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

The linear quadratic cost control problem
\dot{x}(t) = A(t)x(t) + B(t)u(t)
x(0) = x_{0}
with a cost functional
J[u] = \frac{1}{2} \int\min{0}\max{T} [\langlex, Q(t)x\rangle + \langleu, R(t)u\rangle] dt
is considered, supposing
S
is a suitable space of piecewise cubic polynominals on a mesh of norm
h
on the interval
[0, T]
. Then a Ritz type algorithm is developed for minimizing
J [\cdotp]
over
S
. 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 control
u\ast
with order
0(h^{3})
. If
x_{\bar{u}}
is the trajectory generated by
\bar{u}
, then it is shown that
x_{\bar{u}}
approximates the optimal trajectory
x\ast
to
0(h^{3})
. Finally, it is shown that
J[\bar{u}]
approximates
J[u\ast]
to order
0(h^{6})
. The numerical properties of the algorithm, including speed and accuracy comparisons with the conventional numerical approach, are presented in a forthcoming paper.