A Function-Space Framework for BDDC Preconditioning in Control- and State-Constrained Sparse Optimal Control

1.1. Semi-smooth Newton methods and preconditioning.

• Each semi-smooth Newton iteration requires solving a large, sparse, symmetric but indefinite KKT system whose structure changes with the active sets.The paper identifies efficient solution of this linear subproblem as the focus of its preconditioning work.Source: §1.1 · p. 2

1.2. Domain decomposition preconditioners.

• The constrained active-set Schur complement is generally indefinite, preventing the positive-definite interface reduction used in some prior BDDC analyses.The paper therefore uses GMRES for the generally nonsymmetric left-preconditioned system; rigorous GMRES convergence theory remains open.Source: §1.2 · p. 3, §1.3 · p. 3

1.3. Contributions.

• The framework extends nonoverlapping domain decomposition to coupled KKT systems with control constraints, L1-sparsity, and Moreau–Yosida regularized state constraints.It establishes global–local equivalence, local well-posedness, and a two-level additive Schwarz form of BDDC under stated conditions.Source: §1.3 · p. 3
• Numerical experiments for distributed and Neumann boundary control show nearly mesh-independent GMRES iteration counts at fixed H/h and moderate growth with H/h; edge-average enrichment substantially reduces counts and their sensitivity to α and β.For the state-constrained problem, continuation with γ = h^2 yields nearly constant counts.Source: §1.3 · p. 3
• The paper also develops block-operator results on Fredholm index and its preservation under Schur complementation and finite-codimensional restrictions.The article is organized around the continuous problem, domain decomposition, BDDC construction, numerical experiments, and supporting Fredholm theory.Source: §1.3 · p. 3, §1.4 · p. 3

2.1. Function spaces and operators.

• The state and adjoint use the space V, while the control and multiplier use U = L2(U); the weak Laplacian, embedding, and control operator define the PDE blocks.The control operator differs by setting: distributed control uses B = M⋆, whereas Neumann control uses the trace operator.Source: §2.1 · pp. 3–4

2.2. First-order optimality conditions.

• The first-order conditions are expressed as a nonsmooth operator equation, with state- and control-constraint operators defined pointwise.The control complementarity function incorporates the control bounds and L1-sparsity parameter β.Source: §2.2 · p. 4

2.3. Semi-smoothness, active sets, and generalized Jacobians.

• For q > 2, the state and control complementarity operators are semi-smooth between the stated Lq and L2 spaces, supporting locally q-superlinear Newton convergence near a solution with nonsingular generalized Jacobian.The mesh-independence principle gives mesh-independent Newton iteration counts for suitable conforming finite-element discretizations.Source: §2.2 · p. 4, §2.3 · p. 5

2.3.1. Active sets.

• The generalized Jacobians are piecewise constant on partitions into active and inactive regions for the state and control constraints.The control partition consists of active lower-bound, upper-bound, and zero-control sets together with positive and negative inactive sets.Source: §2.3.1 · p. 5

2.3.2. Generalized Jacobians.

• Restriction and extension-by-zero operators express how generalized Jacobians act on the state-active and control-active or inactive sets.The composition of a restriction operator with its adjoint acts as multiplication by the corresponding characteristic function.Source: §2.3.2 · p. 5

2.4. Semi-smooth Newton linearization.

• The semi-smooth Newton step solves a linearized KKT system for the state, adjoint, control, and multiplier increments.The block operator need not be self-adjoint, but eliminating the inactive multiplier using the complementarity equation recasts the step as a self-adjoint indefinite system.Source: §2.4 · p. 6

2.5. Symmetrized system.

• The symmetrized system uses the active multiplier as its fourth unknown after eliminating the inactive multiplier.The global KKT operator is Fredholm of index zero; it is an isomorphism for distributed control, and for Neumann control when σ > 0 or the inactive control set has positive measure.Source: §2.5 · pp. 6–7
• The resulting KKT operator is self-adjoint but indefinite, and its interface formulation couples local subdomain problems through trace conformity and flux balance.The global–local equivalence and local interior well-posedness are established in the later domain-decomposition analysis.Source: §2.5 · p. 7, §3 · p. 8

