Control-constrained parabolic optimal control problems on evolving surfaces – theory and variational discretization
We consider control-constrained linear-quadratic optimal control problems on evolving hypersurfaces in . In order to formulate well-posed problems, we prove existence and uniqueness of weak solutions for the state equation, in the sense of vector-valued distributions. We then carry out and prove convergence of the variational discretization of a distributed optimal control problem. In the process, we investigate the convergence of a fully discrete approximation of the state equation, and obtain optimal orders of convergence under weak regularity assumptions. We conclude with a numerical example.
Mathematics Subject Classification (2010): 58J35 , 49J20, 49Q99, 35D30, 35R01
Keywords: Evolving surfaces, weak solutions, parabolic optimal control, error estimates.
We investigate parabolic optimal control problems on evolving material hypersurfaces in . Following [DE07], we consider a parabolic state equation in its weak form
where is a family of -smooth, compact -dimensional surfaces in , evolving smoothly in time with velocity . Further assume sufficiently smooth and let denote the material derivative of a smooth test function .
We start by defining unique weak solutions for the state equation. The idea is to pull back the problem onto a fixed domain, introducing distributional material derivatives in the sense of [LM68] and a -like solution space. As a consequence, a large part of the theory developed around for fixed domains applies, compare for example [LM68] and [Lio71] .
In [DE10] order-optimal error bounds of type are derived for the discretization of the state equation, assuming a slightly higher regularity of the state than is used in section 5 and 6, where we derive -like bounds. A class of Runge-Kutta methods to tackle the space-discretized problem is investigated in [DLM11], assuming among other things that one can evaluate in a point-wise fashion, i.e. that is well defined. For a fully discrete approach and the according error bounds see [DE11]. There a backwards Euler method is considered for time discretization whose implementation resembles our discontinuous Galerkin approach in Section 6. Yet while the approach in [DE11] ultimately leads to -convergence, we allow for non-smooth controls and thus cannot expect to obtain such strong convergence estimates.
The paper is structured as follows. We begin with a very short introduction into the setting in Section 2. In order to formulate well posed optimal control problems we first proof the existence of an appropriate weak solution in Section 3, complementing the existence results from [DE07]. We then use the the results from Section 3 in order to formulate control constrained optimal control problems in section 4. Afterwards, we examine the space- and time-discretization of the state equation in Sections 5 and 6, before returning to the optimal control problems in Section 7. There we apply variational discretization in the sense of [Hin05] to achieve fully implementable optimization algorithms. We end the paper by giving a numerical example in Section 8.
Before we can properly formulate (1.1), let us introduce some basic tools and clarify what our assumptions are regarding the family .
The hypersurface is -smooth and compact (i.e. without boundary). evolves along a -smooth velocity field with flow , such that its restriction is a diffeomorphism for every .
The assumption gives rise to a second representation of and in particular implies to be orientable with a smooth unit normal field . As a consequence, the evolution of can be described as the level set of the signed distance function such that
as well as and for . Further, we have for some tubular neighborhood of . Due to the uniform boundedness of the curvature of the radius does not depend on . The domain of is which is a neighborhood of in .
Using we can define the projection
which allows us to extend any function to by . Hence we can represent the surface gradient in global exterior coordinates as the euclidean projection of the gradient of onto the tangential space of . In the following we will write instead of , wherever it is clear which surface the gradient relates to.
We are going to exploit existing results on vector-valued distributions, which we recall here for completeness. In order to define weak derivatives consider , the space of real valued -smooth functions with compact support in . Fix . Each defines a vector-valued distribution through the -valued integral .
Its distributional derivative is said to lie in iff it can be represented by in the following sense
and we write . Note that by we denote the representation of the dual which arises from by completion.
For , the space
with scalar product is a Hilbert space.
is compactly embedded into , the space of continuous -valued functions.
Denote by the space of -smooth -valued test functions on . The inclusion is dense.
For two functions the product is absolutely continuous with respect to and
a.e. in , and as a consequence there holds the formula of partial integration
For a proof of the lemma, see [LM68, Ch. I,Thrms. 3.1 and 2.1]. In fact one can use the formula of partial integration to prove the embedding into , see [Eva98, Ch. 5,Thrm 3]. For further references see [Trö05, Thm. 3.10].
