TABLE OF CONTENTS This appendix briefly summarizes the most useful results of optimal control theory. In the first section static problems are analyzed, i.e., the objective function only includes time-independent control variables. This formulation yields a parameter optimization or nonlinear programming problem for which several closed-form and numerical solution algorithms are known. Several excellent textbooks are available on that subject, for instance [19] and [21].
The second section analyzes dynamic optimal control problems. Starting with a brief repetition of the classical variational calculus theory, the concepts of adjoint (Lagrange) states and Hamiltonian formulations are introduced. To be able to deal with the case of constrained input variables, Pontryagin's minimum principle is briefly introduced. As with the first section, the main objective here is to collect the main facts without any proofs and to introduce the notation. Readers interested to learn more about this field are referred to one of the several available textbooks, for instance [24].
Let u = [ u 1, , u m] T ? ? be a vector of arbitrary parameters and ? sufficiently differentiable function (the "performance index") that has to be minimized. [1] Sufficient conditions for a point u o to be a local minimum are
i.e., the gradient of the performance index must be zero at the minimum ( u o is a stationary point), and the Hessian matrix of the performance index must be positive definite...
