Variational discretization of one-dimensional elliptic optimal control problems with BV functions based on the mixed formulation

We consider optimal control of an elliptic two-point boundary value problem governed by functions of bounded variation (BV). The cost functional is composed of a tracking term for the state and the BV-seminorm of the control. We use the mixed formulation for the state equation together with the variational discretization approach, where we use the classical lowest order Raviart-Thomas finite elements for the state equation. Consequently the variational discrete control is a piecewise constant function over the finite element grid. We prove error estimates for the variational discretization approach in combination with the mixed formulation of the state equation and confirm our analytical findings with numerical experiments. 1 Problem formulation We consider the optimal control problem min u∈BV(Ω) J(u) B 1 2‖y − yd‖ 2 L2(Ω) + α ‖u ‖M(Ω), (P) *Mathematisches Institut, Universität Koblenz-Landau, Campus Koblenz, Universitätsstraße 1, 56070 Koblenz, Germany. 1 ar X iv :2 10 6. 14 79 5v 3 [ m at h. O C ] 7 J ul 2 02 1 where y satisfies the one-dimensional elliptic two-point boundary value problem −y ′′ = u in Ω, y = 0 on Γ, (1) where Ω = (0, 1) with boundary Γ = {0, 1}, and α > 0 is a given parameter. We denote the control by u ∈ BV(Ω), the state by y ∈ H1 0(Ω), and the desired state by yd ∈ L∞(Ω). Our work is motivated by [20], where a similar optimal control problem is considered. There, variational discretization (from [23]) combined with the classical piecewise linear and continuous finite element approximation of the state is investigated, and also a fully discrete approach with piecewise constant control approximations. We here propose a variational discrete approach, which automatically delivers piecewise constant control approximations, leading to a simpler and more elegant numerical analysis in the case of piecewise constant control approximations. This is achieved by formulating the elliptic partial differential equation (the respective two-point boundary value problem) in its mixed form. Variational discretization based on the classical Raviart-Thomas discretization of the state equation then delivers piecewise constant control approximations, while keeping the corresponding variationally discrete, reduced optimization problem infinite-dimensional. This in turn simplifies the numerical analysis. We give a brief overview of related literature. An early result in optimization with BV functions and regularization by BV-seminorms is [13]. Further studies involving BV functions are [4, 5, 6]. There exist studies of elliptic optimal control with total variation regularization and control in L∞(Ω), see [15, 24]. Controls from the space BV(Ω) ∩ L∞(Ω) are considered in [8]. Optimal control governed by a semilinear parabolic equation and control cost in a total bounded variation seminorm is discussed in [9], a convergence result is shown and numerical experiments are presented. A similar problem is analyzed in [10], but with semilinear elliptic equation. Numerical results for problems with BV-control are derived in [29]. In [25] the BV source in an elliptic system is recovered. Furthermore, we remark that the inherent sparsity structure of the problem is closely related to the sparsity structure observed in optimal control problems with measures control, see e.g. [11, 12, 21, 22]. We structure this work as follows: In Section 2 we introduce the mixed formulation of the state equation, prove existence of a unique solution to the elliptic optimal control problem and derive its optimality conditions and sparsity structure. We apply variational discretization to the problem in Section 3 and discuss the resulting structure of the non-discretized controls. Then, we proceed analogously to the analysis of the continuous problem by proving existence of a