Voronezh, Voronezh, Russian Federation
UDK 519.63 Численные методы решения дифференциальных уравнений с частными производными
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, cascade decomposition, dynamic system, program control, state function, partial derivatives, matrix coefficient, structural analysis
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