Smooth and Nonsmooth Optimal Lipschitz Control – a Model Problem

Manfred Goebel^{∗}

Abstract. For a semilinear second order ordinary differential equation a control problem with pointwise state constraints is considered. The control of the system is realized via the outer function of the nonlinear term in the state equation. We prove a simple theorem stating the existence of optimal controls within the class of Lipschitz continuous functions and derive two necessary optimality conditions depending on their smoothness properties. For a smooth optimal control function the given optimality condition looks somehow like Pontryagin’s maximum principle. In the general case the optimality condition involves Clarke’s generalized directional derivative. Furthermore a modified problem is considered, for which by means of Ekeland’s variational principle a suboptimality condition can be proved.

1. INTRODUCTION

In our contribution we discuss mainly necessary optimality conditions for the following control problem (P), in which the behaviour is monitored by a semilinear second order ordinary differential equation. Find

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

subject to the state equation

y′′(x) + σ(y(x)) = 0, x ∈ (0, 1),  y(0) = 0, y′(1) = 0,  (1)

the state constraints

0 ≤ y(x) ≤ a ∀ x ∈ [0, 1]  (2)

and the set of admissible controls

\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)

∗ Martin-Luther-Universität Halle-Wittenberg, FB Mathematik und Informatik, D–06099 Halle (Saale), Germany, goebel@maja.mathematik.uni-halle.de

The function h ∈ C[0, 1] and the constants a, b, l ∈ ℝ satisfying 0 < a, 0 < b ≤ 2a and 0 < l < 2 are given before. The difficulties of the problem are not caused by the state constraints (2), but by the fact that the outer function σ of the nonlinear term in (1) acts as the control function. Problems of such a type are studied fairly seldom. We refer to the papers by R. Kluge and his coworkers, see for example R. Kluge [1], to the doctoral thesis A. Rösch [2], and to the very recent papers V. Barbu, K. Kunisch [3, 4], in which similar problems for partial differential equations were considered (mainly from another point of view). In our paper [5] we have also dealt with an optimal control problem of the above type, however for a nonlinear singular integral equation. Here, essentially the same ideas as in [5] are used to derive necessary optimality conditions in case of smooth optimal controls (see [6]). Another set of necessary optimality conditions is found for nonsmooth optimal controls using Clarke’s generalized directional derivative. Moreover, for a modified problem, for which the infimum of the cost functional is the same as for (P), a suboptimality condition is given via Ekeland’s variational principle. In a forthcoming paper [7] these results will be proved for a more general second order ordinary differential equation. State equations of the form (1) play an very important role in different applications. The control problem (P) to be considered has its origion in [8]; it is a least squares formulation of an inverse problem occuring in a certain one-dimensional electrothermal system, which has been solved via Abel integral equations within the class of piecewise continuous functions. In this class to consider control problems of the type above seems to be very challenging.

2. STATE EQUATION AND EXISTENCE OF OPTIMAL CONTROLS

Let G = G(x, ξ) be the uniquely determined Green’s function to the boundary value problem

y′′(x) = 0, x ∈ (0, 1), y(0) = 0, y′(1) = 0,

that means

G(x, ξ) = −ξ for 0 ≤ ξ ≤ x ≤ 1 and G(x, ξ) = −x for 0 ≤ x ≤ ξ ≤ 1.

G is continuous and symmetric on [0, 1] × [0, 1]; it holds

0 ≤ −G(x, ξ) ≤ 1 ∀ x, ξ ∈ [0, 1], 0 ≤ −2 ∫_{0}^{1} G(x, ξ) dξ ≤ 1 ∀ x ∈ [0, 1].

