A Galerkin Approach to the Generalized Karush–Kuhn–Tucker Conditions for the Solution of an Elliptic Distributed Optimal Control Problem with Pointwise State and Control Constraints

A KKT Conditions for the Solution of the Discrete Problem

Let m j (resp. n j ) be the dimension of V j (resp., W j ). We can represent a function y j ∈ V j by the vector 𝒚 j ∈ ℝ m j such that 𝒚 j ⁢ ( k ) = y j ⁢ ( p k ) , where p 1 , … , p m j ∈ 𝒱 j are the interior vertices of 𝒯 j , and we can represent a function u j ∈ W j by the vector 𝒖 j ∈ ℝ n j such that 𝒖 j ⁢ ( k ) = u j | T k , where T 1 , … , T n j are the elements of 𝒯 j . Below the inequalities between vectors are to be understood in the componentwise sense.

The minimization problem (2.1)–(2.4) can be rewritten as

subject to the constraints

where ϕ j , ℓ ∈ ℝ n j ( ℓ = 1 , 2 ) is the vector representing Q j ⁢ ϕ ℓ , 𝝍 j , ℓ ∈ ℝ m j ( ℓ = 1 , 2 ) is the vector representing I j ⁢ ψ ℓ , the matrices 𝑨 j ∈ ℝ m j × m j , 𝑴 j ∈ ℝ m j × m j , 𝑵 j ∈ ℝ n j × n j and 𝑩 j ∈ ℝ m j × n j are defined by

and 𝒚 d , j ∈ ℝ m j is defined by

Let ( 𝒚 ¯ j , 𝒖 ¯ j ) ∈ ℝ m j × ℝ n j be the solution of (A.1)–(A.2). Since the constraints in (A.2) are affine constraints, we can apply a well-known constraint qualification (cf. [2, CQ3 in Section 5.4]) to conclude that there exist Lagrange multipliers 𝒑 j , 𝝁 j , 1 , 𝝁 j , 2 ∈ ℝ m j and 𝝀 j , 1 , 𝝀 j , 2 ∈ ℝ n j such that

for all ( 𝒚 j , 𝒖 j ) ∈ ℝ m j × ℝ n j ,

and for ℓ = 1 , 2 ,

Using (A.3)–(A.7), we can translate (A.8)–(A.10) into (2.24)–(2.26) by taking μ j , ℓ = ∑ p k ∈ 𝒱 j 𝝁 j , ℓ ⁢ ( k ) ⁢ δ p k , where δ p k is the Dirac point measure associated with the vertex p k , and by taking λ j , ℓ = 𝝀 j , ℓ ⁢ ( k ) / | T k | on the element T k ( 1 ≤ k ≤ n j ) of 𝒯 j .