Direct method approximation to the state regulator control problem using a Ritz-Trefftz suboptimal control

The linear quadratic cost control problem with a cost functional is considered, supposing is a suitable space of piecewise cubic polynominals on a mesh of norm on the interval . Then a Ritz type algorithm is developed for minimizing over . The authors have previously discussed [3] certain convergence properties of the algorithm. Here the algorithm is discussed in a form suitable for real-time implementation and additional convergence criteria are presented. In [3] it was shown that the Ritz-Treffiz suboptimal control converges to the optimal control with order . If is the trajectory generated by , then it is shown that approximates the optimal trajectory to . Finally, it is shown that approximates to order . The numerical properties of the algorithm, including speed and accuracy comparisons with the conventional numerical approach, are presented in a forthcoming paper.