Smooth and Nonsmooth Lipschitz Controls

for a Class of Nonlinear

Ordinary Differential Equations of Second Order

Mohamed Akkouchi^{1}, Abdellah Bounabat^{1}, and Manfred Goebel^{2}

Abstract: We study in this paper a control problem associated to a semilinear second order ordinary differential equation with pointwise state constraints. The control is realized via the outer function of the nonlinear part of the state equation. When controls are in the class of Lipshitz functions, we prove under some conditions, the existence of optimal controls and obtain two necessary optimality conditions depending on their smoothness properties. In the case of smooth optimal control this condition looks somehow like Pontryagin’s maximum pinciple. In the general case, this condition makes use of the notion of Clarke’s generalized directional derivative. Using Ekeland’s variational principle, we derive a suboptimality condition for this problem.

AMS (MOS) subject classification: 49K15, 49J15

1. Introduction and statement of the problem

This paper makes a sequel of the papers (Ref. 1, 2), where investigations are made for smooth and nonsmooth optimal lipschitz control for a model problem of a semilinear second order differential equation, in which the nonlinear character is given by the action of a Nemytskij operator. More precisely, the purpose of this work is to treat the following control problem:

For a given function h in the Banach space C([0, 1]), we want to find

inf J (y), J (y) = ∫_{0}^{1} (y(x) − h(x))^{2} dx,

1 Département de Mathématiques. Faculté des Sciences-Semlalia. Université Cadi Ayyad. Avenue du prince My. Abdellah. B.P. S.15. Marrakech. Maroc (Morocco).

2 FB Mathematik und Informatik. Martin-Luther-Universität Halle-Wittenberg. D-06099 Halle (Saale), Germany. Email: goebel@mathematik.uni.halle.de. where the behaviour of y is monitored by the following semilinear second order differential equation

\frac{d}{dx}(p(x)y′(x)) + q(x)y(x) + σ(y(x)) = 0, x ∈ (0, 1), y(0) = 0, y′(1) = 0. (1)

The state constraints being

0 ≤ y(x) ≤ a ∀x ∈ [0, 1],  (2)

and the set of admissible controls taken here is

\begin{aligned}Σ_{ad} = \{σ ∈ C([0, a]) & : 0 ≤ σ(x) ≤ b ∀x ∈ [0, a], \\ & |σ(x_{1}) − σ(x_{2})| ≤ l |x_{1} − x_{2}| , ∀x_{1}, x_{2} ∈ [0, a]\}. \end{aligned}  (3)

where a, b, l are fixed numbers in ]0, ∞[, and p, q are two given functions supposed to be continuously differentiable on [0, 1] and satisfying p(x) > 0, and q(x) ≥ 0, for all x ∈ [0, 1]. In this case there is a unique continuously differentiable function on [0, 1] called θ such that p(x) = 1/θ(x), for all x ∈ [0, 1]. We will use this function θ instead of p and, for the sequel, our contol problem will be denoted by P_{θ}. The Banach space C([0, 1]) will be endowed with its usual norm denoted by ∥.∥_{C([0,1])}.

There are not very much papers dealing with problems considered here. Already in our lecture (Ref. 2) we mentioned the work by R. Kluge (Ref. 3), V. Barbu, K. Kunisch, W. Ring (Ref. 4, 5, and 6), and A. Rösch (Ref. 7) devoted to similar problems for partial differential equations, and M. Goebel, D. Oestreich (Ref. 8), where the state equation is given by a nonlinear singular integral equation.

In the next section, we will give some sufficient conditions ensuring existence for solutions to our problem P_{θ}. These conditions are expressed under the form of inequalities and relations involving the parameters l, a, b, θ, p and q.

2. Solutions of the state equation and existence of optimal controls

2.1 Existence and uniqueness of the solution of the problem P_{θ}

2.1.1 Let G_{θ} = G_{θ}(x, ξ) be the uniquely determined Green’s function to the next boundary problem

\frac{d}{dx} [\frac{y′(x)}{θ(x)}] = 0, x ∈ (0, 1), y(0) = 0, y′(1) = 0.  (1∗)

An easy computation shows that G_{θ} is given by

\{ \begin{matrix}G_{θ}(x, ξ) & = − ∫_{0}^{ξ} θ(s) ds for 0 ≤ ξ ≤ x ≤ 1, \\ G_{θ}(x, ξ) & = − ∫_{0}^{x} θ(s) ds for 0 ≤ x ≤ ξ ≤ 1. \end{matrix}

G_{θ} is continuous and symmetric on [0, 1] × [0, 1] and the following estimations hold

\{ \begin{matrix}0 & ≤ −G_{θ}(x, ξ) ≤ M_{θ} := ∫_{0}^{1} θ(s) ds ∀x, ξ ∈ [0, 1], \\ 0 & ≤ − ∫_{0}^{1} G_{θ}(x, ξ) dξ ≤ R_{θ} := ∫_{0}^{1}(1 − s)θ(s) ds ∀x ∈ [0, 1]. \end{matrix}

In all what follows, we put k := sup{q(x) : 0 ≤ x ≤ 1}, and we will suppose that the two followig conditions

(i) 0 < l + k < \frac{1}{R_{θ}} and (ii) 0 < b ≤ \frac{a}{R_{θ}} (1 − kR_{θ})

are satisfied.

