1. MURI Telecon, Update 7/26/2012 Summary, Part I:  Completed: proving and validating numerically optimality conditions for Distributed Optimal Control (DOC) problem; conservation law analysis; direct method of solution for DOC problems; computational complexity analysis; application to multi-agent path planning.  Submitted paper on developments above to Automatica.  Completed: modeling of maneuvering targets by Markov motion models; derivation of (corresponding) multi-sensor performance function representing the probability of detection of multiple distributed sensors; application to multi- sensor placement.  Submitted paper on developments above to IEEE TC.  In progress: application of methods above to multi-sensor trajectory optimization for tracking and detecting Markov targets based on feedback from a Kalman-Particle filter.  Submitted paper on developments above to MSIT 2012; another journal paper on developments above in preparation.

2. MURI Telecon, Update 7/26/2012 Summary, Part II:  Completed: comparison of information theoretic functions for multi-sensor systems performing target classification.  Published paper on above developments in SMCB –Part B, Vol. 42, No. 1, Feb 2012.  In progress: comparison of information theoretic functions for multi-sensor systems performing (Markov) target tracking and detection.  Submitted paper on above developments to SSP 2012; another journal paper on developments above in preparation.  Completed: derived new approximate dynamic relations for hybrid systems.  Submitted paper on above developments to JDSM.  In progress: integrating DOC for multiple tasks and distributions with consensus based bundle algorithm (CBBA); apply DOC to non-parametric Bayesian models of sensors/targets.  In progress: develop DOC reachability proofs in the presence of communication constraints, for decentralized DOC.

3. 3 DOC Background Classical Optimal Control: Determines the optimal control law and trajectory for a single agent or dynamical system.  Characterized by well-known optimality conditions and numerical algorithms  Applied to a single agent for trajectory optimization, pursuit-evasion, feedback control (auto-pilots)..  Does not scale to systems of hundreds of agents Distributed Systems: A system of multiple autonomous dynamic systems that communicate and interact with each other to achieve a common goal.  Swarms: Hundreds to thousands of systems; homogeneous; minimal communication and sensing capabilities. Decentralized control laws: stable; non- optimal; and, do not meet common goal.  Multi-agent systems: few to hundreds of systems; heterogeneous; advanced sensing and, possibly, communication capabilities. Centralized vs. decentralized control laws: path planning; obstacle avoidance; must meet one or more common goals, subject to agent constraints and dynamics.

4. Benchmark Problem: Multi-agent Path Planning 4 The agent microscopic dynamics are given by the unicycle model with constant velocity, which amounts to the following system of ODEs, Where: Agent: The number of components (m) in the Gaussian mixture is chosen by the used based on the complexity of the initial and goal PDFs.

5. Example with m = 4 5 Initial PDF, p(x i, t 0 ) : Fixed obstacle Goal PDF, h(x i, t f ) Pr(x i )

6. Results: Optimal PDF (m = 4) 6 : Fixed obstacle Pr(x i ): Optimal PDF

7. Agents’ Optimal Trajectories 7 : Fixed obstacle Pr(x i ): Optimal PDF Agent’s control input (Sample) : Individual agent (unicycle) Feedback control of agents via DOC.

8. Example with m = 6 8 Initial PDF, p(x i, t 0 ) : Fixed obstacle Goal PDF, h(x i, t f ) Pr(x i )

9. Results: Optimal PDF (m = 6) 9 : Fixed obstacle Pr(x i ): Optimal PDF

10. Agents’ Optimal Trajectories 10 : Fixed obstacle Pr(x i ): Optimal PDF Agent’s control input (Sample) : Individual agent (unicycle) Feedback control of N = 200 agents via DOC.