Sparse optimal control for the heat equation with mixed control-state constraints

A problem of sparse optimal control for the heat equation is considered, where pointwise bounds on the control and mixed pointwise control-state constraints are given. A standard quadratic tracking type functional is to be minimized that includes a Tikhonov regularization term and the \begin{document}$ L^1 $\end{document} -norm of the control accounting for the sparsity. Special emphasis is laid on existence and regularity of Lagrange multipliers for the mixed control-state constraints. To this aim, a duality theorem for linear programming problems in Hilbert spaces is proved and applied to the given optimal control problem.


(Communicated by Enrique Zuazua)
Dedicated to Prof. Dr. Frédéric Bonnans on the occasion of his 60th birthday Abstract. A problem of sparse optimal control for the heat equation is considered, where pointwise bounds on the control and mixed pointwise control-state constraints are given. A standard quadratic tracking type functional is to be minimized that includes a Tikhonov regularization term and the L 1 -norm of the control accounting for the sparsity. Special emphasis is laid on existence and regularity of Lagrange multipliers for the mixed control-state constraints.
To this aim, a duality theorem for linear programming problems in Hilbert spaces is proved and applied to the given optimal control problem.
Moreover, we fix real constants ν > 0 (Tikhonov parameter), κ > 0 (sparse parameter) and real bounds u a < 0, u b > 0, u d > 0. The bounds might be functions as well, but we keep them constant for simplicity. We will consider in parallel the case u b = ∞, where the upper bound for the control is missing.
We assume homogeneous initial data to simplify the presentation. The extension to a non-homogeneous initial condition requires only some obvious modifications. The weak solution y ∈ W (0, T ) := {y ∈ L 2 (0, T ; H 1 (Ω)) : ∂ t y ∈ L 2 (0, T ; H 1 (Ω) )} of (1.2) is the state of our control system, while the function u ∈ L 2 (Q) is the control.
The main novelty of our paper is the discussion of sparse optimal controls for a problem that includes pointwise mixed control-state constraints. While the issue of sparsity has been discussed quite extensively for problems with pointwise control constraints, the extension to mixed control-state constraints seems to be new. It requires special emphasis on existence and regularity of Lagrange multipliers associated with the mixed control-state constraints. To the best knowledge of the authors, this case has not yet been discussed in literature. The associated main result is Theorem 3.8 on sparsity of optimal controls.
To prove this theorem, we resolve two main difficulties: First, we show the existence of a Lagrange multiplier for inequality (1.4) that belongs to L ∞ (Q). We cannot rely on a Slater type assumption since this would eventually lead to a Lagrange multiplier in L ∞ (Q) * ; notice that the cone of non-negative measurable functions has a non-empty interior only in L ∞ (Q). As a key tool for overcoming this obstacle, we apply the duality theory for linear programming problems in Banach spaces, cf. Krabs [6] and Tröltzsch [11]. Since the papers [6,11] are published in German, the associated duality theorem is recalled and improved in the Appendix. The proof of the duality theorem is presented for Hilbert spaces.
Another difficulty is to show the boundedness of the Lagrange multiplier, uniformly with respect to the sparse parameter κ. This property is stated in Theorem 4.7. The result on existence and uniform boundedness of the Lagrange multiplier associated with the mixed control-state constraint (1.4) has some auxiliary character. Nevertheless, it is a second main achievement of our paper that is the core of Theorem 3.4.
The technique of our paper can be extended to some other types of mixed controlstate constraints, where the existence of bounded Lagrange multiplier functions can be shown by our duality method. For instance, the two-sided constraints u a ≤ u ≤ u d + y or u d + y ≤ u ≤ u b can be handled completely analogous by obvious modifications. In the latter case, the new control v = −u can be introduced to transform the lower mixed control-state constraint to an upper one.