We put (cf. (2))

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

and take a fixed σ ∈ Σ_{ad}. Evidently, y ∈ C^{2}[0, 1] ∩ B_{+}(0, a) is a classical solution to the nonlinear boundary value problem (1) if and only if y ∈ B_{+}(0, a) is a solution to the Hammerstein integral equation

y(x) = − ∫_{0}^{1} G(x, ξ)σ(y(ξ)) dξ, x ∈ [0, 1];  (4)

in the following we will work with (4) instead of (1). The only what will be used from the state equation (1) are the properties of the Green’s function mentioned above. Also for other (linear or nonlinear) boundary value problems occuring in this paper a solution is always to be understood as a solution of the related integral equation obtained by means of the above Green’s function G. Using the Banach fixed point theorem we can prove that for each σ ∈ Σ_{ad} the state equation (1) has a unique solution y = y(σ) ∈ C[0, 1]. Thereby, of course, the assumption 0 < l < 2 is essential. Without supposing this, due to the Schauder fixed point theorem, we have only the solvability of (1). It is quite interesting that the needed contractivity of the integral operator occuring in (4) requires necessarily Lipschitz continuity of the looked for function σ (see [6]). Concerning the map y ↦ σ(y) we see that the estimate

∥y(σ_{1}) − y(σ_{2})∥_{C[0,1]} ≤ \frac{1}{2 − l}∥σ_{1} − σ_{2}∥_{C[0,a]}  (5)

holds for all σ_{1}, σ_{2} ∈ Σ_{ad}, and hence the cost functional j(σ) = J (y(σ)), σ ∈ Σ_{ad}, is Lipschitz continuous, too:

|j(σ_{1}) − j(σ_{2})| ≤ \frac{2}{2 − l}(a + ∥h∥_{C[0,1]})∥σ_{1} − σ_{2}∥_{C[0,a]} ∀ σ_{1}, σ_{2} ∈ Σ_{ad}.

Since Σ_{ad} is compact in C[0,a], we get immediately the existence of optimal controls.

Theorem 1 (Existence) The optimal control problem (P) has an optimal solution σ_{0} ∈ Σ_{ad}.

Thus, here the existence of optimal solutions is proved via the classical Weierstrass theorem using strong convergence, which happens not very often.

3. NECESSARY OPTIMALITY CONDITIONS

3.1 Preliminaries

Let σ_{0} ∈ Σ_{ad} be an optimal control to (P) and σ ∈ Σ_{ad} another admissible control. The respective states are denoted by y_{0} = y(σ_{0}) and y = y(σ). For any λ ∈ [0, 1] we put σ_{λ} = σ_{0} + λ(σ − σ_{0}) and y_{λ} = y(σ_{λ}). From (5) we see that

∥y_{λ} − y_{0}∥_{C[0,1]} ≤ \frac{λ}{2 − l}∥σ − σ_{0}∥_{C[0,a]} ∀ λ ∈ [0, 1],  (6)

such that from

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 ∀ λ ∈ (0, 1]

our first necessary optimality condition follows.

Lemma 1 If (λ_{n}) ⊂ (0, 1] is a zero sequence and ỹ ∈ C[0, 1] with

(∗)  \frac{y_{λ_{n}} − y_{0}}{λ_{n}} ⟶ ỹ in C[0, 1]  then  0 ≤ ∫_{0}^{1} ỹ(y_{0} − h) dx.

In this simple condition the integral essentially represents the directional derivative of j = j(σ) at σ_{0} in direction σ−σ_{0}, provided (∗) holds for any zero sequence (λ_{n}) ⊂ (0, 1] with a fixed function ỹ ∈ C[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 only the uniform boundedness of the quotient (y_{λ} − y_{0})λ^{−1} for all λ belonging to the interval (0, 1]. To find an appropriate zero sequence λ_{n} ⊂ (0, 1] with (∗) becomes the more difficult the less nonsmooth the optimal control σ_{0} is.

3.2 Smooth Optimal Control

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

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 again follows from the Banach fixed point theorem applied to the corresponding (linear) Fredholm integral equation. This allows to characterize the limit function in (∗) as a solution of such a boundary value problem with a paticular right hand side.

Lemma 2 It holds

\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

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.

Thus, Lemma 1 gives the necessary optimality condition

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

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

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.

Theorem 2 If σ_{0} ∈ Σ_{ad} ∩ C^{1}[0, a] is an optimal control to (P), y_{0} ∈ C[0, 1] the related optimal state and z_{0} ∈ C[0, 1] the related adjoint state, then it holds

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

This optimality condition is very clear and perhaps cannot be improved. In particular, it seems to be impossible to derive a pointwise optimality condition. However, since there are not known any criteria assuring the needed smoothness of the optimal control, we have also to consider the case σ_{0} ∉ C^{1}[0, a], what will be done in the next subsection. It is clear that in this case we cannot expect such a nice optimality condition as given in the last theorem.