3. Domain decomposition.

• The section fixes notation for products of Hilbert spaces, their duals, and block-diagonal operators for use in the analysis.It introduces coordinate projections and injections and the associated dual-space identification.Source: §3 · pp. 8–9
• A block-diagonal operator is an isomorphism exactly when each component operator is an isomorphism.The passage also gives the corresponding inverse construction when the component operators are isomorphisms.Source: §3 · p. 9
• The domain Ω is partitioned into mutually disjoint, open, connected subdomains, with Γ denoting their interface.Each subdomain interface portion is denoted Γ_i.Source: §3.1 · p. 9

3.1. Global interior–interface decomposition.

• Only the state and adjoint have interface traces; controls and active multipliers lack interface components because of their regularity.The interior space is the kernel of the trace map and contains state and adjoint functions with vanishing interface trace, together with the control and active multiplier.Source: §3.1 · p. 9
• A bounded lifting of the state–adjoint trace space yields a topological direct-sum decomposition into interior variables and interface traces.The lifting is not unique, but the induced decomposition is unique once a lifting is fixed.Source: §3.1 · pp. 9–10
• The block formulation provides the basis for eliminating interior variables and forming the interface operator.The decomposition and lifting-based equivalence connect the original system to its interface representation.Source: §3.1 · p. 10, §3.2 · p. 10
• With a fixed lifting, the symmetrized KKT system is equivalent to a block operator equation in the interior and interface variables.The global interior KKT operator is independent of the lifting, and the block operators inherit symmetry from the symmetrized KKT operator.Source: §3.1 · p. 10

3.2. Global Schur complement problem.

• Static condensation eliminates interior variables and reduces the global problem to a single equation for the interface trace, assuming the interior KKT operator is an isomorphism.The interior solution is recovered by backsubstitution.Source: §3.2 · pp. 10–11

3.2.1. The Schur complement operator.

• The global Schur complement and reduced right-hand side are independent of the choice of lifting.This permits suppressing the lifting superscript in the interface formulation.Source: §3.2.1 · p. 11

3.2.2. Generalized Harmonic extension.

• The generalized harmonic extension yields a decoupled block system with zero interior-interface coupling and interface block equal to the Schur complement.The extension is linear and bounded, and the solution is recovered by inverting the diagonal blocks.Source: §3.2.2 · pp. 12–13

3.3.1. Local functional setting.

• Anchored subdomains intersect the external boundary; floating subdomains do not.The local state space enforces zero values on the Dirichlet portion of the external boundary, and the local interior space additionally vanishes on the interface.Source: §3.3 · p. 13, §3.3.1 · p. 13

3.3.2. Local operators.

• Local operators and residuals are induced by restricting the global operators to each subdomain, and the global operators decompose additively over subdomains.Non-control subdomains have zero control operator.Source: §3.3.2 · p. 14

3.3.3. Local product spaces.

• The local trace operator maps state and adjoint components to their traces on the subdomain interface.Local product spaces pair these trace variables with the local state, adjoint, control, and active-control-multiplier variables.Source: §3.3.3 · p. 15
• The map identifying global interior variables with the product of local interior spaces is an isometric isomorphism.The product-space construction also supports gluing local components into global functions.Source: §3.3.3 · pp. 15–16

3.4. Local subdomain KKT problems.

• Global and local KKT solutions correspond when the local problems satisfy the interface trace data and their interface fluxes obey a coupling condition.Conversely, local solutions with the prescribed traces and balanced interface fluxes determine a unique global solution.Source: §3.4 · p. 17

3.4.1. Decomposition of the local KKT operator.

• The global interior KKT operator is congruent, through an isometric isomorphism, to the direct sum of the local interior KKT operators.The global interior operator is an isomorphism if and only if every local interior operator is an isomorphism.Source: §3.4.1 · pp. 18–19

3.4.2. Solvability of the local KKT problems.

• Local KKT operators are Fredholm with index zero, while their isomorphism conditions depend on the penalty parameter, anchoring, and inactive-control conditions.The conditions have distinct alternatives for distributed and Neumann boundary control.Source: §3.4.2 · p. 19

3.4.3. Local Schur complement problem.

• Each local Schur complement is Fredholm with index zero and is an isomorphism precisely when the corresponding local KKT operator is.The local Schur complement and reduced load do not depend on the choice of lifting.Source: §3.4.3 · pp. 20–21
• The global Schur complement and reduced load decompose into assembled local contributions, so applying the global operator requires independent local solves and global interface assembly.The local solves are embarrassingly parallel; interface assembly remains global.Source: §3.4.3 · pp. 21–22