2.1.2 Now consider the subset B_{+}(a) of C([0, 1]) defined by

B_{+}(a) := {y ∈ C([0, 1]) : 0 ≤ y(x) ≤ a ∀x ∈ [0, 1]}

and take a fixed control σ ∈ Σ_{ad}. It is clear that an element y ∈ C^{2}([0, 1]) ∩ B_{+}(a) is a solution to the nonlinear boundary value problem (1∗), which is the same as (1), associated to P_{θ} if and only if y ∈ B_{+}(a) is a solution to the Hammerstein integral equation

y(x) = − ∫_{0}^{1} G_{θ}(x, ξ)σ(y(ξ)) dξ − ∫_{0}^{1} G_{θ}(x, ξ)q(ξ)y(ξ) dξ ∀x ∈ [0, 1].  (4)

In all what follows we will work with this integral equation instead of the originally given boundary value problem (1) (or (1∗)). This applies also to any other boundary value problem in this paper. Its solution is always to be understood as a solution to the corresponding integral equation obtained using the above Green’s function.

Now we are able to state and prove our first result.

2.1.3. Theorem: For each control σ ∈ Σ_{ad}, the boundary value problem (1) associated to P_{θ} has a unique solution y_{σ}. Moreover this solution belongs to C^{2}([0, 1]) ∩ B_{+}(a).

Proof: Let σ ∈ Σ_{ad} be fixed and associated to it the map T_{σ} defined from B_{+}(a) to C([0, 1]) by

T_{σ}(y)(x) := − ∫_{0}^{1} G_{θ}(x, ξ)σ(y(ξ)) dξ − ∫_{0}^{1} G_{θ}(x, ξ)q(ξ)y(ξ) dξ ∀x ∈ [0, 1]. (5)

An easy manipulation will show that for all y, z ∈ B_{+}(a), we have

∥T_{σ}(y) − T_{σ}(z)∥_{C([0,1])} ≤ (l + k)R_{θ}∥y − z∥_{C([0,1])}.

Using the relation (ii), we see that T_{σ}(B_{+}(a)) ⊂ B_{+}(a). Moreover, since by asssumption the condition 0 < l + k < 1/R_{θ} holds, the map T_{σ} must be a contraction from B_{+}(a) to itself. Since the set B_{+}(a) is a closed (convex) subset of the Banach space C([0, 1]), we can use the Banach fixed point theorem and assert that T_{σ} has a unique fixed point y_{σ} ∈ B_{+}(a). This ensures the solvability and the uniqueness of the solution to the boundary value problem (1∗).

2.2 Existence of optimal controls for the problem P_{θ}

2.2.1 To each control σ ∈ Σ_{ad} we associate the unique solution S_{θ}(σ) := y_{σ} to the problem (1∗). One can see that S_{θ} is a Lipschitz continuous map from the closed convex subset Σ_{ad} of C([0, a]) to the Banach space C([0, 1]). Indeed, for all controls σ_{1}, σ_{2} in Σ_{ad}, an easy computation will give the following estimation

∥y_{σ_{1}} − y_{σ_{2}}∥_{C([0,1])} ≤ \frac{R_{θ}}{1 − (l + k)R_{θ}} ∥σ_{1} − σ_{2}∥_{C([0,a])}.  (6)

Now, with the help of this inequality, we are in position to state and prove existence of optimal controls to our problem P_{θ}.

2.2.2. Theorem: The optimal control problem P_{θ} has an optimal solution σ_{0} ∈ Σ_{ad}.

Proof: Let us define the cost functional j by

j(σ) := J (y_{σ}) ∀σ ∈ Σ_{ad},  (7)

and consider σ_{1}, σ_{2} ∈ Σ_{ad}. Then an easy computation using the relation (7) will give the following estimation

|j(σ_{1}) − j(σ_{2})| ≤ \frac{2(a + ∥h∥_{C([0,1]})R_{θ}}{1 − (l + k)R_{θ}} ∥σ_{1} − σ_{2}∥_{C([0,a])}.  (8)

This inequality says that the cost functional j is Lipschitz continuous from the compact subset Σ_{ad} of the Banach space C([0, a]) to the set of real numbers. Hence, using the classical Weierstrass theorem, we see that there exists at least one optimal control to our problem P_{θ}.

