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

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Abstract
The linear quadratic cost control problem\dot{x}(t) = A(t)x(t) + B(t)u(t)x(0) = x_{0}with a cost functionalJ[u] = \frac{1}{2} \int\min{0}\max{T} [\langlex, Q(t)x\rangle + \langleu, R(t)u\rangle] dtis considered, supposingSis a suitable space of piecewise cubic polynominals on a mesh of normhon the interval[0, T]. Then a Ritz type algorithm is developed for minimizingJ [\cdotp]overS. 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\bar{u}converges to the optimal controlu\astwith order0(h^{3}). Ifx_{\bar{u}}is the trajectory generated by\bar{u}, then it is shown thatx_{\bar{u}}approximates the optimal trajectoryx\astto0(h^{3}). Finally, it is shown thatJ[\bar{u}]approximatesJ[u\ast]to order0(h^{6}). The numerical properties of the algorithm, including speed and accuracy comparisons with the conventional numerical approach, are presented in a forthcoming paper.

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