Vadim Arkin

Alexander Slastnikov

Real Options and Threshold Strategies

Keywords: real options, investment timing problem, diffusion process, optimal stopping, threshold stopping time, free-boundary problem

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

One of the fundamental problems in real options theory concerns the determination of the optimal time for investment into a given project (see, e.g., the classical monograph [START_REF] Dixit | Investment under Uncertainty[END_REF]).

Let us think of an investment project, for example, a founding of a new firm in the real sector of economy. This project is characterized by a pair (X t , t ≥ 0, I), where X t is the present value of the firm founded at time t, and I is a cost of investment required to implement the project (for example, to found the firm). The input and the output production prices are assumed to be stochastic, so X t , t ≥ 0 is considered as a stochastic process, defined on a general filtered probability space (Ω, F, {F t , t ≥ 0}, P). This model assumes that:

-at any moment, a decision-maker (investor) can either accept the project and proceed with the investment or delay the decision until he obtains new information; -investment are considered to be instantaneous and irreversible so that they cannot be withdrawn from the project any more and used for other purposes.

The investor's problem is to evaluate the project and determine an appropriate time for the investment (investment timing problem). In real options theory investment times are considered as stopping times (adapted to the flow of σalgebras {F t , t ≥ 0}).

In real options theory there are two different approaches to solving investment timing problem (see [START_REF] Dixit | Investment under Uncertainty[END_REF]).

The project value under the first approach is the maximum of the net present value from the implemented project over all stopping times (investment rules):

F = max τ E(X τ -I)e -ρτ , (1)
where ρ is a given discount rate. An optimal stopping time τ * in (1) is viewed as the optimal investment time (investment rule). Within the second approach an opportunity to invest is considered as an American call option -the right but not obligation to buy the asset on predetermined price. The exercise time is viewed as an investment time, and the option value is accepted as the investment project value. In this framework a project is spanned with some traded asset, whose price is completely correlated with the present value X t of the project. In order to evaluate the (rational) value of this real option one can use methods of financial options pricing theory, especially, contingent claims analysis (see, e.g., [START_REF] Dixit | Investment under Uncertainty[END_REF]).

In this paper we follow the approach that the optimal investment timing decision can be mathematically determined as the solution of an optimal stopping problem [START_REF] Alvarez | Reward functionals, salvage values, and optimal stopping[END_REF]. Such an approach originated in the well-known McDonald-Siegel model (see [START_REF] Mcdonald | The value of waiting to invest[END_REF], [START_REF] Dixit | Investment under Uncertainty[END_REF]), in which the underlying present value's dynamics is modeled by a geometric Brownian motion. The majority of results on this problem (optimal investment strategy) has a threshold structure: to invest when the present value of the project exceeds a certain level (threshold). At a heuristic level this is so for the cases of geometric Brownian motion, arithmetic Brownian motion, mean-reverting process and a few others (see [START_REF] Dixit | Investment under Uncertainty[END_REF]). However the following general question arises: For which underlying processes the optimal decision in the investment timing problem will have a threshold structure?

Another investment timing problem (with additional restrictions on stopping times) was considered by Alvarez [START_REF] Alvarez | Reward functionals, salvage values, and optimal stopping[END_REF], who established sufficient (but not necessary) conditions for optimality.

In this paper we focus our attention on finding of necessary and sufficient conditions for optimality of threshold strategies in the investment timing problem. Since this problem is a special case of the optimal stopping problem, a similar question may be addressed in the general optimal stopping problem: Under what conditions (on both process and payoff function) the optimal stopping time will have a threshold structure? Some results in this direction (in the form of necessary and sufficient conditions) were obtained in [START_REF] Arkin | Threshold stopping rules for diffusion processes and Stefan's problem[END_REF], [START_REF] Crocce | Explicit solutions in one-sided optimal stopping problems for one-dimensional diffusions[END_REF], [START_REF] Arkin | Threshold Strategies in Optimal Stopping Problem for One-Dimensional Diffusion Processes[END_REF] under additional assumptions on underlying process and/or payoffs.

