On parametrization of linear pseudo-differential boundary value control systems

TL;DR

This paper explores the parametrization of pseudo-differential boundary value control systems using algebraic and symbolic calculus tools, providing methods for explicit computation and analysis of control solutions.

Contribution

It introduces a framework for parametrizing boundary control systems via algebraic and symbolic techniques, including homological algebra methods for explicit solution characterization.

Findings

Parametrizability linked to projectivity of a factor module.

Explicit computation of parametrization operators S.

Illustrative examples demonstrating computational methods.

Abstract

The paper considers pseudo-differential boundary value control systems. The underlying operators form an algebra D with the help of which we are able to formulate typical boundary value control problems. The symbolic calculus gives tools to form e.g. compositions, formal adjoints, generalized right or left inverses and compatibility conditions. By a parametrizability we mean that for a given control system Au=0 one finds an operator S such that Au=0 if and only if u=Sf. The computation rules of D (or its appropriate subalgebra D') guarantee that in many applications S can be refinely analyzed or even explicitly calculated. We outline some methods of homological algebra for the study of parametrization S. Especially the projectivity of a certain factor module (defined by the system equations) implies the parametrizability. We give some examples to illustrate our computational methods.