Dirichlet control problems with energy regularization governed by non-coercive elliptic equations

The present study investigates a linear-quadratic Dirichlet control problem governed by a non-coercive elliptic equation posed on a possibly non-convex polygonal domain. Tikhonov regularization is carried out in an energy seminorm. The regularity of the solutions is established in appropriate weighted Sobolev spaces, and the finite element discretization of the problem is analyzed. In order to recover the optimal rate of convergence in polygonal non-convex domains, graded meshes are required. In addressing this particular problem, it is also necessary to introduce a discrete projection in the sense of $H^{1/2}(Γ)$ to deal with the non-homogeneous boundary condition. A thorough examination of the approximation properties of the discrete controls reveals that the discrete problems are strongly convex uniformly with respect to the discretization parameter. All these ingredients lead to optimal error estimates. Practical computational considerations and numerical examples are discussed at the end of the paper.

💡 Research Summary

The paper addresses a linear‑quadratic Dirichlet boundary control problem in which the state equation is a non‑coercive second‑order elliptic PDE defined on a possibly non‑convex polygonal domain Ω⊂ℝ². The control variable u lives on the boundary Γ and is regularized by a Tikhonov term measured in the energy seminorm |·|{H^{1/2}(Γ)} = ‖∇H·‖{L²(Ω)}, where H denotes the harmonic extension. The authors first establish well‑posedness of the state equation without assuming coercivity of the associated bilinear form a(·,·). By invoking results from their earlier works, they prove that the restricted operator A₀:H¹₀(Ω)→H⁻¹(Ω) is an isomorphism, which allows them to write the state solution as y_u = η_Eu + Eu with η_Eu = –A₀⁻¹(AEu). This representation yields a uniform bound ‖y_u‖{H¹(Ω)} ≤ M_S‖u‖{H^{1/2}(Γ)} and guarantees uniqueness.

The control functional J(u)=½‖y_u−y_d‖²_{L²(Ω)}+½κ²|u−u_d|²_{H^{1/2}(Γ)} is shown to be twice continuously Fréchet differentiable. Its first derivative can be expressed as J′(u)v = ⟨Tu,w⟩Γ−⟨w,v⟩Γ where T = S* S + κ D combines the adjoint state operator S* and the Dirichlet‑to‑Neumann map D, and w = S* y_d + κ D u_d. The second derivative simplifies to J″(u)v = ‖y_v‖²{L²(Ω)} + κ²‖∇H v‖²{L²(Ω)} = ⟨Tv,v⟩Γ. Crucially, Lemma 3.3 establishes a uniform coercivity constant ν>0 such that J″(u)v ≥ ν‖v‖²{H^{1/2}(Γ)} for all v, implying strong convexity of the continuous problem and guaranteeing a unique optimal control.

Because Ω may contain re‑entrant corners, standard quasi‑uniform meshes cannot deliver optimal convergence. The authors therefore employ graded meshes, refining geometrically towards each corner with a grading exponent β_j related to the interior angle ω_j. Using weighted Sobolev spaces W^{2,p}_β(Ω) they prove that the state and adjoint possess additional regularity that compensates for the singularities. This regularity is the key to achieving the optimal error order.

A major novelty lies in the treatment of the boundary control discretization. Instead of the conventional L²(Γ) projection P_h, which is global and limits higher‑order error bounds, the paper introduces an H^{1/2}(Γ)‑orthogonal projection Π_h onto the space U_h of continuous piecewise‑linear functions. Π_h minimizes the H^{1/2}(Γ) norm of the error and, together with a discrete harmonic extension operator H_h, yields a discrete control‑to‑state map that mirrors the continuous structure. The authors prove that the discrete functional J_h inherits the strong convexity uniformly with respect to the mesh size h (Theorem 6.3).

Error analysis proceeds by inserting an intermediate control u_h^* and exploiting the uniform coercivity of both J″ and J_h″. The main result (Theorem 6.14) states that, for appropriately graded meshes, the control error satisfies ‖u−u_h‖{H^{1/2}(Γ)} ≤ C h^{α}, and the state error satisfies ‖y_u−y{u_h}‖_{L²(Ω)} ≤ C h^{α+½}, where α depends on the grading and equals 1 for optimal grading. These estimates are shown to be sharp.

Two numerical experiments are presented: (i) a non‑convex L‑shaped domain with variable coefficients, and (ii) a domain with mixed boundary conditions. In both cases, the observed convergence rates match the theoretical predictions, confirming the effectiveness of the H^{1/2}(Γ) projection and graded mesh strategy.

The paper concludes with a discussion of practical implementation aspects (assembly of the discrete Dirichlet‑to‑Neumann operator, handling of non‑homogeneous boundary data) and suggests future work on higher‑order elements, adaptive mesh refinement, and extension to nonlinear or time‑dependent control problems.