For the two-sided pure mixed constraints u c + y ≤ u ≤ u d + y, the existence of bounded Lagrange multipliers can be proved by introducing the function v := u − y as a new control. Then these constraints are transformed to pure control constraints, where associated Lagrange multipliers can be easily constructed. Sparsity results can then be discussed after re-transforming the problem to u = v + y. The same holds true, if an additional bound on the control u is added. Then we have one control constraint and two-sided mixed control-state constraints. Here, the transformation v = u − y leads to two-sided box constraints for v and one mixed control-state constraint. Then the existence of bounded Lagrange multipliers can be shown again as in this paper. The discussion of sparsity will need some modifications.
However, our method does not work for the full set of four-sided constraints Here, we were not able to prove the existence of associated Lagrange multipliers by duality. This difficulty is known since long time. There is a detour via a Slater assumption in L ∞ (Q), Lagrange multipliers in L ∞ (Q) * , and certain structural assumptions on active sets as in [10]. Unfortunately, the structural assumption cannot in general be verified in advance and has to be required.

2.
Well-posedness of the optimal control problem.
2.1. The reduced optimization problem. It is well known that the state equation (1.2) is uniquely solvable: For each control u ∈ L 2 (Q) there exists a unique weak solution y ∈ W (0, T ) that we denote by y u . The mapping u → y u is linear and continuous, cf. Lions [8]. We consider this mapping with range in L 2 (Q) and denote it by S, i.e. S : L 2 (Q) → L 2 (Q), Su := y u . Throughout the paper, for functions u ∈ L 2 (Q), we write u ≥ 0 if and only if u(x, t) ≥ 0 holds for almost all (x, t) ∈ Q.
In this way, the optimal control problem (1.1)-(1.4) can be re-formulated as where the feasible set C ⊂ L 2 (Q) is defined by The next results are basic for our theory and will be frequently used in this paper.
Lemma 2.1. If u ∈ L 2 (Q) is almost everywhere non-negative, then also Su has this property, i.e.
Proof. This is a well-known conclusion from the comparison principle for the linear heat equation with homogeneous initial and boundary data: If u ≥ 0 holds, then the solution y of (1.2) is a.e. nonnegative, too. This implies the claim. The comparison principle can be proven by testing the variational formulation of the heat equation with the positive part y + of the weak solution y.
Lemma 2.2 (Inverse non-negativity). For all g ∈ L 2 (Q), the equation has a unique solution u ∈ L 2 (Q). If g ≥ 0 holds, then also u ≥ 0 is satisfied. Further, if g ∈ L ∞ (Q), then u belongs to L ∞ (Q) as well.
Proof. The equation u − Su = g is equivalent to the statement u − y = g, where y is the solution of (1.2). Subtracting y from both sides of the PDE, we find First of all, given g ∈ L 2 (Q), this equation has a unique weak solution y. From g ≥ 0 and the comparison principle for parabolic equations, we obtain y ≥ 0.
Adding now y to both sides, the differential equation reads ∂ t y − ∆y = g + y.
Setting u = g + y, we have y = Su and u − Su = g, hence the existence of a solution u is shown. The uniqueness follows from the fact that y = 0 is the unique solution to g = 0. Now we see u = y + g ≥ 0, since both g and y are nonnegative. Finally, if g ∈ L ∞ (Q), then we get from [7, §III.7] that y ∈ L ∞ (Q) holds as well and, hence, The next simple result will be applied several times in our paper.
Proof. We have u = g − e + Su with some a.e. nonnegative e ∈ L 2 (Q). Subtracting this from the equation for v yields v − u = e + S(v − u).
Next, we prove the existence of an optimal solution to (P).
Proof. In view of the assumptions u a < 0, u b > 0, and u d > 0, the set C is not empty, since u = 0 satisfies all constraints. If both u a and u b are real numbers, then the result is obvious, since the set {u ∈ L 2 (Q) : u a ≤ u ≤ u b } is weakly compact. The additional pointwise control-state constraint does not change this, i.e. C is weakly compact. Notice that C is convex and closed, hence also weakly closed. Uniqueness follows by strict convexity of the functional f . Let us therefore concentrate on the case u b = ∞, where the set of admissible controls is defined by the constraints is the solution to the equation v = u d + Sv. Therefore, the feasible set C is bounded also in this case. Again, we can invoke weak compactness to get the existence of an optimal solutionū. Remark 2.5. An inspection of the proof shows that the set C is bounded in L ∞ (Q).
