-control functions. Graphical visualization of the hierarchical structure of the source space in the form of a diagram is performed. This scheme reflects the essential connections between the components of the subspaces of each decomposition level. Finite-dimensional spaces are considered, which causes the complete completion of the first stage of the algorithm in a finite number of steps not exceeding the dimension of the original space. During the decomposition, the conditions at the start and end points are reduced, so that at the end of each step of the forward stroke, one additional condition appears at each point for each second-level system. In the process of implementing the first stage of the algorithm, the properties of matrix coefficients entailing full controllability or unmanageability of the initial system are established; the properties of functions in the initial conditions necessary for the implementation of the controlled process are also revealed. The criterion of complete controllability of the initial system is deduced. For a fully controlled system, a transition is made to the central stage of the algorithm – the construction of a defining basis function that satisfies all additional conditions for partial derivatives in time at each point resulting from the reduction of the original ones. It is the presence of this defining basic function that lays the prerequisites for constructing the state and control functions of the initial system at the final stage of the algorithm. A diagram visualizing the procedure of step-by-step restoration of the components of the state function in the process of implementing the reverse course is given. The reverse is completed by explicitly constructing first the state function, then the control function. A qualitative analysis of the management structure of the system under study is carried out.

Algorithm for constructing a polynomial solution of a program control problem for a dynamic system in partial derivatives

Zubova

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.