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

Remark 2.1.

The operators Q j : L 2 ⁢ ( Ω ) → W j and I j : C ⁢ ( Ω ¯ ) → V ~ j are projections that preserve signs, i.e.,

Remark 2.2.

The case of (2.15) for d = 1 is the consequence of the Sobolev embedding H 1 ⁢ ( Ω ) ↪ C ⁢ ( Ω ¯ ) (cf. [1]). The proof of the case d = 2 can be found for example in [5, Section 4.9]. The case of d = 3 follows from the Sobolev embedding H 1 ⁢ ( Ω ) ↪ L 6 ⁢ ( Ω ) (cf. [1]) and the inverse estimate (cf. [8, 5])

Lemma 2.3.

Let u ∈ L 2 ⁢ ( Ω ) and u j ∈ W j such that lim j → ∞ ⁡ ∥ u - u j ∥ L 2 ⁢ ( Ω ) = 0 . We have

In particular, the limits (2.16) and (2.17) are valid for u j = Q j ⁢ u .

Proof.

Note that (1.2), (2.2) and (2.12) imply that ( - Δ j ) - 1 ⁢ u j = R j ⁢ ( - Δ ) - 1 ⁢ u j . Since { u j } j = 1 ∞ is a bounded sequence in L 2 ⁢ ( Ω ) , the estimate

follows immediately from (1.7) and (2.13).

Similarly, since α > 1 2 , the estimate

is an immediate consequence of (1.7) and (2.14).

From (1.7) we have

which implies (2.16) because of (2.18).

Similarly, we have, by (1.7) and (1.8),

which together with (2.19) implies (2.17). ∎

Remark 2.4.

To facilitate the presentation, we will assume (by renumbering if necessary) that the discrete Slater condition (2.21) holds for j ≥ 1 .

Remark 2.5.

It follows from (2.24) that

for all ( y j , u j ) ∈ V j × W j that satisfies (2.2), which is a finite-dimensional analog of (1.10).

Remark 2.6.

The Lagrange multipliers λ j , 1 and λ j , 2 are not unique on an element T ∈ 𝒯 j where Q j , T ⁢ ϕ 1 = Q j , T ⁢ ϕ 2 . But their values can be assigned in a canonical way as follows.

First we note that the Lagrange multiplier p j ∈ V j is uniquely determined by the condition

that comes from letting u j = 0 in (2.24).

If we let y j = 0 in (2.24), then we have

which is equivalent to

It follows from (2.25), (2.26) and (2.29) that

for all u j ∈ W j that satisfy the constraints in (2.3).

We deduce from (2.6) (cf. [15, Theorem 2.3 in Chapter I]) that u ¯ j is the projection of - 1 γ ⁢ Q j ⁢ p j on the closed convex subset of W j defined by (2.3), with respect to the inner product of L 2 ⁢ ( Ω ) . Therefore we have

because the characteristic functions of distinct elements of 𝒯 j are orthogonal in L 2 ⁢ ( Ω ) .

It follows from (2.32) that, for any T ∈ 𝒯 j ,

Consequently, if we define the canonical λ j , 1 and λ j , 2 by

then (2.30) (and hence (2.24)) and the sign conditions for λ j , 1 and λ j , 2 in (2.25) are satisfied, and the complementarity condition for λ j , 1 and λ j , 2 in (2.26) follows from (2.33).

Lemma 3.1.

There exists a sequence of positive numbers ϵ j such that ϵ j → 0 as j → ∞ and

Proof.

It suffices to find ϵ j > 0 such that lim j → ∞ ⁡ ϵ j = 0 and ( - Δ j ) - 1 ⁢ [ ( 1 - ϵ j ) ⁢ Q j ⁢ u ¯ + ϵ j ⁢ Q j ⁢ u * ] satisfies the constraints in (2.4), because ( 1 - ϵ j ) ⁢ Q j ⁢ u ¯ + ϵ j ⁢ Q j ⁢ u * satisfies (2.3) for any ϵ j ∈ ( 0 , 1 ) .

Since y ¯ = ( - Δ ) - 1 ⁢ u ¯ , it follows from (2.17) that

In view of the constraints in (1.4) for y ¯ , Remark 2.1 and (2.6), we have

and hence

where δ j = ∥ ( - Δ j ) - 1 ⁢ Q j ⁢ u ¯ - y ¯ ∥ L ∞ ⁢ ( Ω ) . Similarly we have

Let β be the constant in (2.21) and ϵ j = δ j β . Then lim j → ∞ ⁡ ϵ j = 0 by (3.2) and according to (2.21), (3.3) and (3.4), for j sufficiently large so that 0 < ϵ j < 1 ,

Remark 3.2.

Let u j ∈ U ad , j and y j = ( - Δ j ) - 1 ⁢ u j . It follows from (3.8) that

where y ¯ j = ( - Δ j ) - 1 ⁢ u ¯ j . In view of (2.23) and (3.10), we have

Lemma 3.3.

Let u j ∈ W j for j ≥ 1 . If { u j } j = 1 ∞ is a bounded sequence in L 2 ⁢ ( Ω ) , then

If u ∈ L 2 ⁢ ( Ω ) and ∥ u - u j ∥ L 2 ⁢ ( Ω ) → 0 as j → ∞ , then