Our approach to weak material derivatives relies on the following equivalent formulation of condition (2.2)
which defines the weak derivative of a function via its -scalar product with elements of .
3 Weak solutions
The scope of this section is to formulate appropriate function spaces and a related weak material derivative, in order to prove the existence of unique weak solutions of (1.1) for quite weak right-hand sides .
We start by defining the strong material derivative for smooth functions , namely the derivative
along trajectories of the velocity field . The material derivative has the following properties.
Let be sufficiently smooth. Then
with and .
A proof and details can be found in the Appendix of [DE07].
Lemma and Definition 3.2.
Let denote the Jacobian determinant of the matrix representation of with respect to orthogonal bases of the respective tangent space. By Assumption 2.1 and there exists , such that for all
Given Assumption 2.1, consider the family . Then for we introduce the pull-back
which is a linear homeomorphism from into for any . Moreover is a linear homeomorphism from into . Thus finally the adjoint operator, is also a linear homeomorphism. There exist constants independent of , such that for all , or respectively, and for all
and thus finally .
Furthermore there holds .
For we have
and thus , with .
For equivalence consider and choose . Now
and because we can integrate by parts on to obtain with
Note that and that , where depends only on the mean curvature of and the second space derivatives of which are bounded independently of . Now , because as stated above , and . Thus, for some depending only on a global bound on , and , , , there holds
Now and are two equivalent norms on . Hence also their dual norms are equivalent. The norm of can now be expressed by
and the bound on the norm of follows from the equivalence of said -norms.
We need to state one more Lemma concerning continuous time-dependence of the previously defined norms.
Let . For , , the following expressions are continuous with respect to
By the change of variables formula we have
which is a continuous function due to the regularity of stated in Assumption 2.1. Similarly we conclude the continuity of the -norm.
As far as Lemma 3.1 is concerned, for a family of functions , , one can define at simply by . If can be smoothly extended, this is equivalent to (3.1). The following Lemmas aim at defining a weak material derivative of that translates into a weak derivative of the pull-back .
Lemma and Definition 3.4.
Consider the disjoint union . The set of functions , inherits a canonical vector space structure from the spaces (addition and multiplications with scalars). Given Assumption 2.1, for we define
Abusing notation, now and in the following we identify with . Endowed with the scalar product
becomes a Hilbert space.
In the same manner we define the space . For use instead of . All three spaces do not depend on .
For , it is clear how to interpret , namely . We say that has weak material derivative iff there holds
for all , and the definition does not depend on .
In order to define the scalar product of , we must ensure measurability of . Since it suffices to show measurability of for all . By definition of the set we have . Hence, there exists a sequence of measurable simple functions that converge pointwise a.e. to in . Each is the finite sum of measurable single-valued functions, i.e. , , , measurable and disjoint. By Lemma 3.3 the function
is the finite sum of measurable functions and thus measurable. Using the continuity of the operator , as stated in Lemma 3.2, one infers pointwise convergence a.e. of towards which in turn implies measurability of .
Again by Lemma 3.2 we now conclude integrability of and at the same time equivalence of the norms
Completeness of follows, since and are isomorph. Again because of Lemma 3.2, is equivalent to , thus the definition does not depend on the choice of . For and we proceed similarly.
We show that the definition of the weak material derivative does not depend on . On Equation (3.4) reads
for all . For , we now transform the relation into one on , using , and
and because is a linear homeomorphism, it also defines an isomorphism between and .
Strictly speaking the elements of are equivalence classes of functions coinciding a.e. in , just like the elements of .
The definition of the weak derivative of in (3.4) translates into weak derivatives of the pullback . In order to make the connection between the two, we state the following
Let and . Then also lies in and
where is to be understood as .
We show that for the function lies in . The claim then follows by partial integration in .
1. Because and the strong surface gradient are continuous and thus uniformly continuous on the compact set , we infer . Note that . Let , then for sufficiently small one has
2. As to the distributional derivative of , we show that . Observe that the uniform continuity of the strong derivative on allows us to estimate
for sufficiently small. Again by uniform continuity of we conclude . All told, taking into account the continuity of the pointwise multiplication between the respective spaces, we showed