3. Necessary optimality conditions for P_{θ}

3.1 Preliminaries

3.1.1 Let σ_{0} ∈ Σ_{ad} be an optimal control to the problem P_{θ} and σ ∈ Σ_{ad} another admissible control. The respective states are denoted by y_{0} = S_{θ}(σ_{0}) and y = S_{θ}(σ). For any λ ∈ [0, 1] we consider the element σ_{λ} ∈ Σ_{ad} given by σ_{λ} := σ_{0} + λ(σ − σ_{0}), and we will put y_{λ} := S_{θ}(σ_{λ}). From the inequality (6) we see that

∥y_{λ} − y_{0}∥_{C([0,1])} ≤ \frac{λR_{θ}}{1 − (l + k)R_{θ}} ∥σ − σ_{0}∥_{C([0,a])} ∀λ ∈]0, 1].  (9)

We deduce that for all λ ∈]0, 1] we have

0 ≤ \frac{j(σ_{λ}) − j(σ_{0})}{λ} = 2 ∫_{0}^{1} \frac{y_{λ} − y_{0}}{λ} (y_{0} − h) dx + \frac{1}{λ} ∫_{0}^{1}(y_{λ} − y_{0})^{2} dx.  (10)

The following lemma gives our first optimality condition. Its proof is easy and makes use only of the inequalities (9), (10) and Lebesgue’s theorem of dominated convergence.

3.1.2. Lemma:

(i) Let (λ_{n})_{n} ⊂]0, 1] be a zero sequence such that for almost all x ∈ [0, 1] the sequence (\frac{y_{λ_{n}}(x) − y_{0}(x)}{λ_{n}})_{n} converges to a limit denoted by ỹ(x). Then the measurable function ỹ belongs to the space L^{∞}([0, 1]) and satisfies the inequality

0 ≤ ∫_{0}^{1} ỹ(y_{0} − h) dx.  (∗)

(ii) In particular, if (λ_{n})_{n} ⊂]0, 1] is a zero sequence and ỹ ∈ C([0, 1]) are such that, the sequence (\frac{y_{λ_{n}} − y_{0}}{λ_{n}} )_{n} is converging to ỹ in C([0, 1]), then inequality (∗) holds true.

3.1.3 Remarks:

a) If (λ_{n})_{n} ⊂]0, 1] is a zero sequence and ỹ ∈ L^{∞}([0, 1]) are such that the sequence (\frac{y_{λ_{n}} − y_{0}}{λ_{n}} )_{n} is converging to ỹ weakly-* in L^{∞}([0, 1]), then (∗) holds true.

b) Suppose that (μ_{n})_{n} ⊂]0, 1] is a zero sequence. Since the set {(y_{μ_{n}} − y_{0})μ_{n}^{−1} : n ∈ IN} is bounded then by Alaoglu’s theorem, there exists a subsequence (μ_{n_{k}})_{k} of (μ_{n})_{n} such that the subsequence ((y_{μ_{n_{k}}} −y_{0})μ_{n_{k}}^{−1})_{n} converges weakly-* to an element ỹ ∈ L^{∞}([0, 1]).

Now, put λ_{k} := μ_{n_{k}} for all integers k, then (λ_{n})_{n} is a zero sequence of elements in ]0, 1], for which the sequence (\frac{y_{λ_{n}} − y_{0}}{λ_{n}} )_{n} is converging to ỹ weakly-* in L^{∞}([0, 1]). From the relation (10), we obtain for all integers n

0 ≤ 2 ∫_{0}^{1} \frac{y_{λ_{n}} − y_{0}}{λ_{n}} (y_{0} − h) dx + λ_{n} ∫_{0}^{1} \frac{(y_{λ} − y_{0})^{2}}{λ_{n}^{2}} dx,

from which we deduce again the validity (∗).

c) Therefore, we may always find a zero sequence (λ_{n})_{n} ⊂]0, 1] and ỹ ∈ L^{∞}([0, 1]) such that the sequence (\frac{y_{λ_{n}} − y_{0}}{λ_{n}} )_{n} converges to ỹ weakly-* in L^{∞}([0, 1]), and we know from

above that we must have the necessary condition 0 ≤ ∫_{0}^{1} ỹ(y_{0} − h) dx. This condition essentially will represent the directional derivative of j = j(σ) at σ_{0} in the direction σ − σ_{0}, provided that (\frac{y_{λn} −y_{0}}{λ_{n}})_{n} converges in the Banach space C([0, 1]) to a fixed element ỹ ∈ C([0, 1]), for any zero sequence (λ_{n})_{n} ⊂]0, 1]. However, this property could be verified only under an additional smoothness assumption concerning the optimal control σ_{0}. In general, because of (6), we have the uniform boundedness of the quotient (y_{λ} − y_{0})λ^{−1} for all λ ∈]0, 1]. Hence, we may find a zero sequence (λ_{n})_{n} ⊂]0, 1] and ỹ ∈ L^{∞}([0, 1]) such that the sequence ((y_{λ_{n}} − y_{0})λ_{n}^{−1})_{n} converges to ỹ weakly-* in L^{∞}([0, 1]). But, certainly, to find an appropriate zero sequence (λ_{n})_{n} ⊂]0, 1], for which the sequence ((y_{λ_{n}} − y_{0})λ_{n}^{−1})_{n} is converging in C([0, 1]) to an element ỹ ∈ C([0, 1]), becomes the more difficult the less nonsmooth the optimal control σ_{0} is.