The paper is organized as follows. After a formal description of the investment timing problem and the assumptions on the underlying process (Section 2.1), we turn to study the threshold strategies in this problem. Since the investment timing problem by threshold strategies is reduced to one-dimensional maximization problem, then a related problem is to find the optimal threshold. In Section 2.2 we give necessary and sufficient conditions for the optimal threshold (over all thresholds). Solving a free-boundary problem (based on smooth-pasting principle) is the most commonly used method (but this is not the only method, see, e.g., [START_REF] Peskir | Optimal Stopping and Free-Boundary Problems[END_REF]) that allows to find a solution to the optimal stopping problem.

In Section 2.3 we discuss the connection between solutions to the investment timing problem and the free-boundary problem. Finally, in Section 2.4 we prove the main result on necessary and sufficient conditions under which the optimal investment time is generated by a threshold strategy.

Investment Timing Problem

Let I be the cost of investment required for implementing a project, and X t the present value from the project started at time t. As usual the investment is supposed to be instantaneous and irreversible, and the project -infinitely-lived.

At any time a decision-maker (investor) can either accept the project and proceed with the investment or delay the decision until she/he obtains new information regarding its environment (prices of the product and resources, demand etc.). The goal of a decision-maker in this situation is to use the available information and find the optimal time for investing in the project (investment timing problem), i.e., find a time τ that maximizes the net present value from the project:

E x (X τ -I) e -ρτ 1 {τ <∞} → max τ ∈M . ( 2
)
Here E x is the expectation for the process X t starting from the initial state x, 1 A is indicator function of the set A, and the maximum is taken over all stopping times τ from a certain class M of stopping times1 . We treat the interesting case I<r; otherwise the optimal time in (2) will be +∞.

Mathematical Assumptions

Let X t , t > 0 be a diffusion process with values in the interval D ⊆ R 1 with boundary points l and r, where -∞ ≤ l < r ≤ +∞, open or closed (i.e. it may be (l, r), [l, r), (l, r], or [l, r]), which is the solution to the stochastic differential equation:

dX t = a(X t )dt + σ(X t )dw t , X 0 = x, (3)
where w t is a standard Wiener process, a : D → R 1 and σ : D → R 1 + are the drift and the diffusion coefficients, respectively. Denote I = int(D) = (l, r).

The process X t is assumed to be regular; this means that, starting from an arbitrary point x ∈ I, this process reaches any point y ∈ I in finite time with positive probability.

It is known that the following local integrability condition:

x+ε x-ε 1 + |a(y)| σ 2 (y) dy < ∞ for some ε > 0, (4)
at any x ∈ I guarantees the existence of a weak solution of equation ( 3) and its regularity (see, e.g. [START_REF] Karatzas | Brownian Motion and Stochastic Calculus[END_REF]).

The process X t is associated with the infinitesimal operator

Lf (x) = a(x)f (x) + 1 2 σ 2 (x)f (x). ( 5
)
Under the condition (4) there exist (unique up to constant positive multipliers) increasing and decreasing functions ψ(x) and ϕ(x) with absolutely continuous derivatives, which are the fundamental solutions to the ODE

Lf (p) = ρf (p) (6)
almost sure (with respect to Lebesque measure) on the interval I (see, e.g. [10, Chapter 5, Lemma 5.26]). Moreover, 0 < ψ(p), ϕ(p) < ∞ for p ∈ I. Note, if the functions a(x), σ(x) are continuous, then ψ, ϕ ∈ C 2 (I).

Optimality of Threshold Strategies

Let us define τ p =τ p (x)= inf{t≥0 : X t ≥p} -the first time when the process X t , starting from x, exceeds level p. The time τ p is said to be a threshold stopping time generated by a threshold strategy -to stop when the process exceeds threshold p. Let M th ={τ p , p∈I} be a class of all such threshold stopping times.

