Symbolic computations for identities of linear operators

Many processes in science and engineering can be modeled by so-called linear functional systems. To manipulate and analyze such systems, one computes with the corresponding matrices and linear operators. Properties of systems and operators are expressed by identities. Instead of working with concrete matrices and operators, symbolic computation works with mathematical objects represented by symbols. The main goal of the project “Symbolic computations for identities of linear operators” is to automatize such formal computations with operators and classes of systems beyond what is currently possible on the computer. In particular, we are interested in symbolic methods and computer algebra tools for proving and discovering identities of linear operators and for solving operator equations. In the project, we develop methods to analyze classes of dynamical systems in engineered processes and their control. These systems and their transformations are usually modeled by differential, delay, and integral operators. To compute with such operators, we work out a unique way of representing them. Based on these normal forms, we will prove and discover identities of operators automatically by computer algebra software, which we develop in the course of the project. If input and output of operators or matrices have different dimensions, they cannot be added and composed in arbitrary ways. This restricts computations with operators and matrices. In the project, we will work out new symbolic methods to deal with these restrictions. The idea is to first compute symbolically without restrictions and then justify the result independent of how it was obtained.