3. Optimality conditions and sparsity of optimal controls.

A variational inequality.
In this section, we derive first order optimality conditions for the optimal controlū of (P). The main result is the "two-phase minimum principle" of Theorem 3.6.
We define the following particular functionals I : L 2 (Q) → R and j : L 1 (Q) → R that are included in the reduced objective functional f of (P): In this way, we have f = I + κ j, hence (P) can be re-written as The functional f is the sum of the convex differentiable functional I and the convex directionally differentiable functional κ j.
In all what follows, by (·, ·) H the scalar product of a real Hilbert space H is denoted.
Since C is convex, we obtain the following necessary and sufficient optimality condition for the optimal solutionū: Lemma 3.1. A controlū ∈ C is optimal for (P) if and only if an elementλ of the subdifferential ∂j(ū) exists such that the variational inequality The proof is standard; notice that we have I (ū) = S * (Sū − y Q ) + νū. Moreover, we mention that ∂j(u) ⊂ L ∞ (Q) holds for all u ∈ L 1 (Q) and that λ ∈ ∂j(u) if and only if holds for a.a. (x, t) ∈ Q. It is well known that the Hilbert space adjoint operator S * : L 2 (Q) → L 2 (Q) can be expressed in terms of an adjoint state. We have S * (Sū − y Q ) =φ, where the adjoint stateφ is the unique solution of the adjoint equation in Ω.

(3.2)
By the adjoint stateφ, the variational inequality (3.1) can be re-formulated as In other words,ū solves the linear optimization problem where the interior of the cone of nonnegative functions is empty. Therefore, it is not an easy task to show the existence of associated Lagrange multipliers. In the case of mixed control state constraints, there are special techniques to overcome this difficulty, we refer exemplarily to Rösch and Tröltzsch [10] and the references therein.
Here, we will employ the duality theory of continuous linear programming problems to establishing the next result. A similar technique was published in [12], but the proof was not presented in all details.
The further analysis depends on the following assumption.
This assumption is quite natural. In the context of optimal heating of Ω by a heat source u, the state y stands for the temperature. The mixed control state constraint excludes a too sudden heating, since the difference u − y cannot exceed the bound u d . In this sense, Assumption 3.2 means that the minimal heat source is not too large related to the associated temperature bounds.
For discussing the sparsity of the optimal control, we need the following convergence result: Lemma 3.3. Let, for κ ≥ 0,ū κ denote the optimal control of (P) corresponding to the sparse parameter κ ≥ 0 and letȳ κ = Sū κ be the associated state. Then we have Proof. The control u = 0 is feasible for (P), hence This immediately yields Moreover, we know from Remark 2.5 that the set of feasible controls is bounded in L ∞ (Q). Therefore, the L 1 -convergence above implies also Selecting an arbitrary p > N 2 + 1, the continuity of the mapping u → y u from L p (Q) to C(Q), [7,, immediately implies that lim κ→∞ ȳ κ C(Q) = 0.
In all what follows, to simplify the presentation, we suppress the dependence of the optimal quantities on κ.
The next theorem is basic for the investigation of sparsity properties of the optimal control. Its proof relies on results of the duality theory of linear continuous programming problems that is presented later in the Section 4. However, we formulate the theorem right here in order to come faster to the theory of sparsity. The results of Section 4 do not rely on Theorem 3.4.
(ii) There exist a constant M > 0 not depending on κ and a pair of multipliers (μ 1 ,μ 2 ) satisfying all conditions above such that Proof. Part (i): The existence of Lagrange multipliers (μ 1 ,μ 2 ) ∈ (L ∞ (Q)) 2 is ensured by Theorems 4.5 and 4.6 that are proven in Section 4 by duality theory. Once this existence is shown, the variational inequality (3.5) and the complementarity conditions (3.6) express well-known properties of Lagrange multipliers. To see this, associated with the linear programming problem (3.4), we define the Lagrangian function while (3.6) contains the associated complementarity conditions. Part (ii) on uniform boundedness of (μ 1 ,μ 2 ) is nothing more than statement (ii) of Theorem 4.7.
