- Open Access
A posteriori error estimates of fully discrete finite-element schemes for nonlinear parabolic integro-differential optimal control problems
© Lu; licensee Springer. 2014
Received: 12 August 2013
Accepted: 11 December 2013
Published: 15 January 2014
The aim of this work is to study the optimality conditions and the adaptive multi-mesh fully discrete finite-element schemes for quadratic nonlinear parabolic integro-differential optimal control problems. We derive a posteriori error estimates in -norm and -norm for both the coupled state and control approximation. Such estimates can be used to construct reliable adaptive finite-element approximation for nonlinear parabolic integro-differential optimal control problems.
Parabolic integro-differential optimal control problems are very important for modeling in science. They have various physical backgrounds in many practical applications such as population dynamics, visco-elasticity, and heat conduction in materials with memory. The finite-element approximation of parabolic integro-differential optimal control problems plays a very important role in the numerical methods for these problems. The finite-element approximation of an optimal control problem by piecewise constant functions has been investigated by Falk  and Geveci . The discretization for semilinear elliptic optimal control problems is discussed by Arada et al. in . In , Brunner and Yan analyzed the finite-element Galerkin discretization for a class of optimal control problems governed by integral equations and integro-differential equations. Systematic introductions of the finite-element method for optimal control problems can be found in [5–10].
The adaptive finite-element approximation is the most important method to boost the accuracy of the finite-element discretization. It ensures a higher density of nodes in a certain area of the given domain, where the solution is discontinuous or more difficult to approximate, using an a posteriori error indicator. A posteriori error estimates are computable quantities in terms of the discrete solution and measure the actual discrete errors without the knowledge of exact solutions. They are essential in designing algorithms for a mesh which equidistribute the computational effort and optimize the computation. The literature for this is huge. Some techniques directly relevant to our work can be found in [11, 12]. Recently, in [13–16], we derived a priori error estimates and superconvergence for linear quadratic optimal control problems using mixed finite-element methods. A posteriori error estimates of mixed finite-element methods for general semilinear optimal control problems were addressed in .
In this paper, we adopt the standard notation for Sobolev spaces on Ω with a norm given by , a semi-norm given by . We set . For , we denote , , and , . We denote by the Banach space of all integrable functions from J into with norm for , and the standard modification for . The details can be found in .
The plan of this paper is as follows. In the next section, we construct the optimality conditions and present the finite-element discretization for nonlinear parabolic integro-differential optimal control problems. A posteriori error estimates of finite-element solutions for those problems are established in Section 3. Finally, we analyze the conclusion and future work in Section 4.
2 Finite elements for integro-differential optimal control
where the inner product in or is indicated by .
where is the adjoint operator of B.
Let us consider the finite-element approximation of the optimal control problem (2.5)-(2.7). Again here we consider only n-simplex elements and conforming finite elements.
For ease of exposition we will assume that Ω is a polygon. Let be regular partition of Ω. Associated with is a finite-dimensional subspace of , such that are polynomials of order m () and . It is easy to see that . Let denote the maximum diameter of the element τ in , . In addition C or c denotes a general positive constant independent of h.
Due to the limited regularity of the optimal control u in general, there will be no advantage in considering higher-order finite element spaces rather than the piecewise constant space for the control. So, we only consider piecewise constant finite elements for the approximation of the control, though higher-order finite elements will be used to approximate the state and the co-state.
where , is an approximation of .
where , .
where is an approximation of .
Now we restate the following well-known estimates in .
where l is the edge of the element.
3 A posteriori error estimates
In this section we will obtain a posteriori error estimates in and for the coupled state and control approximation. Firstly, we estimate the error .
where n is the unit normal vector on outwards of .
This completes the proof. □
Analogously to the proof of Theorem 3.1, we can obtain the following estimates.
where n is the unit normal vector on outwards of .
where is the solution of equations (2.37)-(2.39).
where u and are the solutions of equations (3.14) and (3.15), respectively. We will assume the above inequality throughout this paper.
Then, it is clear that the three subsets do not intersect, and , .
Hence, we combine Theorems 3.1-3.3 to conclude that
where - are defined in Theorems 3.1-3.3, respectively.
4 Conclusion and future work
In this paper we discuss the finite-element methods of the nonlinear parabolic integro-differential optimal control problems (1.1)-(1.4). We have established a posteriori error estimates for both the state, the co-state, and the control variables. The posteriori error estimates for those problems by finite-element methods seem to be new.
In our future work, we shall use the mixed finite-element method to deal with nonlinear parabolic integro-differential optimal control problems. Furthermore, we shall consider a posteriori error estimates and superconvergence of mixed finite-element solution for nonlinear parabolic integro-differential optimal control problems.
The author expresses his gratitude to the referees for their helpful suggestions, which led to improvements of the presentation. This work is supported by National Science Foundation of China (11201510), Chongqing Research Program of Basic Research and Frontier Technology (cstc2012jjA00003), Scientific and Technological Research Program of Chongqing Municipal Education Commission (KJ121113), and Science and Technology Project of Wanzhou District of Chongqing (2013030050).
