Singular surfaces typically arise in optimal control problems or differential games, where either the value function of the underlying Hamilton-Jacobi-Bellman-Isaacs equation is non-smooth or the initial or terminal conditions are non-smooth. In that case, the method of regular characteristics does not allow for construction of the optimal phase portrait in the entire space. The method of singular characteristics has been explored towards the construction of optimal trajectories in the entire phase space. Pursuit-evasion games on manifolds with non-unique geodesic lines present a nice application for the method of singular characteristics. Some recent applications include connectivity maintenance among non-cooperative agents, pursued in collaboration with Sourabh Bhattacharya and Tamer Başar.
- Hovakimyan, N. and Melikyan, A., “Geometry of Pursuit-Evasion on Second-Order Rotation Surfaces”, in Dynamics and Control, Vol. 10, pp. 297-312, 2000.
- Melikyan, A., Akhmetzhanov, A. and Hovakimyan, N., “Initial Value and Terminal Value Problems for Hamilton-Jacobi Equation,” in Systems & Control Letters, Vol. 56, pp. 714-721, 2007. [pdf]
- Bhattacharya, S., Başar, T and Hovakimyan, N., “Singular Surfaces in Multi-Agent Connectivity Maintenance Games,” in IEEE Conference on Decision and Control, Orlando, FL, 2011. [pdf]
|nhovakim (at) illinois (dot) edu
sbhattac (at) illinois (dot) edu