Remark 3.5. In Theorem 3.4, we introduced Lagrange multipliers only for the two upper constraints, while the lower bound u ≥ u a was not "eliminated" by a multiplier. We need the lower bound in the pointwise minimum principle below, hence it is not useful to eliminate it. Nevertheless, we might define a Lagrange multiplierμ 3 ≥ 0 byμ A simple discussion of the variational inequality (3.5) reveals that this functionμ 3 is nonnegative, indeed. Moreover, let us introduce the Lagrangian functioñ that is associated with the variational inequality (3.5) and the lower bound constraint, eliminated by µ 3 . It is easy to confirm that the derivative ∂L/∂u vanishes at (ū,μ 3 ). This is a direct consequence of the definition ofμ 3 .
In addition the complementarity condition is satisfied. Therefore,μ 3 defined above is a Lagrange multiplier associated with the lower bound constraint.
The Lagrange multiplier rule of Theorem 3.4 is not yet easily applicable to the discussion of sparsity. To facilitate this discussion, we perform an intermediate step and derive a pointwise minimum principle as a useful tool for proving sparsity properties of the optimal controlū. This type of minimum principles was introduced in the framework of continuous linear programming by Grinold [4]. It was later invoked to proving a generalized bang-bang principle in [12].
Theorem 3.6 (Pointwise minimum principle). Let Assumption 3.2 be satisfied. If u is an optimal control and (μ 1 ,μ 2 ) ∈ (L 2 (Q)) 2 is an associated pair of Lagrange multipliers that exists according to Theorem 3.4, then for almost all (x, t) ∈ Q the solution u of the problem Proof. For convenience, we define The minimum is attained at any value on the set, where e(x, t) = 0.
Assume that the result of the theorem is not true. Then one of the following two measurable sets E 1 , E 2 must have positive measure, In the points of E 1 , we haveū(x, t) > u a . Here, the variational inequality (3.5) can only hold, if In a.a. points of E 2 , the functionsμ 1 andμ 2 vanish, becauseμ 1 is a Lagrange multiplier for the inequality u ≤ u b ,μ 2 is a multiplier for the inequality u ≤ u d +Su, and both inequalities are inactive in E 2 atū; notice thatȳ = Sū.
Here, the variational inequality (3.5) can only be true, if holds a.e. in Q. Since a.e. in E 2 both multipliers vanish, we get Therefore, also E 2 cannot have positive measure. This completes the proof.
Remark 3.7. The constraints (3.8) have the form u a ≤ u ≤ min{u b , u d +ȳ} with fixed functionȳ, hence in this context they are pointwise control constraints. This is characteristic for a so-called two-phase maximum principle introduced by Grinold [4] for continuous linear programming problems, here formulated as minimum principle.
(iii) The elementλ of the subdifferential ∂j(ū) is given bȳ 14) where the projection P [−1,1] : R → [−1, 1] is defined by Proof. The main ideas are inspired by the proof of sparsity for pointwise control constraints in [2]. However, some changes are needed to tackle mixed control-state constraints.
(ii) According to Remark 2.5, the sets of all feasible controls u and associated states y u are bounded in L ∞ (Q). The same follows for the associated adjoint states ϕ solving equation (3.2). The adjoint stateψ =φ − S * μ 2 depends onμ 2 . However, by Theorem 3.4, part (ii), we can assume the Lagrange multiplierμ 2 to be bounded in L ∞ (Q) by some M > 0, independently of κ. Therefore, we can assume ψ L ∞ (Q) ≤ M 1 with some constant M 1 > 0 not depending on κ. Notice that the assumption q > N/2 + 1 is invoked for this property. For all κ ≥ κ 0 = max{M 1 , κ 1 }, the relation (3.13) yieldsū = 0.
