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
V=\int\min{t}\max{\infty}[q(x)+h(u)]d_{\tau}
are obtained. It is shown that if the value of the optimum V
⁰
is required to be a quadratic form
V^{o} = \frac{1}{2}x'Mx
of the current state
x
, and if certain state variables cannot be measured, then
M
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
V^{0}(x)
.