Generalized Finite-Horizon Linear-Quadratic Optimal Control

The linear quadratic problem is the prototype of a large number of optimal control problems, including the fixed end-point, the point-to-point and several H_2-H_infinity control problems, as well as their dual counterparts. In the past fifty years this multitude of problems have been addressed using different techniques, each tailored to their specific structure. It is only in the last ten years that it was recognized that a unifying framework is available both for the continuous and the discrete case, that enables these important extensions to be dealt with using a single approach. This framework hinges on formulae that parameterize the solutions of the Hamiltonian differential equation in the continuous time and the solutions of the extended symplectic system in the discrete time. Whereas traditional techniques involve the solutions of Riccati differential or difference (i.e., recursive) equations, the formulae used here to solve the finite-horizon LQ problem only rely on solutions of the algebraic Riccati equations.