For the class M th the investment timing problem (2) can be written as follows:

(p -

I) E x e -ρτp → max p∈(l,r) . (7)
Such a problem appeared in [START_REF] Dixit | A Markup Interpretation of Optimal Investment Rules[END_REF] as the heuristic method for solving a general investment timing problem (2) over the class of all stopping times.

We say that the threshold p * is optimal for the investment timing problem [START_REF] Dixit | A Markup Interpretation of Optimal Investment Rules[END_REF] if the threshold stopping time τ p * is optimal in [START_REF] Dixit | A Markup Interpretation of Optimal Investment Rules[END_REF]. The following result gives necessary and sufficient conditions for the optimal threshold. Theorem 1. Threshold p * ∈ I is optimal in the problem (7) for all x ∈ I, if and only if the following conditions hold:

p -I ψ(p) ≤ p * -I ψ(p * ) if p < p * ; (8) p -I ψ(p) does not increase for p ≥ p * , (9)
where ψ(p) is an increasing solution to the ODE (6).

Proof. Let us denote the left-hand side in (7) by V (p; x). Obviously, V (p; x) = x -I for x ≥ p. Along with the above stopping time τ p let us define the first hitting time to the threshold p: T p = inf{t≥0 : X t = p}, p ∈ (l, r).

For x < p, clearly, τ p = T p and using known formula E x e -ρTp = ψ(x)/ψ(p) (see, e.g., [START_REF] Ito | Diffusion Processes and Their Sample Paths[END_REF], [START_REF] Borodin | Handbook of Brownian Motion -Facts and Formulae[END_REF]) we obtain:

V (p; x) = (p -I)E x e -ρτp 1 {τp<∞} = (p -I)E x e -ρTp = p -I ψ(p) ψ(x). (10)
Denote h(x) = (x -I)/ψ(x). i) Necessity. Let p * ∈ I be an optimal threshold in the problem (7) for all x ∈ I. Then for p < p * we have

V (p; p) = p -I ≤ V (p * ; p) = p * -I ψ(p * ) ψ(p), i.e. ( 8
) holds. If p * ≤ p 1 < p 2 , then V (p 2 ; p 1 ) = h(p 2 )ψ(p 1 ) ≤ V (p * ; p 1 ) = p 1 -I = h(p 1 )ψ(p 1 ),
and it follows that ( 9) is true. ii) Sufficiency. Now, suppose that conditions ( 8)-( 9) hold.

Let p < p * . If x ≥ p * , then V (p; x) = x -I = V (p * ; x). If p ≤ x < p * , then, due to (8), V p (x)=x-I=h(x)ψ(x)≤h(p * )ψ(x)=V (p * ; x).
Finally, if x < p, then, using ( 8) and ( 10), we obtain:

V (p; x) = h(p)ψ(x) ≤ h(p * )ψ(x) = V (p * ; x). Consider the case p > p * . If x ≥ p, then V (p; x) = x -I = V (p * ; x). Whenever p * ≤ x < p, then, due to (9), V (p; x) = h(p)ψ(x) ≤ h(x)ψ(x) = x -I = V (p * ; x). When x < p * , then V (p; x)=h(p)ψ(x)≤h(p * )ψ(x)=V (p * ; x), since h(p)≤h(p * ).
Theorem is completely proved.

Remark 1. The condition ( 9) is equivalent to the inequality

(p -I)ψ (p) ≥ ψ(p) for p ≥ p * .
This relation implies, in particular, that the optimal threshold p * must be strictly greater than the cost I (because the values ψ(p * ), ψ (p * ) are positive).

Remark 2. Assume that log ψ(x) is a convex function, i.e. ψ (x)/ψ(x) increases. In this case there exists a unique point p * which satisfies the equation

(p * -I)ψ (p * ) = ψ(p * ). (11)
This value p * constitutes the optimal threshold in the problem (7) for all x ∈ I. Indeed, the sign of the derivative of the function (p -I)/ψ(p) coincides with the sign of ψ(p) -(p -I)ψ (p). Therefore, in the considered case the conditions (8)-( 9) in Theorem 1 are true automatically.