- Falk FS: Approximation of a class of optimal control problems with order of convergence estimates. J. Math. Anal. Appl. 1973, 44: 28–47. 10.1016/0022-247X(73)90022-XMATHMathSciNetView ArticleGoogle Scholar
- Geveci T: On the approximation of the solution of an optimal control problem governed by an elliptic equation. RAIRO. Anal. Numér. 1979, 13: 313–328.MATHMathSciNetGoogle Scholar
- Arada N, Casas E, Tröltzsch F: Error estimates for the numerical approximation of a boundary semilinear elliptic control problem. Comput. Optim. Appl. 2005, 31: 193–219. 10.1007/s10589-005-2180-2MathSciNetView ArticleGoogle Scholar
- Brunner H, Yan NN: Finite element methods for optimal control problems governed by integral equations and integro-differential equations. Appl. Numer. Math. 2003, 47: 173–187. 10.1016/S0168-9274(03)00054-0MathSciNetView ArticleGoogle Scholar
- French DA, King JT: Approximation of an elliptic control problem by the finite element method. Numer. Funct. Anal. Optim. 1991, 12: 299–315. 10.1080/01630569108816430MATHMathSciNetView ArticleGoogle Scholar
- Gunzburger MD, Hou SL: Finite dimensional approximation of a class of constrained nonlinear control problems. SIAM J. Control Optim. 1996, 34: 1001–1043. 10.1137/S0363012994262361MATHMathSciNetView ArticleGoogle Scholar
- Mossino J: An application of duality to distributed optimal control problems with constraints on the control and the state. J. Math. Anal. Appl. 1975, 50: 223–242. 10.1016/0022-247X(75)90019-0MATHMathSciNetView ArticleGoogle Scholar
- Neittaanmaki P, Tiba D: Optimal Control of Nonlinear Parabolic Systems: Theory, Algorithms and Applications. Dekker, New York; 1994.Google Scholar
- Lu Z: A posteriori error estimates of finite element methods for nonlinear quadratic boundary optimal control problem. Numer. Anal. Appl. 2011, 4: 210–222. 10.1134/S1995423911030037View ArticleGoogle Scholar
- Chen Y, Liu WB: A posteriori error estimates for mixed finite element solutions of convex optimal control problems. J. Comput. Appl. Math. 2008, 211: 76–89. 10.1016/j.cam.2006.11.015MATHMathSciNetView ArticleGoogle Scholar
- Liu WB, Yan NN: A posteriori error estimates for distributed convex optimal control problems. Numer. Math. 2005, 101: 1–27. 10.1007/s00211-005-0608-3MathSciNetView ArticleGoogle Scholar
- Verfurth R: A posteriori error estimates for nonlinear problems. Math. Comput. 1994, 62: 445–475.MathSciNetView ArticleGoogle Scholar
- Chen Y, Lu Z: Error estimates of fully discrete mixed finite element methods for semilinear quadratic parabolic optimal control problems. Comput. Methods Appl. Mech. Eng. 2010, 199: 1415–1423. 10.1016/j.cma.2009.11.009MATHView ArticleGoogle Scholar
- Chen Y, Lu Z: Error estimates for parabolic optimal control problem by fully discrete mixed finite element methods. Finite Elem. Anal. Des. 2010, 46: 957–965. 10.1016/j.finel.2010.06.011MathSciNetView ArticleGoogle Scholar
- Lu Z, Chen Y:-error estimates of triangular mixed finite element methods for optimal control problem govern by semilinear elliptic equation. Numer. Anal. Appl. 2009, 2: 74–86. 10.1134/S1995423909010078View ArticleGoogle Scholar
- Chen Y, Lu Z, Huang Y: Superconvergence of triangular Raviart-Thomas mixed finite element methods for bilinear constrained optimal control problem. Comput. Math. Appl. 2013, 66: 1498–1513. 10.1016/j.camwa.2013.08.019MathSciNetView ArticleGoogle Scholar
- Lu Z, Chen Y: A posteriori error estimates of triangular mixed finite element methods for semilinear optimal control problems. Adv. Appl. Math. Mech. 2009, 1: 242–256.MATHMathSciNetGoogle Scholar
- Lions JL: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin; 1971.MATHView ArticleGoogle Scholar
- Liu WB, Yan NN: A posteriori error estimates for control problems governed by nonlinear elliptic equation. Adv. Comput. Math. 2001, 15: 285–309. 10.1023/A:1014239012739MATHMathSciNetView ArticleGoogle Scholar
- Sloan IH, Thome V: Time discretization of an integro-differential equation of parabolic type. SIAM J. Numer. Anal. 1986, 23: 1052–1061. 10.1137/0723073MATHMathSciNetView ArticleGoogle Scholar
- Yanik EG, Fairweather G: Finite element methods for parabolic and hyperbolic partial integro-differential equations. Nonlinear Anal. 1988, 12: 785–809. 10.1016/0362-546X(88)90039-9MATHMathSciNetView ArticleGoogle Scholar
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