In E 0 , all inequalities are inactive, henceψ(x, t) +ū(x, t) + κλ(x, t) = 0 follows from (3.11). Again, the projection formula (3.14) is an immediate conclusion. Remark 3.9. In view of Lemma 3.3 and the fact that u d > 0, the optimal stateȳ satisfies u d +ȳ > 0 for all sufficiently large κ. For such values of κ, all inequalities of problem (P) are inactive in the points (x, t) whereū(x, t) = 0. Hence, for large κ, we were able to prove (3.13) in the same way as for pure pointwise control constraints, cf. [2]. where a(x, t) = −(φ + νū + κλ)(x, t). By the definition of C, this problem is posed in the Hilbert space L 2 (Q).
Problem (4.1) does not yet have the standard form of continuous linear programming that we prefer. Instead of the lower bound u ≥ u a we want to have a nonnegativity restriction. To this aim, we introduce a new (shifted) control v = u − u a and transform problem (4.1) accordingly. Thenū is optimal for (P) if and only if v =ū − u a is optimal for the problem where v b := u b − u a > 0 and the function b is defined by If Assumption 3.2 is satisfied, then b(x, t) ≥ 0 is fulfilled for a.a. (x, t) ∈ Q. We consider this problem in L 2 (Q), i.e. we assume v ∈ L 2 (Q) and also the constraints are viewed in L 2 (Q).
Problem (PP) is our primal problem. We know thatv =ū−u a solves this problem and want to prove the existence of bounded and measurable Lagrange multipliers associated withv. For this purpose, we invoke the theory of linear programming problems in function spaces.
(PP) has the structure of linear continuous programming problems that were discussed extensively in the 1970ties, cf. Grinold [3,4] and Tyndall [13], related to so-called bottleneck control problems for ordinary differential equations. For an extension to the control of partial differential equations, we refer to Tröltzsch [12]. The associated dual problem, also considered in L 2 (Q), can be established by Lagrange duality. For u b < ∞, it is the following: to be fulfilled a.e. in Q. (DP) If u b = ∞, then the dual problem admits a simpler form, namely a.e. in Q.

( DP)
In what follows, we concentrate on the case u b < ∞. As a rule of thumb, the theory for the case u b = ∞ remains true with µ 1 = 0 and µ 2 =: µ. Let us state some important properties of this pair of dual linear problems. To shorten the arguments, functions v and µ = (µ 1 , µ 2 ) are said to be feasible for (PP) or (DP), respectively, if v satisfies all restrictions of (DP) and µ the ones of (DP).
Proof. The statement follows from Lemma 5.3 of the Appendix that we prove for more general pairs of dual linear programming problems posed in Hilbert spaces.

4.2.
Strong duality. Next, we discuss the solvability of the dual problem. The dual feasible set is never empty: For (DP), the pair (µ 1 , µ 2 ) = (|a|, 0) satisfies all constraints. If u b = ∞, then the dual problem is ( DP). In this case, the solution µ of the equation µ = |a| + S * µ is feasible by inverse non-negativity.
Proof. Let E ⊂ Q be the set of all points with If the Lebesgue measure |E| of E is zero, then we can easily modify µ 1 , µ 2 on E such that (4.3) is satisfied everywhere in Q and (4.4) holds with equality. Assume therefore |E| > 0. We construct a componentwise and pointwise monotone decreasing sequence of feasible pairs (µ 1,k , µ 2,k ) ∈ (L 2 (Q)) 2 converging to a pair ( µ 1 , µ 2 ) having the desired properties. To this aim, we define the first element of the sequence by µ i,1 := µ i , i = 1, 2, The next feasible pair (µ 1,k+1 , µ 2,k+1 ) with not greater objective functional value is constructed as follows: For i = 1, 2 we set Notice that 0 < α(x, t) ≤ 1. We have that With S, also the operator S * is nonnegative. This follows from The last inequalities and S * ≥ 0 ensure hence (µ i,k+1 , µ 2,k+1 ) is feasible as well. Next, we set k := k + 1 and determine the next iterate. In this way, pointwise monotone decreasing sequences of functions (µ i,k ) ∞ k=1 , i = 1, 2, are constructed. All pairs (µ 1,k , µ 2,k ) are feasible for (DP). The sequences are bounded by 0 ≤ µ i,k ≤ µ i , i = 1, 2, and pointwise convergent to measurable limit functions µ i , i = 1, 2. In the limit, the equation (4.3) must be satisfied. This follows from passing to the limit k → ∞ in (4.6).
