# Examples of Time-Dependent Surveillance-Evasion Games

These examples are from the following paper: E. Cartee, L. Lai, Q. Song and A. Vladimirsky, "Time-Dependent Surveillance-Evasion Games," 2019 IEEE 58th Conference on Decision and Control (CDC), Nice, France, 2019, pp. 7128-7133, doi: 10.1109/CDC40024.2019.9029329

In all of our examples, we assume:

• The domain is a unit square, sometimes containing impenetrable and occluding obstacles.
• The square is discretized on a $201 \times 201$ grid ($h = 0.005$).
• The Evader's speed is uniform ($f(\boldsymbol{x}) = 1$) and the timestep is chosen according to the CFL stability condition: $\Delta t = h$.
• The Evader's deadline for reaching the target is $T=4$, but all Nash equilibrium controls lead to a much earlier arrival.
(The cumulative observability is computed up to the arrival time $T_{\boldsymbol{a}}$ only.)
• The magenta diamond is the Evader's starting location, and the green triangle is the Evader's target.
• The Observer's patrol trajectories shown in black.
• The Evader's trajectories are shown in color.
• All trajectories are superimposed on the expected pointwise observability $K^{\lambda}$.

#### Example 1:

• No obstacles
• Two patrol trajectories
• Omnidirectional sensor
• $\lambda_* = (0.67, 0.33)$
• $\theta_* = 1$

#### Example 2:

• One obstacle
• Two patrol trajectories
• Omnidirectional sensor
• $\lambda_* = (0.48, 0.52)$
• $\theta_* = (0.691, 0.309)$

#### Example 3:

• Three rectangular obstacles
• Two patrol trajectories
• Sector-restricted sensor ($\alpha=2\pi/3$)
• $\lambda_* = (0.5, 0.5)$
• $\theta_* = (0.5, 0.5)$

#### Example 4:

• Maze-like domain
• Four different starting locations along the same patrol trajectory
• Sector-restricted sensor ($\alpha=2\pi/3$)
• $\lambda_*=(0.077, 0.127, 0.452, 0.344)$
• $\theta_*=(0.592, 0.084, 0.145, 0.180)$