4.1. Broken interface operators.

• The broken Schur complement is Fredholm of index zero, with kernel equal to the product of the local Schur-complement kernels.It is an isomorphism when every local kernel is trivial.Source: §4.1 · p. 23

4.2. Partially assembled interface operators.

• Partial assembly identifies selected bounded functionals of state and adjoint traces across incident subdomains, leaving other trace components independent.This restores interface coupling absent from the fully broken operator while retaining local parallel solves.Source: §4.2.1 · p. 23, §4.1 · p. 23

4.2.1. Primal constraints.

• A primal constraint family consists of consistent trace functionals shared by subdomains, with compatibility requiring adjoint constraints to be at least as rich as state constraints.The constraints may act on state or adjoint traces, or both.Source: §4.2.1 · pp. 23–24

4.2.2. Partially assembled Schur complement.

• For a compatible primal family, the partially assembled operator is an isomorphism exactly when its trace space intersects the broken operator’s kernel only at zero.This gives a kernel-based criterion alongside the sufficient graph condition.Source: §4.2.2 · p. 25
• A sufficient graph criterion for the partially assembled Schur complement to be an isomorphism is that every constraint-graph component contain a vertex in N0.Under a compatible primal family, the partially assembled operator is an isomorphism.Source: §4.2.2 · pp. 25–26

4.3. The BDDC preconditioner.

• BDDC application reduces to independent local dual-subspace solves and a finite-dimensional global coarse solve.The primal component carries inter-subdomain coupling, while the dual component is eliminated by local solves.Source: §4.3 · p. 27, §4.4 · p. 27

4.4. The primal-dual splitting.

• The dual subspace and its operator decompose into local factors, and the dual operator is an isomorphism exactly when every local block is.This separates the dual solve into independent subdomain problems.Source: §4.4 · p. 29

4.4.1. The coarse basis and block factorization.

• Eliminating the dual component by static condensation leaves a coarse Schur complement on the primal variables.Applying the coarse basis operator involves independent local dual solves followed by assembly of primal residuals; the coarse solve is global and finite-dimensional.Source: §4.4 · p. 29, §4.4.1 · p. 30

4.4.2. Well-posedness of the local dual subspace problems.

• The local dual-subspace operator is an isomorphism precisely when condition (4.35) holds for every subdomain.The combined dual operator is an isomorphism under the same condition on all subdomains.Source: §4.4.2 · p. 31

4.4.3. Two-level additive Schwarz form.

• When the dual and coarse operators are isomorphisms, BDDC has a two-level additive Schwarz form with N independent local dual solves and a global coarse solve of dimension at most 2NΠ.The numerical studies report average GMRES iterations per Newton step and assess mesh dependence, H/h growth, primal constraints, and parameter sensitivity.Source: §4.4.3 · p. 32, §5 · p. 32

5.1. Discretization and primal constraints.

• Experiments use uniform triangular meshes on the unit square and subdomains with H/h in {4, 8, 16}; state and adjoint use Vh, while controls and multipliers use Uh.The state-constrained study refines from ℓ=4 through ℓ=11 while maintaining H/h=16 and setting γ=h^2.Source: §5.1 · p. 32, §5.2.3 · p. 38

5.1.2. Symmetrized Newton system.

• The discrete symmetrized Newton system is assembled on the finite-element spaces using active-set index sets and has a symmetric matrix Kh.Active state nodes and control cells define the discrete active sets.Source: §5.1.2 · p. 33

5.1.3. Subdomain restrictions and interface unknowns.

• Controls and multipliers have no interface degrees of freedom, so the state and adjoint traces form the interface unknowns and each Newton step reduces to a Schur-complement system solved by preconditioned GMRES.Local interface Schur complements are formed from local interior factorizations.Source: §5.1.3 · p. 34, §5.1.4 · p. 34

5.1.4. The BDDC preconditioner.

• The BDDC coarse constraints enforce continuity of corner values and, in the enriched variant, edge averages; static condensation yields independent local dual solves and a coarse problem.The coarse problem has dimension at most 2NΠ.Source: §5.1.4 · pp. 34–35

