Abstract. In the stability analysis of linear systems depending on several parameters, the D-partition method is often used, also known as the D-decomposition method in the literature. This method describes the stability region of a characteristic polynomial via the equation of its boundary. A constructive D-partition method proposed below identifies individual parts of curves and straight lines on the parameter plane that form the boundaries of the D-partition regions and, in particular, the stability region. A characteristic polynomial linearly dependent on two parameters and a stability region with a piecewise rational parametric boundary are considered. In this case, the boundary of each D-partition region is a finite set of arcs of rational curves and segments, rays, or straight lines that can be found explicitly. The rational curve arcs are parameterized on intervals whose limits are found by calculating the real roots of auxiliary polynomials. A D-partition, bounded (localized) on a compact set, consists of a finite number of segments and arcs of rational curves parameterized on the segments.
Keywords: constructive D-partition, D-decomposition, root localization, root clustering, rational curves, localized D-partition.
Acknowledgments. The research presented in Sections 3 and 4 was supported by the Russian Science Foundation, project no. 21-71-30005, https://rscf.ru/en/project/21-71-30005/. The author is grateful to the reviewers for careful reading of the manuscript and helpful remarks.
Constructive D-Partition for Two Parameters Entering a Polynomial Linearly. Part I: Description of the Boundaries of D-Partition Regions