Convergence Analysis of an Adaptive Finite Element Method for Distributed Control Problems with Control Constraints
We develop an adaptive finite element method for a class of distributed optimal control problems with control constraints. The method is based on a residual-type a posteriori error estimator and incorporates data oscillations. The analysis is carried out
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onvergence Analysis of an Adaptive Finite Element Method for Distributed Control Problems with Control Constraints A. Gaevskaya, R.H.W. Hoppe, Y. Iliash and M. Kieweg Abstract. We develop an adaptive finite element method for a class of distributed optimal control problems with control constraints. The method is based on a residual-type a posteriori error estimator and incorporates data oscillations. The analysis is carried out for conforming P1 approximations of the state and the co-state and elementwise constant approximations of the control and the co-control. We prove convergence of the error in the state, the costate, the control, and the co-control. Under some additional non-degeneracy assumptions on the continuous and the discrete problems, we then show that an error reduction property holds true at least asymptotically. The analysis uses the reliability and the discrete local efficiency of the a posteriori estimator as well as quasi-orthogonality properties as essential tools. Numerical results illustrate the performance of the adaptive algorithm. Mathematics Subject Classification (2000). Primary 65K10; Secondary 49M15. Keywords. Distributed optimal control, control constraints, adaptive finite elements, residual-type a posteriori error estimators, convergence analysis.
1. Introduction We present a convergence analysis of adaptive finite element approximations of a distributed optimal control problem with control constraints. In particular, assuming Ω ⊂ R2 to be a bounded, polygonal domain with boundary Γ := ∂Ω and given data y d ∈ L2 (Ω) and f ∈ L2 (Ω), ψ ∈ H 1 (Ω) ∩ L∞ (Ω) as well as a parameter 0 < α ≤ 1, we consider the following distributed optimal control problems with The second author has been partially supported by the NSF under Grant No. DMS-0411403 and Grant No. DMS-0511611. The fourth author acknowledges the support by the elite graduate school TopMath.
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A. Gaevskaya, R.H.W. Hoppe, Y. Iliash and M. Kieweg
bound constrained controls 1 α y − y d 20,Ω + u 20,Ω 2 2 over (y, u) ∈ H01 (Ω) × L2 (Ω),
minimize J(y, u) :=
subject to − ∆ y = f + u ,
(1.1a)
(1.1b)
u ∈ K := {v ∈ L (Ω) | v ≤ ψ a.e. in Ω} . 2
(1.1c)
It is well known (cf., e.g., [15, 20, 21]) that (1.1a)–(1.1c) admits a unique solution (y, u) ∈ H01 (Ω) × L2 (Ω). The optimality conditions involve the existence of a costate p ∈ H01 (Ω) and a co-control σ ∈ L2+ (Ω) such that y, p, u, σ satisfy a(y, v) = (f + u, v)0,Ω
,
v ∈ H01 (Ω) ,
a(p, v) = − (y − y , v)0,Ω , v ∈ 1 u = (p − σ) ∈ K , α (σ, u − v)0,Ω ≥ 0 , v ∈ K .
H01 (Ω)
d
(1.2a) ,
(1.2b) (1.2c) (1.2d)
Here, (·, ·)0,Ω refers to the standard L inner product and a(·, ·) stands for the bilinear form a(w, z) := ∇w · ∇z dx , w, z ∈ H01 (Ω) . 2
Ω
We note that the variational inequality (1.2d) can be equivalently stated as the complementarity condition σ ∈ L2+ (Ω) , ψ − u ∈ L2+ (Ω) , (σ, ψ − u)0,Ω = 0 .
(1.3)
We define the active control set A(u) as the maximal open set A ⊂ Ω such that u(x) = ψ(x) f.a.a. x ∈ A and the inactive control set I(u) according to I(u) := * ε>0 Bε , where Bε
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