5.2. Numerical experiments.

• At fixed H/h, counts are nearly constant under mesh refinement; edge-average constraints reduce counts by factors of 2.8–4.1 in Experiment 1 and 2.4–3.3 in Experiment 2.For Experiment 3, counts stabilize along the tested continuation path, with corner-and-edge BDDC remaining below 10 iterations.Source: §5.2 · p. 36

5.2.1. Experiment 1: Control-constrained distributed control.

• In distributed control, increasing H/h from 4 to 16 approximately doubles counts across methods, while corner-and-edge BDDC has at most 2.8 iterations of variation as α decreases from 10−4 to 10−8.Its counts are reduced by factors of 2.8–4.1 relative to BNN and corner-only BDDC.Source: §5.2.1 · p. 36

5.2.2. Experiment 2: Control-constrained Neumann control.

• In Neumann control, corner-and-edge BDDC has counts ranging from 10.0–11.4 at H/h=4 to 13.4–15.6 at H/h=16, and its sensitivity to α and β is weaker than corner-only BDDC.Adding edge-average constraints reduces counts pointwise by factors of 2.4–3.3.Source: §5.2.2 · p. 38

5.2.3. Experiment 3: State-constrained distributed control.

• Along the state-constrained continuation γ=h^2, counts become essentially independent of further refinement after an initial increase on coarser meshes.From ℓ=7 onward, BNN and corner-only BDDC remain near 29–30 iterations, while corner-and-edge BDDC stays between 8 and 9; h and γ vary together.Source: §5.2.3 · p. 38

6. Conclusion.

• The analysis characterizes kernels of local and broken Schur complements and gives conditions for well-posed partially assembled and local dual-subspace problems.The framework establishes global–local equivalence and assembly of the interface operator.Source: §6 · p. 39
• Across distributed- and Neumann-control experiments, iteration counts are nearly independent of mesh refinement and subdomain count at fixed H/h, and grow only moderately as H/h increases.Edge-average constraints lower counts and weaken sensitivity to α and β.Source: §6 · p. 39
• For the state-constrained problem, γ=h^2 yields nearly constant iteration counts along the tested continuation path.The mesh size and penalty parameter vary simultaneously, so the result concerns the coupled path.Source: Figure 1 · p. 40
• A rigorous convergence theory for GMRES on these indefinite preconditioned interface systems remains open.The conclusion also lists parallel scalability, parameter dependence, adaptive primal constraints, multilevel BDDC, and more complex models as further directions.Source: §6 · p. 40

• The appendix records results on assembled homogeneous spaces, block-diagonal representations, Fredholm theory, saddle-point operators, Schur complements, and finite-codimensional restrictions.Source: Appendix A · p. 42
• When selected local trace quantities must agree across interfaces, local homogeneous components satisfy those compatibility conditions automatically.Source: Appendix A · p. 42
• Restriction therefore identifies the global homogeneous space with the product of the corresponding local kernels.Source: Appendix A · p. 42

A.1. Homogeneous decompositions of assembled spaces.

• Under the stated isometry and gluing assumptions, restriction maps the global homogeneous space bijectively onto the assembled product of local kernels.The resulting corestriction is an isometric isomorphism.Source: §A.1 · p. 43
• The proposition defines induced maps from each local kernel into the global homogeneous space using the inverse corestriction and the local injections.Source: §A.1 · p. 43
• The proof uses the gluing condition to show both inclusions: global homogeneous elements restrict to local kernels, and local kernel components assemble into a global element.Source: §A.1 · p. 43

A.2. Fredholm theory of block operators.

• Under its stated operator assumptions, the block-operator result establishes Fredholmness with index zero.The proof uses Fredholmness of the diagonal blocks and stability under compact perturbations.Source: §A.2 · pp. 44–45
• When A is an isomorphism, the block operator is Fredholm exactly when its Schur complement is, with the same index; it is an isomorphism exactly when the Schur complement is.The result also identifies their kernel dimensions.Source: §A.2 · p. 45
• Restricting an index-zero Fredholm operator to a closed finite-codimensional subspace preserves Fredholmness and index zero.The proof decomposes the space into the subspace and a finite-dimensional complement, then uses a finite-rank perturbation.Source: §A.2 · p. 46