3.2 Smooth Optimal Control

3.2.1 First, let us suppose the optimal control σ_{0} to be an element of the space C^{1}([0, a]). Then, for an arbitrary right hand side f ∈ C([0, 1]) the linear boundary value problem

\frac{d}{dx} [\frac{z′(x)}{θ(x)}] + q(x)z(x) + σ′_{0}(y_{0}(x))z(x) = f(x), x ∈ (0, 1), z(0) = 0, z′(1) = 0,

has a unique solution z ∈ C([0, 1]), which is true, since we can apply as before the Banach fixed point theorem to the corresponding linear Fredholm integral equation. This will allow us to characterize the limit function ỹ in (ii) of 3.1.2., as a solution of such a boundary value problem having a particular right hand side. More precisely, we have the following lemma.

3.2.2. Lemma: If σ_{0} ∈ Σ_{ad} ∩ C^{1}([0, 1]), then it holds that

\frac{y_{λ} − y_{0}}{λ} ⟶ ỹ in C([0, 1]) as λ ⟶ 0+,

where ỹ = ỹ(σ_{0}, σ) ∈ C([0, 1]) denotes the unique solution to the boundary value problem

\begin{matrix}\frac{d}{dx} [\frac{z′(x)}{θ(x)}] + q(x)z(x) + σ′_{0}(y_{0}(x))z(x) = −σ(y_{0}(x)) + σ_{0}(y_{0}(x)), x ∈ (0, 1), \\ z(0) = 0, z′(1) = 0. \end{matrix}

Proof: Due to the definition of y_{0}, y_{λ}, and ỹ the following identities are valid for all x in the interval [0, 1]:

y_{0}(x) = − ∫_{0}^{1} G_{θ}(x, ξ)σ_{0}(y_{0}(ξ)) dξ − ∫_{0}^{1} G_{θ}(x, ξ)q(ξ)y_{0}(ξ) dξ,
y_{λ}(x) = − ∫_{0}^{1} G_{θ}(x, ξ)σ_{λ}(y_{λ}(ξ)) dξ − ∫_{0}^{1} G_{θ}(x, ξ)q(ξ)y_{λ}(ξ) dξ
= − ∫_{0}^{1} G_{θ}(x, ξ)σ_{0}(y_{λ}(ξ)) dξ − λ ∫_{0}^{1} G_{θ}(x, ξ)[σ(y_{λ}(ξ)) − σ_{0}(y_{λ}(ξ))] dξ
− ∫_{0}^{1} G_{θ}(x, ξ)q(ξ)y_{λ}(ξ) dξ,
ỹ(x) = − ∫_{0}^{1} G_{θ}(x, ξ)σ′_{0}(y_{0}(ξ))ỹ(ξ) dξ − ∫_{0}^{1} G_{θ}(x, ξ)[σ(y_{0}(ξ)) − σ_{0}(y_{0}(ξ))] dξ
− ∫_{0}^{1} G_{θ}(x, ξ)q(ξ)ỹ(ξ) dξ.

For all λ ∈]0, 1] we define the function Δ_{λ} on the interval [0, 1] by setting

Δ_{λ} := \frac{y_{λ} − y_{0}}{λ} − ỹ.

This function may also be written in the form

Δ_{λ} = f_{λ} + g_{λ} + r_{λ}, (11)

where the functions f_{λ}, g_{λ} and r_{λ} are defined for all x ∈ [0, 1] by

\begin{aligned}f_{λ}(x) & = − ∫_{0}^{1} G_{θ}(x, ξ) [\frac{σ_{0}(y_{λ}(ξ)) − σ_{0}(y_{0}(ξ))}{λ} − σ′_{0}(y_{0}(ξ))ỹ(ξ)] dξ, \\ g_{λ}(x) & = − ∫_{0}^{1} G_{θ}(x, ξ) [(σ(y_{λ}(ξ)) − σ(y_{0}(ξ))) + (σ_{0}(y_{0}(ξ)) − σ_{0}(y_{λ}(ξ)))] dξ, \\ r_{λ}(x) & = − ∫_{0}^{1} G_{θ}(x, ξ)q(ξ)Δ_{λ}(ξ) dξ. \end{aligned}

Because of the classical mean value theorem, for all ξ ∈]0, 1] and λ ∈]0, 1] there exists a real number ω = ω(ξ, λ) ∈ (0, 1), such that

\frac{σ_{0}(y_{λ}(ξ)) − σ_{0}(y_{0}(ξ))}{λ} = σ′_{0} (y_{0}(ξ) + ω(y_{λ}(ξ) − y_{0}(ξ))) \frac{y_{λ}(ξ) − y_{0}(ξ)}{λ}.

