On the inverse optimum control problem for a class of nonlinear autonomous systems

Some aspects of the inverse optimum control problem are considered for a class of nonlinear autonomous systems. A closed-loop system with a known control law is given; the problem is to determine performance criteria for which the given control law is optimum. Algebraic conditions that must be satisfied by a class of scalar performance criteria of the form are obtained. It is shown that if the value of the optimum V⁰is required to be a quadratic form of the current state , and if certain state variables cannot be measured, then cannot be positive definite. The inverse optimum control problem corresponding to the problem of Lur\´e is considered. Examples are given to illustrate the techniques and to compare the properties of a linear and nonlinear system having the same optimum performance .