A Direct Method Approximation to the Linear Parabolic Regulator Problem Over Multivariate Spline Bases

In this paper we develop the Ritz-Trefftz algorithm in the context of a useful class of distributed parameter control problems. More particularly, we treat a typical linear parabolic control problem ∂ ν/∂ t = A(x, t) ∂2ν/∂ x2+ B(x, t)u, with$\lim_{t \rightarrow 0^+} \nu(x, t) = v_0(x), \alpha v(0, t) + (\partial \nu/\partial x)(0, t) = cf_1(t)$, and β ν(1, t) + (∂ ν/∂ x)(1, t) = df2(t). The cost functional J is given by$J \lbrack u, f_1, f_2 \rbrack = \frac{1}{2} \int^T_0 \int^1_0 \lbrack\langle v, Q, (x, t)\nu \rangle + \langle u, R(x, t)u \rangle\rbrack dt dx + \frac{1}{2} \int^T_0 \lbrack \langle f_1, r(t)f_1 \rangle + \langle f_2, s(t)f_2 \rangle\rbrack dt$. Let Σ be a suitable space of piecewise bicubic polynomials on a uniform mesh of norm h on the rectangle [ 0, 1 ] × [ 0, T ]. It is then shown that the Ritz-Trefftz method for minimizing J[ · ] over Σ leads to an approximation to J[ · ] of order O(h4). It is also shown that the computed quadruple$(\bar u, \bar f_1, \bar f_2, \bar v)$converges to the optimal quadruple (u*, f*1, f*2, v*) with an order of O(h2). Similar statements are made for piecewise polynomial approximations of arbitrary positive order. Numerical results will appear in a forthcoming paper.