Error Bounds of High Order Accuracy for the State Regulator Problem via Piecewise Polynomial Approximations

Consider the linear quadratic cost control problem x = A(t)x + B(t)u, x(0) = x₀,with a cost functional J[u] = 1/2∫₀T [ x,Q(t)x + u,R(t)u ]dt. Let S be a suitable space of piecewise cubic polynomials on a mesh of norm h on the interval $[0,T]$. Then it is shown that a so-called Ritz–Trefftz method for minimizing J[ · ] over S leads to an approximation to J[ · ] of order O(h⁷ ). Further, a computable error bound can be exhibited. It is also shown that the computed pair ( u, x) converges to the optimal pair (u^ * ,x^ * ) with order O(h³ ). Similar statements are made for piecewise polynomial approximation of arbitrary positive order.