Error estimates for parabolic optimal control problems with control and state constraints
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Error estimates for parabolic optimal control problems with control and state constraints Wei Gong · Michael Hinze
Received: 1 July 2011 / Published online: 20 February 2013 © Springer Science+Business Media New York 2013
Abstract The numerical approximation to a parabolic control problem with control and state constraints is studied in this paper. We use standard piecewise linear and continuous finite elements for the space discretization of the state, while the dG(0) method is used for time discretization. A priori error estimates for control and state are obtained by an improved maximum error estimate for the corresponding discretized state equation. Numerical experiments are provided which support our theoretical results. Keywords Optimal control problem · Finite element method · A priori error estimate · Parabolic equation · Control and state constraints
1 Introduction In this paper we consider the optimal control problem 1 α J (y, u) = y − yd 2L2 (Ω ) + u2U T y∈KY , u∈KU 2 2 min
W. Gong () LSEC, Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China e-mail: [email protected] W. Gong · M. Hinze Bereich Optimierung und Approximation, Universität Hamburg, Bundestrasse 55, 20146 Hamburg, Germany M. Hinze e-mail: [email protected]
(1.1)
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subject to
W. Gong, M. Hinze
⎫ yt − Δy + y = Bu in ΩT , ⎪ ⎪ ⎪ ⎬ ∂y =0 on ΓT , ⎪ ∂n ⎪ ⎪ ⎭ in Ω, y(0) = y0
(1.2)
where ΩT = Ω × (0, T ], ΓT = ∂Ω × (0, T ], Ω is an open bounded domain in R2 with sufficiently smooth boundary Γ = ∂Ω, and α > 0, T > 0, yd ∈ L2 (ΩT ) are fixed. The precise smoothness requirements on Γ are given in the next section. The initial value y0 is specified in Sect. 2. The constraints on control and state are specified through the closed and convex subsets u ∈ KU := u ∈ L2 (ΩT ) : a ≤ u(x, t) ≤ b, for a.a. (x, t) ∈ ΩT ⊂ U (1.3) for controls with U := L2 (ΩT ) and y ∈ KY := y ∈ L∞ (ΩT ) : y(x, t) ≤ φ, for a.a. (x, t) ∈ ΩT
(1.4)
for states, where for simplicity we assume that a, b and φ denote constants with a < b. Furthermore, B : L2 (ΩT ) → L2 (0, T ; H 1 (Ω)∗ ) denotes the injection. State constrained optimal control problems are important from the practical point of view. The numerical analysis for these problems is involved since the multipliers associated to constraints on the state in general are Borel measures. To the best of the authors’ knowledge, there are only a few contributions to numerical analysis of parabolic optimal control problems with state constraints. Lavrentiev regularization of state constrained parabolic optimal control problems is studied in [24]. Recently, error estimates for state constrained parabolic control problem with controls of the form m (Bu)(x, t) := ui (t)fi (x) (x, t) ∈ ΩT , (1.5) i=1
are derived in [9], where f1 , . . . , fm ∈ H 1 (Ω) ∩ L∞ (Ω) are given functions. Error analysis for optimal control problems with final state constraints and control constraints is considered in [29]. Finally, in [22] a p
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