Solving analysis and synthesis problems for a spatially two-dimensional distributed object represented with an infinite system of differential equations

We prove theorems that define an algorithm for passing from differential equations with partial derivatives with respect to two spatial variables and time to an infinite-dimensional system of ordinary differential equations in Cauchy form. We study the convergence of resulting solutions and show that it is possible to pass from an infinite system in Cauchy form to a finite one, which opens up the possibilities to use state space methods for controller design in distributed systems. Based on the quadratic quality criterion, we design a controller for the case when controlling influences are applied at the boundaries of the control object. We obtain the solution of this system analysis problem in the form of Fourier series with respect to spatial variables based on orthogonal systems of trigonometric functions and Bessel functions.