New a posteriori error estimates of mixed finite element methods for quadratic optimal control problems governed by semi
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New a posteriori error estimates of mixed finite element methods for quadratic optimal control problems governed by semilinear parabolic equations with integral constraint Zuliang Lu1,2* , Shaohong Du3 and Yuelong Tang4 * Correspondence: [email protected] 1 School of Mathematics and Statistics, Chongqing Three Gorges University, Chongqing, 404000, P.R. China 2 Laboratory for Applied Mathematics, Beijing Computational Science Research Center, Beijing, 100084, P.R. China Full list of author information is available at the end of the article
Abstract In this paper, we investigate new L∞ (L2 ) and L2 (L2 )-posteriori error estimates of mixed finite element solutions for quadratic optimal control problems governed by semilinear parabolic equations. The state and the co-state are discretized by the order one Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. We derive a posteriori error estimates in L∞ (J; L2 ())-norm and L2 (J; L2 ())-norm for both the state and the control approximation. Such estimates, which are apparently not available in the literature, are an important step towards developing reliable adaptive mixed finite element approximation schemes for the optimal control problem. MSC: 49J20; 65N30 Keywords: a posteriori error estimates; quadratic optimal control problems; semilinear parabolic equations; mixed finite element methods; integral constraint
1 Introduction In this paper we consider quadratic optimal control problems governed by the semilinear parabolic equations T min p – pd + y – yd + u dt , u∈K⊂U yt (x, t) + div p(x, t) + φ y(x, t) = f (x, t) + u(x, t), x ∈ , t ∈ J,
(.)
x ∈ , t ∈ J,
(.)
p(x, t) = –A(x)∇y(x, t), y(x, t) = ,
x ∈ ∂, t ∈ J,
y(x, ) = y (x),
x ∈ ,
(.)
(.)
where the bounded open set ⊂ R is a convex polygon with the boundary ∂. J = [, T]. Let K be a closed convex set in the control space U = L (J; L ()), p, pd ∈ (L (J; H ())) , y, yd ∈ L (J; H ()), f , u ∈ L (J; L ()), y (x) ∈ H (). For any R > , the function φ(·) ∈ W ,∞ (–R, R), φ (y) ∈ L () for any y ∈ H (), and φ (y) ≥ . Assume that the coefficient ¯ R× ) is a symmetric × -matrix and there are conmatrix A(x) = (aij (x))× ∈ C ∞ (; stants c , c > satisfying for any vector X ∈ R , c XR ≤ Xt AX ≤ c XR . We assume ©2013 Lu et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Lu et al. Boundary Value Problems 2013, 2013:230 http://www.boundaryvalueproblems.com/content/2013/1/230
that the constraint on the control is an obstacle such that K = u ∈ L J; L () : u(x, t) dx ≥ , a.e. in × J .
Optimal control problems have been successfully utilized in scientific and engineering numerical sim
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