## Optimal Strategies for Uncertain Differential Games with Applications

The major goal of this project is to develop and test ingenious solutions for controlling multiple unmanned vehicles having a common goal under the influence of external disturbances and uncertainties. The external disturbances can include winds (in the case of aerial vehicles), water currents (in the case of marine vehicles), traffic flow (in the case of self-driving cars). The difference in the actual and perceived behavior of other adversarial agents(asymmetric information)in the environment account to the uncertainties part of the research. More specifically, the project intends to find decentralized strategies for the players in a multi-player pursuit-evasion game with external disturbances and uncertainties using algorithms that are computationally efficient.

As a part of initial study, two-pursuer/single-evader pursuit-evasion problems with different information structures, assuming both pursuers to be superior to the evader in terms of their speed capabilities, and no external disturbances are examined. The time optimal evading strategies have been identified. The regions of non-degeneracy proposed in this research can be used to investigate the utility of employing two pursuers to more efficiently capture the evader. If the initial positions of the players are such that the problem is degenerate, then one of the pursuers does not play any role in the game, and the optimal evading strategy is pure evasion from the other pursuer. Optimal evading strategies against relay pursuit are also investigated by keeping one pursuer stationary.

Regions of degeneracy and non-degeneracy in the case of pursuers' strategy being constant bearing

Regions of degeneracy and non-degeneracy in the case of pursuers' strategy being pure pursuit

We address the problem of steering a discrete-time linear dynamical system from an initial Gaussian distribution to a final distribution in a game-theoretic setting. One of the two players strives to minimize a payoff which is quadratic while trying to meet a given mean and covariance constraint at the final time-step. The other player maximizes the payoff, but it is assumed to be indifferent to the constraint of arriving at a given terminal distribution. At first, the unconstrained version of the game is examined, and the necessary conditions for the existence of a saddle point are obtained. We then show that obtaining a solution for the one-sided constrained dynamic game is not guaranteed, and subsequently the players’ best responses are analyzed. Finally, we propose to solve the problem of steering the distribution under adversarial scenarios by converting it to a convex programming problem.

A case where the terminal covariance constraint is met.

A case where the terminal covariance constraint is not met.

We proposed an approach to address optimal control problems with parametric uncertainties using the idea of Desensitized Optimal Control (DOC). A sensitivity function provides information about the first order variation of the state under parameter variations at a given time instant along a trajectory. It is demonstrated that the sensitivity function can be employed to effectively desensitize either an optimal trajectory or the state at a particular time instant (for example, the final state) along the optimal trajectory.

Desensitized trajectories for a time-optimal Zermelo’s problem ($\alpha$ is weight on the sensitivity cost).

Monte-Carlo Simulations.