1. Dynamic Adversarial Conflict with Restricted Information Jason L. Speyer Research Asst.: Ashitosh Swarup Mechanical and Aerospace Engineering Department UCLA MURI, Review June 4 2002

2. Cooperative and Adversarial Static Strategies with Restricted Information Static Stochastic Teams [Radner] Each team member’s strategy is a function of only its local noisy measurement of the state of the world. Minimize the expected value of a convex function of the team strategies and the state of the world. All a priori statistics and functions are known. Solution Stationary conditions for general convex cost [Radner]. Locally finiteness condition relaxed [Krainak, Speyer, Marcus] Solutions available for the LQG and LEG. Static Stochastic Nonzero-sum Games [Basar] Solutions available for LQG.

3. Dynamic Team and Game Strategies with Nonclasical Information Patterns LQG and LEG team strategies with one-step delayed-information pattern available. All information except for the current measurement is shared. Solution constructed by dynamic programming where a static game is solved at each step in the backward recursion. Since information can not be shared, few results are available for game strategies [Willman]. Formal (possible) solution to LQG games of conflict. We interpret his results and discuss new directions.

4. Formulation of the Dynamic LQG Game With Restricted Information Consider the quadratic cost criterion The discrete-time system dynamics are The measurements are

5. Strategies for Games with Restricted Information Define the measurement history of the pursuer and evader as Define the strategies as general linear functions Since these strategies are adversarial, there is no cooperation and therefore, no possibility of cheating. Consider the Saddle Point Inequality as where ( ) * denotes the saddle point strategies. If one player uses a linear strategy, then the resulting LQG problem produces the other linear optimal strategy.

6. Construction of the Linear Strategies The cost can be formed through the following nesting Assume the pursuer knows the functional form of the evader’s strategy as Substitute evader’s strategy into the dynamic system where is a ((i+1)·n+i·e)×(i·n+(i-1)·e) growing matrix.

7. Construction (Continued) Knowing the evaders strategy, solve the LQG problem of minimizing where is the conditional mean propagated as The pursuer’s strategy given the evaders strategy is This strategy, known to the evader, must be reduced to the form

8. Convergence to Saddle Point Strategies Strategies are a complex function of the opponents gains. Strategy gains are determined by an iterative procedure. Begin by assuming an adversaries strategy. Solve for the opponents strategy given an adversaries strategy. A sequence of LQG minimization and maximizations oscillate about the saddle point. This sequence may converge to the saddle point strategies. Require conditions for the existence of a saddle point of pure strategies. Require conditions for convergence to the saddle point.

9. Special Cases Consider the full state information LQR differential game. If there exits a solution to the saddle point Riccati eq., then the Riccati eqs. associated with sequential min and max operations converge to the saddle point Riccati eq. Convergence has been proved. Consider a scalar three stage dynamic game with restricted information. Cost criterion converges to a fixed point for some parameters. Saddle point strategies converge to a fixed point. Results indicate a hedging policy by the adversaries over their full state strategies.

10. Application to SEAD Requires generalization of LQG strategies for adversarial conflict. Apply to suppression of enemy air defenses (SEAD) UCAVs allocate resources based on their distributed sensor information. SAM site allocates resources based on its sensor information.

11. Coordinated Flight Aerodynamically Coupled Formation Flight of Aircraft Chichka, Wolfe, & Speyer Applications: Autonomous Aerial Refueling. Autonomous Formation Flight for Drag Reduction. UCAV Clusters.