3.3 Nonsmooth Optimal Control

Here we briefly deal with the more interesting case that the optimal control σ_{0} does not belong to C^{1}[0, a]. Since the map j : Σ_{ad} → ℝ is Lipschitz continuous (but surely not convex), we have the optimality condition

0 ∈ ∂j(σ_{0}) + N_{Σ_{ad}}(σ_{0}),  (7)

where j is now a Lipschitz continuous extension of j : Σ_{ad} → ℝ to the whole space C[0, a], ∂j(σ_{0}) denotes 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 [9, p. 52]). Unfortunately, we failed to interpret (7) in terms of given datas describing (P), so that we have to go another way. Clearly, we try to verify (∗) of Lemma 1. To this end we put

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

and define two functions φ_{∓} by setting

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

Note, both functions depend also on the element σ ∈ Σ_{ad} chosen before. The set Φ(σ) is bounded (see (6)), equi-continuous, and therefore relatively compact in C[0, 1]. As a first consequence of this, the functions φ_{−} and φ_{+} are continuous over [0, 1]. By

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

we introduce another set in C[0, 1], which, due to the relative compactness of Φ(σ), is nonempty and bounded. Using these notations we may formulate the following optimality condition.

Theorem 3 Let σ_{0} ∈ Σ_{ad} be an optimal control to (P) 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(x, ξ)[σ_{0}^{0}(y_{0}(ξ); φ(ξ)) − (σ(y_{0}(ξ)) − σ_{0}(y_{0}(ξ)))] dξ ∀ x ∈ [0, 1],
(iii)  0 ≤ ∫_{0}^{1} φ(x)(y_{0}(x) − h(x)) dx.

The assertions (i) and (iii) are obvious. In (ii) σ_{0}^{0}(x; ξ) denotes Clarke’s directional derivative of σ_{0} : ℝ → ℝ at x in direction ξ, where now σ_{0} is a Lipschitz continuous extension of the originally given σ_{0} to all of ℝ. Of course, the proof of (ii) requires some knowledge of subgradient calculus as given, for example, in [9, 10]. Particularly, instead of the classical mean value theorem, which we have applied in the proof of Lemma 2, now we have to make use of the mean value theorem for subgradients. The unexpected inequality sign in (ii) comes just from this theorem. For several reasons Theorem 3 seems to be only an interime result. First, we could not find an appropriate adjoint state, by means of which the optimality condition could be somehow simplified. Secondly, it is desirable that an optimality condition for a nonsmooth optimal control (like in Theorem 3) reduces in case of a smooth optimal control to the optimality condition just proved under this stronger assumption (like in Theorem 2). But here we don’t have such a relationship. Only going step by step through the proof of Theorem 3 it can be seen that it generalizes Theorem 2. Note, since we know that there exist optimal controls σ_{0} ∈ Σ_{ad} to (P) Theorem 3 includes also the statement that the sublinear ”Fredholm integral inequality” in (ii) is solvable in C[0, 1].

4. A MODIFIED OPTIMAL CONTROL PROBLEM

In the foregoing section we have given necessary optimality conditions for an existing optimal control to (P). However, in the first condition we had to assume smoothness of the optimal control and the second one seems to be not very convenient. In this section we announce some results for a slightly modified optimal control problem, for which the infimum of the cost functional is the same as for the originally given control problem (see (8) below). Since we cannot verify that for the modified problem there exists an optimal control, we restrict our consideration to suboptimal solutions, for which we describe a necessary condition. The results to be presented rely heavily on Ekeland’s variational principle quoted as the following Lemma 3 (see, for example, [11]).

Lemma 3 Let (Σ, d) be a complete metric space and j : Σ ⟶ ℝ a lower semicontinuous functional bounded from below. For any α > 0 there is an element σ_{α} ∈ Σ with:

(a) j(σ_{α}) ≤ inf {j(σ) : σ ∈ Σ } + α,

(b) j_{α}(σ_{α}) ≤ j_{α}(τ) ∀ τ ∈ Σ, where j_{α}(τ) = j(τ) + αd(σ_{α}, τ).