Then we get

\begin{aligned}f_{λ}(x) & = − ∫_{0}^{1} G_{θ}(x, ξ)σ′_{0} (y_{0}(ξ) + ω(y_{λ}(ξ) − y_{0}(ξ))) Δ_{λ}(ξ) dξ \\ & − ∫_{0}^{1} G_{θ}(x, ξ) [σ′_{0} (y_{0}(ξ) + ω(y_{λ}(ξ) − y_{0}(ξ))) − σ′_{0}(y_{0}(ξ)] ỹ(ξ) dξ \end{aligned}

and therefore, for all x ∈ [0, 1] and λ ∈]0, 1], we obtain that

|f_{λ}(x)| ≤ l ∫_{0}^{1} |G_{θ}(x, ξ)| |Δ_{λ}(ξ)| dξ
+ ∫_{0}^{1} |G_{θ}(x, ξ)| |σ′_{0} (y_{0}(ξ) + ω(y_{λ}(ξ) − y_{0}(ξ))) − σ′_{0}(y_{0}(ξ)| |ỹ(ξ)| dξ
≤ lR_{θ}∥Δ_{λ}∥_{C([0,1])} + μ_{λ}(x),

where μ_{λ} ⟶ 0 in the space C([0, 1]) provided λ ⟶ 0 + . Since the controls σ and σ_{0} are supposed to be Lipschitz continuous, then for all x ∈ [0, 1] and λ ∈]0, 1], we obtain the estimate

|g_{λ}(x)| ≤ 2lR_{θ}∥y_{λ} − y_{0}∥_{C([0,1])}.

Since the estimate

|r_{λ}(x)| ≤ kR_{θ}∥Δ_{λ}∥_{C([0,1])}

is true, then after combining the last three inequalities with (11) we get

0 ≤ (1 − (l + k)R_{θ})∥Δ_{λ}∥_{C([0,1])} ≤ ∥μ_{λ}∥_{C([0,1])} + 2lR_{θ}∥y_{λ} − y_{0}∥_{C([0,1])} ∀λ ∈]0, 1].

Since we have assumed that 0 < 1 − (l + k)R_{θ}, we deduce that ∥Δ_{λ}∥_{C([0,1])} ⟶ 0 provided λ ⟶ 0 + . So our lemma is proved. Thus, Lemma 3.2.2 gives the necessary optimality condition

0 ≤ ∫_{0}^{1} ỹ(x)(y_{0}(x) − h(x)) dx ∀σ ∈ Σ_{ad},  (12)

which after introducing the adjoint state z_{0} ∈ C([0, 1]) as the unique solution to the linear boundary value problem

\frac{d}{dx} [\frac{z′(x)}{θ(x)}]+q(x)z(x)+σ′_{0}(y_{0}(x))z(x) = 2(h(x)−y_{0}(x)), x ∈ (0, 1), z(0) = 0, z′(1) = 0,

yields eventually the following necessary optimality condition.

3.2.3. Theorem: If σ_{0} ∈ Σ_{ad} ∩ C^{1}([0, 1]) is an optimal control to problem P_{θ}, y_{0} ∈ C([0, 1]) the related optimal state and z_{0} ∈ C([0, 1]) the corresponding adjoint state, then it holds

0 ≤ ∫_{0}^{1} (σ(y_{0}(x)) − σ_{0}(y_{0}(x))) z_{0}(x) dx ∀σ ∈ Σ_{ad}.

Proof: Let σ ∈ Σ_{ad} be fixed. We know that inequality (12) holds, where ỹ ∈ C([0, 1]) denotes the unique solution to the boundary value problem given in Lemma 3.2.2. Using the two identities (valid for all x ∈ [0, 1])

z_{0}(x) = − ∫_{0}^{1} G_{θ}(x, ξ)σ′_{0}(y_{0}(ξ))z_{0}(ξ) dξ
− ∫_{0}^{1} G_{θ}(x, ξ)q(ξ)z_{0}(ξ) dξ + 2 ∫_{0}^{1} G_{θ}(x, ξ)[h(ξ) − y_{0}(ξ)] dξ,
ỹ(x) = − ∫_{0}^{1} G_{θ}(x, ξ)σ′_{0}(y_{0}(ξ))ỹ(ξ) dξ
− ∫_{0}^{1} G_{θ}(x, ξ)q(ξ)ỹ(ξ) dξ − ∫_{0}^{1} G_{θ}(x, ξ)[σ(y_{0}(ξ)) − σ_{0}(y_{0}(ξ))] dξ,

it can be easily obtained that

2 ∫_{0}^{1} ỹ(x)(y_{0}(x) − h(x)) dx = ∫_{0}^{1} (σ(y_{0}(x)) − σ_{0}(y_{0}(x))) z_{0}(x) dx,

which proves our theorem.

The optimality condition given in Theorem 3.2.3 is very clear and perhaps cannot be improved. In particular, it seems to be impossible to derive a pointwise optimality condition. However, since we do not have any criteria assuring the needed smoothness condition of the optimal control, we must deal with the case σ_{0} ∉ C^{1}([0, 1]). This will be treated in the next subsection. It is clear that in this case one cannot expect such a nice condition of optimality. To find sufficient conditions ensuring existence of smooth optimal controls is surely an interesting problem to treat.

