Application of Numerical Methods to the Hamilton-Jacobi-Isaacs Equation in Various Dynamical Systems

Thumbnail Image
Journal Title
Journal ISSN
Volume Title
University of Alabama Libraries

The field of differential games has broad applicability to topics of economics, engineering, business, and warfare. Given the increasing levels of autonomy implemented in man-made systems in these fields, competition-based analysis may be the best option for understanding behavioral bounds when such systems interact. Differential games are governed by the Hamilton-Jacobi-Isaacs PDE, and many solution techniques are explored before identifying a gap in the existing literature. This dissertation develops a new approach to analyzing differential games based on a saddle-point solution technique. In a 2D system, the standard algorithmic approach produces both a value function interpolation and an approximate control map. Additionally, analysis of the observation error indicates that future analysis should prefer a problem formulation with relative motion. In the Circular Restricted Three-Body Problem, the same algorithms are applied to a system with real-world implications. The value and control interpolations produce a near-optimal trajectory, but the radial basis function approach suffered from high data density and did not exactly recreate the nominal solution. A perturbation analysis indicated that any mid-flight disturbance to the game state is most likely to benefit the pursuer, extending the works of Isaacs to a new domain. Ultimately, the proposed method is demonstrated to be a valuable tool for future differential games research.

Electronic Thesis or Dissertation