We modify the control problem (P) simple by taking another set of admissible controls. So, let Σ_{ad} be replaced by Σ_{ad}^{1} defined by

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

The new optimal control problem will be shortly denoted by (P_{1}). First of all we note that

inf {j(σ) : σ ∈ Σ_{ad}^{1} } = inf {j(σ) : σ ∈ Σ_{ad} } = j(σ_{0}) with σ_{0} ∈ Σ_{ad}.  (8)

Here the last equality sign is due to Theorem 1. To verify the first equality sign we introduce the Steklov function σ_{0}^{h} corresponding to σ_{0} by

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

where before we have extended σ_{0} to a Lipschitz continuous function on the whole space ℝ having the same Lipschitz constant as before (namely l). Then σ_{0}^{h} ∈ Σ_{ad}^{1} for all h > 0 and σ_{0}^{h} → σ_{0} in C[0, a] as h tends to zero. Substituting σ = σ_{0}^{h} into the obvious estimate

j(σ) ≥ inf {j(σ) : σ ∈ Σ_{ad}^{1} } ≥ j(σ_{0}) ∀ σ ∈ Σ_{ad}^{1},

and letting h → 0+ we get the first equality sign in (8). The modified problem has the great advantage that all admissible controls are smooth, so that in principle we may apply the same analytical methods as in section 3.2. Particularly, the classical directional derivative δj(σ_{α}, σ − σ_{α}) of the cost functional j can be evaluated for any σ_{α} ∈ Σ_{ad}^{1} as

δj(σ_{α}, σ − σ_{α}) = 2 ∫_{0}^{1} ỹ(y_{α} − h) dx ∀ σ ∈ Σ_{ad}^{1},

where y_{α} = y(σ_{α}) is the state related to σ_{α} and ỹ = ỹ(σ_{α}, σ) ∈ C[0, 1] denotes the unique solution to the linear boundary value problem

z′′(x) + σ′_{α}(y_{α}(x))z(x) = −σ(y_{α}(x)) + σ_{α}(y_{α}(x)), x ∈ (0, 1), z(0) = 0, z′(1) = 0.

Besides Lemma 3 this is the crucial point in proving Theorem 4 below. Since, furthermore, (Σ_{ad}^{1}, d) with d(σ, τ) = ∥σ − τ∥_{C^{1}[0,a]}/(b + l), σ, τ ∈ Σ_{ad}^{1}, is a complete metric space with d(σ, τ) ≤ 2 for all σ, τ ∈ Σ_{ad}^{1} and since the cost functional j : Σ_{ad}^{1} ⟶ ℝ is Lipschitz continuous and hence also bounded from below, Lemma 3 can be applied to the control problem (P_{1}). This gives the condition

0 ≤ ∫_{0}^{1} ỹ(y_{α} − h) dx + α ∀ σ ∈ Σ_{ad}^{1}.

Here y_{α} is the state corresponding to the suboptimal solution σ_{α} ∈ Σ_{ad}^{1} for the modified problem (P_{1}). After some further steps like in subsection 3.2 we obtain the following necessary suboptimality condition.

Theorem 4 For any α > 0 there exists an element σ_{α} ∈ Σ_{ad}^{1}, such that with the associated state y_{α} = y(σ_{α}) ∈ C[0, 1] it holds:

(a)  j(σ_{α}) ≤ inf {j(σ) : σ ∈ Σ_{ad}^{1} } + α,
(b)  0 ≤ ∫_{0}^{1} (σ(y_{α}(x)) − σ_{α}(y_{α}(x)))z_{α}(x) dx + α ∀ σ ∈ Σ_{ad}^{1},

where z_{α} ∈ C[0, 1] denotes the unique solution to

z′′(x) + σ′_{α}(y_{α}(x))z(x) = 2(h(x) − y_{α}(x)), x ∈ (0, 1), z(0) = 0, z′(1) = 0.

Condition (b) is anlogous to the optimality condition given in Theorem 2. At least under some additional assumptions it should be possible to let α tend to zero.

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