# 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 (\(\alpha=2\pi/3\))
- \(\lambda_*=(0.077, 0.127, 0.452, 0.344)\)
- \(\theta_*=(0.592, 0.084, 0.145, 0.180)\)