3.3 Nonsmooth Optimal Control

3.3.1 As it was done in the papers (Ref. 1, 2), we treat here the more interesting case where the control σ_{0} ∉ C^{1}([0, 1]). We start by pointing out that, since the map j : Σ_{ad} → IR is Lipschitz continuous (but surely not convex), we have the optimality condition

0 ∈ ∂j(σ_{0}) + N_{Σ_{ad}}(σ_{0}),  (13)

where j is now a Lipschitz continuous extension of the map j : Σ_{ad} → IR to the whole Banach space C([0, 1]). The notation ∂j(σ_{0}) designates Clarke’s subdifferential of j at σ_{0} and N_{Σ_{ad}}(σ_{0}) is the normal cone to Σ_{ad} at σ_{0}, see (Ref. 9, p. 52). Although we do not have an interpretation of (13) in terms of given datas desribing P_{θ}, we will try, as in the foregoing section, to find a zero sequence (λ_{n})_{n} of elements in ]0, 1] for which ((y_{λ_{n}} − y_{0})λ_{n}^{−1})_{n} converges in C([0, 1]). To this end, for each admissible control σ we introduce the set Φ(σ) given by

Φ(σ) := \{ϕ_{λ} ∈ C([0, 1]) : ϕ_{λ} = \frac{y_{λ} − y_{0}}{λ} , λ ∈]0, 1]\},

where y_{λ} = y_{σ_{λ}} is the same as before, and define two new functions ϕ_{−} and ϕ_{+} by setting

ϕ_{−}(x) := lim inf_{λ→0+} ϕ_{λ}(x), ϕ_{+}(x) := lim sup_{λ→0+} ϕ_{λ}(x), x ∈ [0, 1].

The set Φ(σ) is bounded (see (9)) and one can see that Φ(σ) is also equicontinuous, and therefore this set is relatively compact in the Banach space C([0, 1]). As a consequence, we deduce that the functions ϕ_{−} and ϕ_{+} are continuous on [0, 1]. Another consequence is that, if we introduce the new set Φ_{0}(σ) given by

Φ_{0}(σ) := {ϕ ∈ C([0, 1]) : ϕ = lim_{n→+∞} ϕ_{λ_{n}} in C([0, 1]), where

(ϕ_{λ_{n}})_{n} ⊂ Φ(σ) and (λ_{n})_{n} ⊂ [0, 1] with λ_{n} → 0 + },

then Φ_{0}(σ) is nonempty and bounded. Now, let us denote by σ_{0} any given Lipschitz continuous extension of the (possibly nonsmooth) control σ_{0} : [0, 1] → IR to the whole IR, and let σ_{0}^{0}(x, ξ) designates Clarke’s directional derivative of σ_{0} at x in the direction ξ for all x, ξ ∈ IR. Then with these notations we have the following result:

3.3.2. Theorem: Let σ_{0} ∈ Σ_{ad} be an optimal control to P_{θ} and y_{0} ∈ C([0, 1]) the related optimal state. Then for all σ ∈ Σ_{ad} and all ϕ ∈ Φ_{0}(σ) it holds

(i) ϕ(x) ∈ [ϕ_{−}(x), ϕ_{+}(x)] ∀x ∈ [0, 1], (ii) ϕ(x) ≤ − ∫_{0}^{1} G_{θ}(x, ξ) [σ_{0}^{0}(y_{0}(ξ); ϕ(ξ)) − (σ(y_{0}(ξ)) − (σ_{0}(y_{0}(ξ)))] dξ

− ∫_{0}^{1} G_{θ}(x, ξ)q(ξ)ϕ(ξ) dξ ∀x ∈ [0, 1],

(iii) 0 ≤ ∫_{0}^{1} ϕ(x)(y_{0}(x) − h(x)) dx.

Proof: Let σ ∈ Σ_{ad}. We see first that the assertions (i), (ii) and (iii) hold if σ = σ_{0}. Indeed, in this case, we have Φ(σ_{0}) = {0}, and therefore Φ_{0}(σ_{0}) = {0}, and ϕ_{−} = ϕ_{+} = 0. Thus we may suppose that σ ≠ σ_{0}. Consider an element ϕ ∈ Φ_{0}(σ). Then by the definition of the set Φ_{0}(σ), there exists a zero sequence (λ_{n})_{n} ⊂]0, 1] and a sequence (ϕ_{λ_{n}})_{n} ⊂ Φ(σ) such that ϕ = lim_{n→+∞} ϕ_{λ_{n}} in C([0, 1]). Assertion (i) is obvious and assertion (iii) is a consequence of Lemma 3.1.2. Thus it remains to prove assertion (ii). Due to the definition of ϕ_{λ_{n}}, for all x ∈ [0, 1] we have

\begin{aligned}ϕ_{λ_{n}}(x) & = \frac{y_{λ_{n}}(x) − y_{0}(x)}{λ_{n}} \\ & = − ∫_{0}^{1} G_{θ}(x, ξ)\frac{σ_{0}(y_{λ_{n}}(ξ)) − σ_{0}(y_{0}(ξ))}{λ_{n}} dξ \\ & − ∫_{0}^{1} G_{θ}(x, ξ)[σ(y_{λ_{n}}(ξ)) − σ_{0}(y_{λ_{n}}(ξ))] dξ \\ & − ∫_{0}^{1} G_{θ}(x, ξ)q(ξ)ϕ_{λ_{n}}(ξ) dξ. \end{aligned}  (14)

Because of (9) we have

lim_{n→+∞} σ(y_{λ_{n}}) = σ(y_{0}) in C([0, 1]),

and for all x ∈ [0, 1] we have the estimate

|\frac{σ_{0}(y_{λ_{n}}(ξ)) − σ_{0}(y_{0}(ξ))}{λ_{n}} | ≤ \frac{l}{λ_{n}} |y_{λ_{n}}(x) − y_{0}(x)| ≤ \frac{lR_{θ}}{1 − (l + k)R_{θ}} ∥σ − σ_{0}∥_{C([0,a])}.

