Document Type : Research Article
Author
Department of Basic Sciences, Technical and Vocational University (TVU), Tehran, Iran.
Abstract
This paper analyzes systems of linear first-order ordinary differential equations (ODEs) with parametric coefficients, a class of problems that arises in control theory, optimization, and applied mathematics. We introduce the notion of a comprehensive solution system for such parametric ODEs, constructed using Gröbner systems from computer algebra. Our approach partitions the parameter space into finitely many cells and associates an explicit solution with each cell. Furthermore, we present an algorithm that computes a comprehensive solution system for any given parametric system. To address the computational challenges inherent in Gröbner systems, we adopt the GES algorithm, a parametric variant of Gaussian elimination, which eliminates the need for Gröbner bases. This method builds upon the LDS algorithm proposed in 2017. Both algorithms have been implemented in Maple, and we illustrate the structural framework of the main algorithm with a straightforward example. The results highlight the practicality and effectiveness of the proposed methods for solving parametric linear first-order ODE systems.
Highlights
- Investigation of linear first-order ODE systems with parametric coefficients, motivated by applications in control theory, optimization, and applied mathematics.
- Definition of a comprehensive solution system (CSS) for parametric linear ODEs, which organizes parameter space into finite cells, with each cell paired to an explicit closed-form solution.
- Development of the CSS algorithm to compute comprehensive solution systems for any given parametric linear first-order ODE system.
- Adoption of the GES algorithm (parametric Gaussian elimination) to tackle computational challenges in Gröbner-system-based approaches, reducing reliance on Gröbner bases.
- The CSS framework builds upon prior LDS methodology and integrates parametric linear algebra techniques to efficiently handle parameter dependencies.
- All algorithms are implemented in Maple, enabling practical experimentation and reproducibility.
- The Improved-CSS algorithm (based on GES) outperforms traditional Gröbner-based methods in efficiency across benchmark examples, indicating potential for real-time and large-scale parameter analyses.
- Potential impact on control theory and optimization through systematic, parameter-aware analysis, including gain scheduling, robust stability analysis, and design of controllers or estimators with stability guarantees across parameter ranges.
Keywords
- Parametric linear ODE systems
- Gröbner system
- Gaussian elimination system (GES) algorithm
- Comprehensive solution system
- Parameter space
- Improved-CSS algorithm
Main Subjects
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