**PhD Proposal by Julian Brew***(Advisor: Prof. Holzinger)*

*will propose a doctoral thesis entitled,*

**Using sample-based techniques to efficiently compute subspace reachable sets of high-dimensional systems**

*On*

**Wednesday, November 7 at 3:00 p.m.
Montgomery Knight Building 317**

**Abstract**

For a given continuous-time dynamical system with control input constraints and prescribed state boundary conditions, one can compute the reachable set at a specified time horizon. Forward reachable sets contain all states that can be reached using a feasible control policy at the specified time horizon. Alternatively, backwards reachable sets contain all initial states that can reach the prescribed state boundary condition using a feasible control policy at the specified time horizon. The computation of reachable sets has been applied to problems such as vehicle collision avoidance, operational safety planning, system capability demonstration, and even economic modeling and weather forecasting.

To compute exact or convex over-approximations of reachable sets for general nonlinear systems, level sets of the value function of the Hamilton-Jacobi-Bellman partial differential equation are commonly evolved numerically. However, the computational costs suffer from the curse of dimensionality and limit the feasibility of computing exact reachable sets for high-dimensional (> 4D) systems without making assumptions on the structure of the reachable set (polytope, ellipsoidal, etc.). This research investigates computational techniques for alleviating the curse of dimensionality by computing reachable sets on subspaces of the full state dimension and computing point solutions for the reachable set boundary. To compute these point solutions, optimal control problems are reduced to initial value problems using continuation methods and then solved. This results in substantial reduction in computational load in computing reachable sets compared to the typical level-set approach. Furthermore, since the computation of each reachable point solution is independent on other point solutions, the problem is embarrassingly parallel.

As the reachable set boundaries are described using independent point solutions, the evolution of these reachable sets over time results in sparse and dense collections of point solutions. Proposed work includes framing the subspace reachability problem as a distributed control problem with independent point solution agents where the goal of the distributed system is to achieve uniform surface coverage. Additionally, proposed work includes developing theory connecting reachability surfaces with pareto-optimal surfaces.

**Committee**

- Prof. Marcus Holzinger – Aerospace Engineering Sciences Dept., University of Colorado Boulder (advisor)
- Prof. Glenn Lightsey– School of Aerospace Engineering, Georgia Institute of Technology
- Stefan Schuet – Research Engineer, NASA Ames Research Center
- Prof. Panagiotis Tsiotras– School of Aerospace Engineering, Georgia Institute of Technology
- Prof. Jonathan Rogers – School of Aerospace Engineering, Georgia Institute of Technology