Approximate Hamilton-Jacobi Reachability Analysis for a Class of Two-Timescale Systems, with Application to...

Area of Science:

• Control Theory
• Dynamical Systems
• Computational Mathematics

Background:

• Hamilton-Jacobi reachability (HJR) is crucial for controlling nonlinear, uncertain systems.
• The curse of dimensionality significantly limits HJR's scalability.
• Singular perturbation methods offer model reduction for multi-timescale systems but are difficult to apply within HJR's differential game context.

Purpose of the Study:

• To address the computational challenges of HJR in complex systems.
• To develop a method for reducing the dimensionality of systems analyzed by HJR.
• To identify a specific class of systems amenable to model reduction within the HJR framework.

Main Methods:

• Leveraging existing research on singularly perturbed differential games.
• Identifying a class of systems where singular perturbation methods can be effectively applied to HJR.
• Relating the identified system properties to key HJR quantities.

Main Results:

• A method for reducing the dimensionality of systems in HJR was identified.
• The approach facilitates the application of singular perturbation techniques to HJR.
• The utility of the method was demonstrated on biological system examples.

Conclusions:

• The developed approach enhances the applicability of HJR to complex, multi-timescale systems.
• This work provides a pathway to overcome the curse of dimensionality in HJR.
• The findings are particularly relevant for control problems in biological systems.