Optimal control with quadratic performance index and fixed terminal time

The conventional solution for the optimal control of a linear-stationary regulator with quadratic performance index and fixed terminal time leads to a linear control law with time-varying gain coefficients [1]. In addition to the usual disadvantages of time-variable controllers, some of the time-varying gain coefficients approach infinity as the specified terminal time is approached. In the present paper, it is shown that the optimal control for the above problem can be expressed in the form of a time-invariant non-linear control law. Certain parameters in the nonlinear control law are functions of the initial time and initial state of the system. The conventional time-varying linear control law can be obtained directly from the time-invariant nonlinear control law. The results of the present paper are applicable to a more general class of optimal control problems involving linear and nonlinear systems. Two examples are given to illustrate the method.