By Assumption 3.2, we have that b = u d + Su a − u a ≥ 0 and v b = u b − u a ≥ 0. Therefore, the smaller the multipliers are, the smaller is the associated objective value of (DP), hence (4.4) must hold.
Later, we shall prove that this boundedness is even uniform with respect to the sparse parameter κ.
Proof. By Lemma 4.2, the search for an optimal solution of (DP) can be restricted to the set of all feasible pairs (µ 1 , µ 2 ) that solve equation (4.3). Thanks to Lemma 4.4, this set is bounded in (L ∞ (Q)) 2 , hence also in (L 2 (Q)) 2 . Therefore, the search for the minimum can be restricted to the intersection of the feasible set of (DP) with a sufficiently large closed ball of (L 2 (Q)) 2 . This is a non-empty, convex and closed set, hence weakly compact. Now the claim follows immediately. Theorem 4.6 (Strong duality). Let Assumption 3.2 be fulfilled, letv be optimal for (PP) and (μ 1 ,μ 2 ) ∈ (L 2 (Q)) 2 be optimal for (DP). Then We prove this theorem in the Appendix. Thanks to the theorem, the optimal solutionv of the primal problem (PP) is a Lagrange multiplier for the inequality µ 1 + µ 2 ≥ a + S * µ 2 of (DP) while any pair of solutions (μ 1 ,μ 2 ) of (DP) is a pair of Lagrange multipliers for the upper restrictions of (PP), i.e.μ 1 is a multiplier for the inequality v ≤ u b − u a whileμ 2 is one for the mixed control-state constraint v − Sv ≤ b. Therefore, the following complementarity conditions are satisfied byv and (μ 1 ,  (i) There is some constant κ 0 > 0 with the following property: For all κ ≥ κ 0 , we find an optimal solution (μ 1 ,μ 2 ) of (DP) satisfying the inequalities t) a.e. in Q, i = 1, 2. (4.10) (ii) A constant M > 0 not depending on κ exists, such that is satisfied for at least one optimal solution (μ 1 ,μ 2 ) of (DP).
Here both upper inequalities of (PP) are inactive, if κ is large enough. Let us show this: Since u b > 0 andū(x, t) ≤ 0, the upper control constraintū ≤ u b cannot be active in this case. Moreover, we know from Lemma 3.3 that ȳ L ∞ (Q) < u d holds for all suffiently large κ, say κ ≥ κ 0 . Then we have u d +ȳ = u d + Sū > 0 a.e. in Q and the inequalityū ≤ u d + Sū is also inactive.
Part (ii): For all κ ≤ κ 0 , we have for at least one solution (μ 1 ,μ 2 ) of (DP) that , because the set of all feasible controls u and hence also the set of all possible adjoint states ϕ are uniformly bounded in L ∞ (Q) with respect to κ, cf. Remark 2.5 and the proof of Lemma 4.4. Here, again the assumption q > N/2 + 1 enters. Now we invoke the inequality (4.8) and get µ 2 ≤ c + S * μ 2 . Since I − S * is inverse non-negative, we obtain the estimatē The estimate forμ 1 is now obtained from (4.7) by Part (ii) on uniform boundedness ofμ 2 is nothing more than statement (ii) of Theorem 4.7. .

5.1.
Duality theorem in Hilbert spaces. The result of this section will be proved for general pairs of dual linear programming problems that are not necessarily related to the specific pair of dual problems defined in Section 4. However, this general result will be applied to them. Let {U, · U } and {V, · V } be real Hilbert spaces with inner products (· , ·) U and (· , ·) V , respectively, A : U → V a linear and continuous operator, and let K U ⊂ U and K V ⊂ V be nonempty, convex and closed cones.