Hence, taking in (14) the limit superior and applying Fatou’s lemma we get

\begin{aligned}ϕ(x) & ≤ − ∫_{0}^{1} G_{θ}(x, ξ) lim sup_{n→+∞} \frac{σ_{0}(y_{λ_{n}}(ξ)) − σ_{0}(y_{0}(ξ))}{λ_{n}} dξ \\ & − ∫_{0}^{1} G_{θ}(x, ξ)(σ(y_{0}(ξ)) − σ_{0}(y_{0}(ξ))) dξ \\ & − ∫_{0}^{1} G_{θ}(x, ξ)q(ξ)ϕ(ξ) dξ ∀x ∈ [0, 1]. \end{aligned}  (15)

Since σ_{0} : [0, a] → IR is Lipschitz continuous, it can be extended to the whole IR in a Lipschitz continuous function denoted again by σ_{0} having the same Lipschitz constant l. For this extension and for all x, ξ ∈ IR we will denote Clarke’s directional derivative of σ_{0} at x in the direction ξ by σ_{0}^{0}(x; ξ), and ∂σ_{0}(x) will be the corresponding subdifferential (see for example (Ref. 9) or (Ref. 10)). By using the mean-value theorem for subdifferentials (see for example (Ref. 10, p. 40)) we get the following property: For any natural number n ∈ IN and any x ∈ [0, 1] there are two real numbers ζ_{n}(x), z_{n}(x) ∈ IR verifying the properties

ζ_{n}(x) ∈ ∂σ_{0}(z_{n}(x)) ∀n ∈ IN, lim_{n→+∞} z_{n}(x) = y_{0}(x) (in IR),
\frac{σ_{0}(y_{λ_{n}}(ξ)) − σ_{0}(y_{0}(ξ))}{λ_{n}} = ζ_{n}(x)ϕ_{λ−n}(x) ≤ σ_{0}^{0}(z_{n}(x); ϕ_{λ_{n}}(x)).

Hence, by using the upper semicontinuity of Clarke’s directional derivative we get

lim sup_{n→+∞} \frac{σ_{0}(y_{λ_{n}}(ξ)) − σ_{0}(y_{0}(ξ))}{λ_{n}} ≤ σ_{0}^{0}(y_{0}(x); ϕ(x)) ∀x ∈ [0, 1].  (16)

Now, by using the estimate

σ_{0}^{0}(y_{0}(x); ϕ(x)) ≤ l |ϕ(x)| ∀x ∈ [0, 1],

we deduce that the function σ_{0}^{0}(y_{0}(.); ϕ(.)) is bounded. Furtheremore, since it is upper semicontinuous on [0, 1] it is measurable and, hence it is integrable on [0, 1]. The property (ii) follows then from (15) and (16).

This Theorem is a generalization of our previous Theorem 3.2.3, and since it does not give a more simple optimality condition it may be said that it is only an interime result. But since we do not have sufficient conditions under which optimal controls could be smooth, this result is more convenient. In the next section, we will look for suboptimality conditions.

4. Necessary suboptimality conditions for P_{θ}

4.1 Recalls and preliminaries

4.1.1 Although we cannot ensure smoothness conditions for optimal controls, we are able to use smooth controls to derive a suboptimality condition. To this end, we introduce the closed subset Σ_{ad}^{1} of C^{1}([0, a]) given by

Σ_{ad}^{1} := {σ ∈ C^{1}([0, a]) : 0 ≤ σ(x) ≤ b, |σ′(x)| ≤ l ∀x ∈ [0, a]}.

The Banach space C^{1}([0, a]) will be endowed with its usual norm denoted by ∥.∥_{C^{1}([0,a])}, and it is clear that the restricted maps σ ↦ y_{σ} and σ ↦ j(σ) are still Lipschitz continuous on this space. Indeed, for all σ_{1}, σ_{2} ∈ Σ_{ad}^{1} we have

\begin{aligned}∥y_{σ_{1}} − y_{σ_{2}}∥_{C([0,1])} & ≤ \frac{R_{θ}}{1 − (l + k)R_{θ}} ∥σ_{1} − σ_{2}∥_{C^{1}([0,a])}, \\ |j(σ_{1}) − j(σ_{2})| & ≤ \frac{2(a + ∥h∥_{C([0,1]})R_{θ}}{1 − (l + k)R_{θ}} ∥σ_{1} − σ_{2}∥_{C^{1}([0,a])}. \end{aligned}

The results obtained here rely heavily on Ekeland’s variational principle that we quote as the following Lemma 4.1.2., see (Ref. 11).

