An Eigenvector Solution of the Optimal Linear Regulator Problem

The problem of optimal linear regulation considered is the return of a stablo linear constant coefficient dynamical systom to its equilibrium position with minimum value of a prescribed functional of system variables and source outputs. Pontryagin's Maximum Principle is used to set up the equations governing optimally regulated motion. A solution to these equations is obtained in terms of certain eigenvalues and eigenvectors of a matrix obtained from the equations defining optimal motion. It is shown that an equivalent linear constant coefficient dynamical system exists whose free motion is identical to the optimally regulated motion of the given system. Explicit expressions are obtained for the optimally regulated motion, optimal vnluo of performance functional and optimal source outputs.