Mathematical modeling of linear dynamic systems with distributed parameters describes processes that depend on time and spatial coordinates by means of initial boundary value problems for partial derivative equations [1, 2]. In the presence of uncertainties, pseudo-solution methods are applied, which makes it possible to find stable solutions in identification and control problems that have no classical solution.

An approach to solving problems of mathematical modeling of the dynamics of linear spatially distributed systems was proposed. It is based on a linear differential model of the process, supplemented by initial conditions [3]. There are no restrictions on the order and structure of the differential operator of the model, as well as on the number and quality of external dynamic observations with which it is supplemented. The latter can be both discretely and continuously specified with discretely and continuously defined functions by which they are modeled according to the mean-square criterion.

These functions are defined from systems of algebraic and integral equations, for which methods of constructing pseudo-solutions with an assessment of their accuracy and unambiguity are proposed. By the same criterion and the same accuracy, the consistency of the exact mathematical solution of the differential model of the system with the available initial boundary value observations is evaluated. To extend the methods for mathematical modeling of the dynamics of incompletely observed spatially distributed systems to processes and phenomena not described by a differential model, identification algorithms for constructing kernels of their differential models have been developed. The methodology of mathematical modeling of solutions to direct problems of dynamics of spatially distributed systems proposed in this paper can be easily extended to the problems of their control. In this case, as in the case of observations, the desired state of the system is set both discretely and continuously by linear differential transformations of the state function of the latter. The results obtained are useful for modeling processes and phenomena in various fields of science and technology.

References:

1. Smetankina N. Dynamic response of laminate composite shells with complex shape under low-velocity impact / N. Smetankina, A. Merkulova, D. Merkulov, O. Postnyi // Integrated Computer Technologies in Mechanical Engineering – 2020. ICTM 2020. Lecture Notes in Networks and Systems. – 2023. – Vol. 188. – P. 267–276. Режим доступу: https://doi.org/10.1007/978-3-030-66717-7_22

2. Smetankina N.V. Optimal design of layered cylindrical shells with minimum weight under impulse loading / N.V. Smetankina, O.V. Postnyi, S.Yu. Misura, A.I. Merkulova, D.O. Merkulov // In: 2021 IEEE 2nd KhPI Week on Advanced Technology (KhPIWeek). – 2021. – P. 506–509. Режим доступу: https://doi.org/10.1109/KhPIWeek53812.2021.9569982

3. Kurennov S. Stress-strain state of a double lap joint of circular form. Axisymmetric model / S. Kurennov, N. Smetankina // Integrated Computer Technologies in Mechanical Engineering – 2021. ICTM 2021. Lecture Notes in Networks and Systems. – Vol. 367. – 2022. – P. 36–46. https://doi.org/10.1007/978-3-030-94259-5_4