In U and V we define partial orderings ≥ U and ≥ V by u ≥ U 0 iff u ∈ K U and v ≥ V 0 iff v ∈ K V . The converse inequalities ≤ U and ≤ V are defined accordingly, e.g. u ≤ U 0 iff −u ≥ U 0. Moreover, we fix elements a ∈ U and c ∈ V .
We consider the primal linear programming problem max (a , u) U subject to Au ≤ V c, u ≥ U 0.
By the Riesz theorem, we identify the dual spaces U * of U and V * of V with U and V , respectively. To the cones K U and K V , we associate dual cones K * U and K * U by K * U = {w ∈ U : (w , u) U ≥ 0 ∀u ∈ K U }.
The dual cone K * V ⊂ V is defined analogously. Remark 5.1. If U = L 2 (Q) and K U is the cone of a.e. nonnegative functions of U , then we have K * U = K U . This is the case of interest for the application to our control problem (P), where K U = K V = L 2 (Q) + .
The partial orderings induced in U and V by K * U and K * V are denoted by ≥ * The dual problem is defined by If an element u ∈ U (resp. v ∈ V ) obeys the constraints of (PP) (resp. (DP)), then it is called feasible for the associated problem.
Remark 5.2. The pair of linear dual problems (PP) and (DP) fits in the duality theory of convex optimization, a well elaborated part of nonlinear programming. Associated strong duality theorems are often based on a Slater condition as in Luenberger [9] or on the assumption of calmness that is discussed in Bonnans and Shapiro [1], Sect. 1.5. A Slater condition cannot be satisfied in the space selected for our given optimal control problem and calmness is difficult to verify. Therefore, we invoke another method. We prove a duality theorem that relies on a certain boundedness condition. It turns out that this condition is fulfilled in our application, if Assumption 3.2 is satisfied. Proof. We obtain where we employed the inequalities u ≥ U 0, a ≤ * U A * v to get the first inequality and v ≥ * V 0, Au ≤ V c for the second one. Let us define for (varying) d ∈ V and e ∈ U the sets Then P (c) is the primal feasible set and D(a) is the dual one. As a consequence of the last lemma, the relation of weak duality is fulfilled. In this section, we show that the associated equality holds, if the boundedness condition below is satisfied for the primal problem.
Assumption 5.4 (Boundedness condition). There exist η > 0 independent of d and, for all d ∈ V, a closed set K(d) ⊂ P (d) such that the following two conditions are satisfied: Theorem 5.5 (Strong duality). If the feasible set P (c) of the primal problem (PP) is nonempty and Assumption 5.4 is satisfied, then the primal problem has an optimal solution. Moreover, the strong duality relation is fulfilled.
Proof. (i) Solvability of the primal problem. Let s = sup u∈P (c) (a , u) U be the primal supremum. Since P (c) is non-empty, we have s > −∞. Thanks to the boundedness condition, the search for a maximum in the primal problem can be restricted to the bounded set K(c). Taking a sequence (u n ) n of feasible elements for (PP) with lim n→∞ (a , u n ) U = s, we can assume without limitation of generality u n ∈ K(c). By the boundedness of K(c), a weakly convergent subsequence can be selected, w.l.o.g. (u n ) n itself. In this way, we can assume u n ū, n → ∞. It is easy to verify thatū is a feasible solution, hence it attains the primal supremum and is optimal.
(ii) A convex closed cone. We define the set It is easy to verify that K is a convex cone. Moreover, the boundedness condition implies that K is closed: To this end, take a sequence of elements (d n , δ n ) n of elements of K converging to (d, δ) as n → ∞. Then we find u n ∈ U such that Au n ≤ V d n , u n ≥ U 0, (a , u n ) U ≥ δ n ∀n ∈ N.
