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A posteriori error estimates of fully discrete finite-element schemes for nonlinear parabolic integro-differential optimal control problems
Advances in Difference Equations volume 2014, Article number: 15 (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 problems that we are interested in are the following nonlinear parabolic integro-differential optimal control problems:
subject to the state equations
where the bounded open set is a 2 regular convex polygon with boundary ∂ Ω, , , , , α is a positive constant, and B is a continuous linear operator from K to . For any the function , for any , and . We assume the coefficient matrix is a symmetric positive definite matrix and there is a constant satisfying for any vector , . Here, K denotes the admissible set of the control variable, defined by
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
We shall now construct the optimality conditions and the finite element discretization of the nonlinear parabolic integro-differential optimal control problem (1.1)-(1.4). Let , . Let
Then the nonlinear parabolic integro-differential optimal control problem (1.1)-(1.4) can be restated as
where the inner product in or is indicated by .
It is well known (see, e.g., ) that the optimal control problem has a solution , and that if a pair is the solution of equations (2.5)-(2.7), then there is a co-state such that the triplet satisfies the following optimality conditions:
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.
Let denote the piecewise constant space over τ. Then we take . By the definition of the finite-element subspace, the finite-element discretization of equations (2.5)-(2.7) is as follows: compute such that
where , is an approximation of .
Again, it follows that the optimal control problem (2.13)-(2.15) has a solution , and that if a pair is the solution of equations (2.13)-(2.15), then there is a co-state such that triplet satisfies the following optimality conditions:
where , .
We now consider the fully discrete finite-element approximation for the semidiscrete problem. Let , , and , . Also, let
For , construct the finite-element spaces with the mesh (similar to ). Similarly, construct the finite-element spaces with the mesh (similar as ). Let denote the maximum diameter of the element in . Define mesh functions and mesh size functions such that , . For ease of exposition, we shall denote and by τ and , respectively. Then the fully discrete finite-element approximation of equations (2.13)-(2.15) is as follows: compute , , such that
where is an approximation of .
Now, it follows that the optimal control problem (2.21)-(2.23) has a solution , , and that if a pair , , is the solution of (2.21)-(2.23), then there is a co-state , , such that triplet satisfies the following optimality conditions:
For , let
For any function , let , . Then the optimality conditions (2.24)-(2.28) can be restated as
In the rest of the paper, we shall use some intermediate variables. For any control function , we first define the state solution which satisfies
Now we restate the following well-known estimates in .
Lemma 2.1 Let be the Clément type interpolation operator defined in . Then for any and all elements τ,
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 .
Theorem 3.1 Let and be the solutions of equations (2.37)-(2.39) and equations (2.32)-(2.34), respectively. Then
where l is a face of an element τ, is the size of face l, is the A-normal derivative jump over the interior face l, defined by
where n is the unit normal vector on outwards of .
Proof Let , and let be the Clément type interpolator of defined in Lemma 2.1. Note that
From equation (3.2), we have
By using the assumptions of A and ϕ, thus we can obtain the following result:
By using equations (2.32), (2.37), and (3.4), we infer that
Let us bound each of the terms on the right-hand side of equation (3.5). By Lemma 2.1 we have
Next, using Lemma 2.1, we get
For the right-hand terms - of equation (3.5), the Schwarz inequality implies
Let δ be small enough, and add inequalities (3.5)-(3.11) to obtain
This completes the proof. □
Analogously to the proof of Theorem 3.1, we can obtain the following estimates.
Theorem 3.2 Let and be the solutions of equations (2.37)-(2.39) and equations (2.32)-(2.34), respectively. Then
where - are defined in Theorem 3.1, l is a face of an element τ, is the A-normal derivative jump over the interior face l, defined by
where n is the unit normal vector on outwards of .
For given , let M be the inverse operator of the state equation (2.8), such that is the solution of the state equation (2.8). Similarly, for given , is the solution of the discrete state equation (2.32). Let
It is clear that S and are well defined and continuous on K and . Also the functional can be naturally extended on K. Then equations (2.5) and (2.21) can be represented as
It can be shown that
where is the solution of equations (2.37)-(2.39).
In many applications, is uniform convex near the solution u (see, e.g., ). The convexity of is closely related to the second-order sufficient conditions of the control problems, which are assumed in many studies on numerical methods of the problems. If is uniformly convex, then there is a , such that
where u and are the solutions of equations (3.14) and (3.15), respectively. We will assume the above inequality throughout this paper.
In order to have sharp a posteriori error estimates, we divide Ω into some subsets:
Then, it is clear that the three subsets do not intersect, and , .
Theorem 3.3 Let u and be the solutions of equations (2.5) and (2.36), respectively. Then
Proof It follows from the inequality (3.16) that
It is easy to see that
Since is piecewise constant, if is not empty. If , there exist and , such that , and . For example, one can always find such a required β from one of the shape functions on s. Hence, , where as and otherwise . Then, it follows from equation (2.36) that
Note that on , and from equation (3.20) we have
Let be the reference element of s, , and be a part mapped from . Note that , are both norms on . In such a case for the function β fixed above, it follows from the equivalence of the norm in the finite-dimensional space that
where the constant C can be made independent of β since it is always possible to find the required β from the shape functions on s. Thus
It follows from the definition of that on . Note that , we have
It is easy to show that
Hence, we combine Theorems 3.1-3.3 to conclude that
Theorem 3.4 Let and be the solutions of equations (2.8)-(2.12) and equations (2.32)-(2.36), respectively. Then
where - are defined in Theorems 3.1-3.3, respectively.
Proof From equations (2.8)-(2.11) and (2.37)-(2.40), we obtain the error equations
for all and . Thus it follows from equations (3.28)-(3.29) that
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.
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-X
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.
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/s10589-005-2180-2
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)00054-0
French DA, King JT: Approximation of an elliptic control problem by the finite element method. Numer. Funct. Anal. Optim. 1991, 12: 299–315. 10.1080/01630569108816430
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/S0363012994262361
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)90019-0
Neittaanmaki P, Tiba D: Optimal Control of Nonlinear Parabolic Systems: Theory, Algorithms and Applications. Dekker, New York; 1994.
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/S1995423911030037
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.2006.11.015
Liu WB, Yan NN: A posteriori error estimates for distributed convex optimal control problems. Numer. Math. 2005, 101: 1–27. 10.1007/s00211-005-0608-3
Verfurth R: A posteriori error estimates for nonlinear problems. Math. Comput. 1994, 62: 445–475.
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.2009.11.009
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.2010.06.011
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/S1995423909010078
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.2013.08.019
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.
Lions JL: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin; 1971.
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:1014239012739
Sloan IH, Thome V: Time discretization of an integro-differential equation of parabolic type. SIAM J. Numer. Anal. 1986, 23: 1052–1061. 10.1137/0723073
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)90039-9
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).
The author declares that they have no competing interests.
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Lu, Z. A posteriori error estimates of fully discrete finite-element schemes for nonlinear parabolic integro-differential optimal control problems. Adv Differ Equ 2014, 15 (2014). https://doi.org/10.1186/1687-1847-2014-15
- nonlinear parabolic integro-differential optimal control problems
- adaptive multi-mesh finite-element methods
- a posteriori error estimates
- fully discrete