4.1.2. Lemma: If (Σ, d) denotes a complete metric space and j : Σ → IR a lower semicontinuous functional bounded from below, then for any ε > 0 there exists an element σ_{ε} ∈ Σ verifying the two following conditions:

(a) j(σ_{ε}) ≤ inf{j(σ) : σ ∈ Σ} + ε,

(b) j_{ε}(σ_{ε}) ≤ j_{ε}(σ) ∀σ ∈ Σ, where j_{ε}(σ) := j(σ) + εd(σ_{ε}, σ).

Now, we can state our result giving necessary suboptimal conditions. More precisely, we have the following theorem.

4.2. Theorem: Let σ_{0} be an optimal control to the problem P_{θ}. Then we have:

(i) j(σ_{0}) = inf{j(σ) : σ ∈ Σ_{ad}^{1}}. (ii) For any ε > 0 there exists an element σ_{ε} ∈ Σ_{ad}^{1}, such that with the associated state

y_{ε} ∈ C([0, 1]) it holds:

(a) j(σ_{ε}) ≤ inf{j(σ) : σ ∈ Σ_{ad}^{1}} + ε,

(b) 0 ≤ ∫_{0}^{1}(σ_{0}(y_{ε}(x)) − σ_{ε}(y_{ε}(x)))z_{ε}(x) dx + ε, where z_{ε} ∈ C([0, 1]) denotes the unique solution to

\begin{matrix}\frac{d}{dx} [\frac{z′(x)}{θ(x)}] + q(x)z(x) + σ′_{ε}(y_{ε}(x))z(x) = 2(h(x) − y_{ε}(x)), x ∈ (0, 1) & , \\ z(0) = 0, z′(1) = 0 & . \end{matrix}

Proof: (i) We start by writing the following obvious estimate:

j(σ_{0}) ≤ inf{j(τ) : τ ∈ Σ_{ad}^{1}} ≤ j(σ) ∀σ ∈ Σ_{ad}^{1}.

Now, we extend σ_{0} to a Lipschitz continuous function on the whole space IR having the same Lipschitz constant as before (namely l). We denote this extension again by σ_{0}, and consider for it the Steklov function σ_{0}^{t} defined by

σ_{0}^{t}(x) := \frac{1}{2t} ∫_{x−t}^{x+t} σ_{0}(ξ) dξ, x ∈ [0, a], t > 0.

It is easy to see that σ_{0}^{t} ∈ Σ_{ad}^{1}, for all t > 0 and that σ_{0}^{t} converges to σ_{0} in C([0, a]) provided t → 0 + . After substituting σ = σ_{0}^{t} into the last estimate, we get

j(σ_{0}) ≤ inf{j(τ) : τ ∈ Σ_{ad}^{1}} ≤ j(σ_{0}^{t}) ∀t > 0,

which gives the desired equality by tending t to zero from the right. (ii) We define a new metric d on the space Σ_{ad}^{1}, by setting

d(σ, τ) := ∥σ − τ∥_{C^{1}([0,1])}/(b + l) ∀σ, τ ∈ Σ_{ad}^{1}.

We can verify that (Σ_{ad}^{1}, d) is a complete metric space and that d(σ, τ) ≤ 2 for all σ, τ ∈ Σ_{ad}^{1}. Since the map j : Σ_{ad}^{1} → IR is Lipschitz continuous and hence also bounded from below, Lemma 4.1.2 can be applied to j and the metric space (Σ_{ad}^{1}, d). Thus, for any ε > 0, there exists an element σ_{ε} ∈ Σ_{ad}^{1}, for which condition (a) of our theorem holds and such that

j(σ_{ε}) ≤ j(τ) + εd(σ_{ε}, τ) ∀τ ∈ Σ_{ad}^{1}.

Now, after replacing τ by σ_{ε} +λ(σ −σ_{ε}) with arbitrary σ ∈ Σ_{ad}^{1} and λ ∈]0, 1], we obtain the inequality

0 ≤ j(σ_{ε} + λ(σ − σ_{ε})) − j(σ_{ε}) + \frac{ελ}{b + l} ∥σ − σ_{ε}∥_{C^{1}([0,1])} ∀σ ∈ Σ_{ad}^{1}, ∀λ ∈]0, 1],

from which we deduce that

0 ≤ \frac{j(σ_{ε} + λ(σ − σ_{ε})) − j(σ_{ε})}{λ} + 2ε ∀σ ∈ Σ_{ad}^{1}, ∀λ ∈]0, 1].

After letting λ tend to 0+ and taking account of the fact that the classical directional derivative δj(σ_{ε}, σ − σ_{ε}) of the cost functional j, defined by

δj(σ_{ε}, σ − σ_{ε}) := lim_{λ→0+} \frac{j(σ_{ε} + λ(σ − σ_{ε})) − j(σ_{ε})}{λ},

can be evaluated for σ_{ε} as

2 ∫_{0}^{1} ỹ(y_{ε} − h) dx,

we obtain by the same arguments as in the proof of Theorem 3.2.3. the statement (b) needed.

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