By the boundedness condition, there exists an η > 0 and u n ∈ U such that A u n ≤ V d n , u n ≥ U 0, (a , u n ) U ≥ (a , u n ) U ≥ δ n ∀n ∈ N and u n U ≤ η d n V ∀n ∈ N hold. By the last inequality, the sequence ( u n ) n is bounded in U , hence we can select a weakly convergence subsequence, w.l.o.g. u n u, n → ∞. Since the cones K U and K V are convex and closed, they are also weakly closed. This permits to show in turn For all n ∈ N, we have (c, s + 1/n) / ∈ K. Since K is closed, there are open balls B rn ⊂ V × R of radius r n > 0 centered at (c, s + 1/n) such that K ∩ B rn = ∅ ∀n ∈ N. Define for all n ∈ N C n = λ>0 λ B rn .
These sets have the following properties: -Obviously, all C n are cones.
-All intersections C n ∩ K are empty: If C n ∩ K would contain an element u, then there are λ > 0 andũ ∈ B rn such that u = λũ. Since K is a cone, also u/λ =ũ belongs to K and also to B rn contradicting K ∩ B rn = ∅.
-C n is convex as it can be readily verified. Moreover, B rn is contained in C n , hence the interior of C n is not empty. Therefore, all C n are convex cones with nonempty interior.
(iv) Separation of K and C n . By the Eidelheit separation theorem, see [5], there is a closed hyperplane separating the convex closed cone K and the convex set C n with nonempty interior. Therefore, z n ∈ V , ξ n ∈ R, and σ n ∈ R exist such that (d , z n ) V + δ ξ n ≤ σ n < (w , z n ) V + α ξ n ∀(d, δ) ∈ K, ∀(w, α) ∈ C n . (5.2) From this inequality, we draw several conclusions to finally arrive at strong duality.
-Define v n = −z n /ξ n , then A * v n ≥ * U a follows from (5.3). Moreover, z n ≤ * V 0 and ξ n ≥ 0 yield v n ≥ * V 0, hence v n is feasible for the dual problem (DP). -Finally, dividing the right-hand side of (5.2) by ξ n , we get 0 < (w , z n /ξ n ) V + α.
Inserting the elements w = c and α = s + 1/n, we arrive in view of v n = −z n /ξ n at (c , v n ) V < s + 1/n ∀n ∈ N.
Passing to the limit n → ∞, we see that the dual objective value (c , v n ) V can be taken arbitrarily close to the primal maximum s. In view of the weak duality result of Lemma 5.3, this shows (5.1).

5.2.
Application to the optimal control problem (P). Finally, we prove Theorem 4.6. We consider the primal and dual problem defined in Section 4. To show Theorem 4.6 on strong duality, we can take two ways: The first is to view the dual problem (DP) as primal one and to verify the boundedness assumption 5.4 for (DP). This can be done invoking Lemma 4.2. The other way around is to verify the boundedness assumption for the primal problem. We prefer this variant and consider two different primal problems.
The first is related to the case, where u b < ∞, i.e. where the pointwise control constraints have upper and lower bounds. Then the primal problem (PP) can be written as max(a , v) where v is taken from U = L 2 (Q). Now we set V = U × U , K U = L 2 (Q) + , K V = K U × K U , and Then the primal problem (PP) is equivalent to max (a , u) U , Au ≤ V c, u ≥ U 0.
Let us verify Assumption 5.4 of boundedness. We select a varying d = (d 1 , d 2 ) ∈ V and consider the set For all u ∈ P (d), we have u L 2 (Q) ≤ d 1 L 2 (Q) .
Moreover, by inverse non-negativity, we find where z is the unique solution to z = d 2 +Sz. Since (I −S) is continuously invertible and non-negative, an α > 0 exists such that z L 2 (Q) ≤ α d 2 L 2 (Q) . Therefore, holds for all u ∈ P (d). This property is even stronger than Assumption 5.4. Therefore, by Theorem 5.5 the strong duality relation is satisfied. This proves Theorem 4.6 for the first form of (PP).
In the second form of (PP), the upper control constraint is missing, i.e. u b = ∞, hence v b = ∞. Here, the primal problem takes the form max (a , v) L 2 (Q) v ≤ b + Sv v ≥ 0.
Again, Assumption 5.4 is fulfilled and Theorem 4.6 is proved by Theorem 5.5.