Invariant optimal control problems on the 3D Lorentz group SO(1,2)0

Invariant optimal control problems cover a wide variety of mathematical problems inspired by the irreversible nature of time-evolving events. The 3D Lorentz group is both the transformation group of reduced Minkowski spacetime and the symmetry group of hyperobolic plane geometry and as such allows some strong visual and mathematical insights into the kinematic laws of special relativity. Making use of the invariance properties, we deal with the optimal control problems geometrically using the powerful machinery of geometric control theory. We cover generally controllability properties and determination of equivalence classes of all such left-invariant dynamical systems. Choosing specific examples we determine explicit expressions of extremals in terms of Jacobi elliptic functions and the nonlinear stability of their equilibrium points. All the required background on hyperbolic plane geometry, introductory Lie theory and geometric control theory is covered and clearly presented to make this a stand-alone monograph on invariant control systems on the 3D Lorentz group.