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 ∂ v/∂ t = A(x,t)∂ ² v/ ∂ x² + B(x,t)u, with lim t → 0^ + v(x,t) = v₀ (x),α v(0,t) + (∂ v / ∂ x)(0,t) = cf₁ (t), and β v(1,t) + (∂ v/∂ x)(1,t) = df₂ (t). The cost functional J is given by $J[u,f_1 ,f_2 ] = 1/2∫_0^T {∫_0^1 {[ {v,Q,(x,t)v} + {u,R(x,t)u} ]dt\,dx + 1/2∫_0^T {[ {f_1 ,r(t)f_1 } + {f_2 ,s(t)f_2 } ]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 $O(h^4 )$. It is also shown that the computed quadruple $( u, f_1 , f_2 , v)$ converges to the optimal quadruple $(u^ * ,f_1^ * ,f_2^ * ,v^ * )$ with an order of $O(h^2 )$. Similar statements are made for piecewise polynomial approximations of arbitrary positive order. Numerical results will appear in a forthcoming paper.