There are a number of cases of diffusion processes X t which are more or less realistic for modeling the present values of a project. Some of them are listed below.

1) Geometric Brownian motion (GBM):

dX t = X t (αdt + σdw t ). ( 12
)
In this case ψ(x) = x β , where β is the positive root of the equation

1 2 σ 2 β(β -1) + αβ -ρ = 0.
2) Arithmetic Brownian motion (ABM):

dX t = x + αdt + σdw t . (13)
In this case ψ(x) = e βx , where β is the positive root of the equation

1 2 σ 2 β 2 + αβ -ρ = 0.
3) Mean-reverting process (or geometric Ornstein-Uhlenbeck process):

dX t = α(x -X t )X t dt + σX t dw t . (14)
In this case

ψ(x) = x β 1 F 1 β, 2β + 2αx σ 2 ; 2α σ 2 x
, where β is the positive root of equation 1 2 σ 2 β(β -1)+αxβ -ρ = 0, and 1 F 1 (p, q; x) is the confluent hypergeometric function satisfying Kummer's equation xf (x) + (q -x)f (x) -pf (x) = 0. 4) Square-root mean-reverting process (or Cox-Ingersoll-Ross process):

dX t = α(x -X t )dt + σ X t dw t . (15)
In this case

ψ(x) = 1 F 1 ρ α , 2αx σ 2 ; 2α σ 2 x .
The above processes are well studied in the literature (in connection with real options and optimal stopping problems see, e.g., [START_REF] Dixit | Investment under Uncertainty[END_REF], [START_REF] Johnson | A discretionary stopping problem with applications to the optimal timing of investment decisions[END_REF]).

For the first two processes, ( 12) and ( 13), Theorem 1 gives explicit formulas for the optimal threshold in the investment timing problem:

p * = β β -1
I for the GBM, and p * = I + 1 β for the ABM.

On the contrary, for mean-reverting processes ( 14) and (15) the function ψ(x) is represented as an infinite series, and the optimal threshold can be find only numerically.

So, Theorem 1 states that optimal threshold p * is a point of maximum for the function h(x) = (x -I)/ψ(x). This implies the first-order optimality condition h (p * ) = 0, i.e. the equality [START_REF] Mcdonald | The value of waiting to invest[END_REF], and smooth-pasting principle:

V x (p * ; x) x=p * = 1.
In the next section we discuss smooth-pasting principle and appropriate freeboundary problem more closely.

Threshold Strategies and Free-Boundary Problem

There is a common opinion (especially among engineers and economists) that the solution to a free-boundary problem always gives a solution to an optimal stopping problem.

A free-boundary problem in the case of threshold strategies in the investment timing problem can be formulated as follows: find the threshold p * ∈ (l, r) and a twice differentiable function H(x), l < x < p * , such that LH(x) = ρH(x), l < x < p * ;

(16)

H(p * -0) = p * -I, H (p * -0) = 1. ( 17
)
If ψ(x) is a twice differentiable function, then the solution to the problem ( 16)-( 17) has the form

H(x) = p * -I ψ(p * ) ψ(x), l < x < p * . ( 18
)
Here ψ(x) is an increasing solution to the ODE ( 6) and p * satisfies the smoothpasting condition [START_REF] Mcdonald | The value of waiting to invest[END_REF]. We call such p * a solution to a free-boundary problem.

According to Theorem 1 the optimal threshold in problem ( 7) must be the point of maximum of the function h(x) = (x -I)/ψ(x). However the smoothpasting condition [START_REF] Mcdonald | The value of waiting to invest[END_REF] provides only a stationary point for h(x). Thus, we can apply standard second-order optimality conditions to derive relations between the solutions to the investment timing problem and the free-boundary problem.

Let p * be a solution to the free-boundary problem ( 16)-( 17). If p * is also an optimal threshold in the investment timing problem [START_REF] Dixit | A Markup Interpretation of Optimal Investment Rules[END_REF], then, of course, h (p * ) ≤ 0. This means that

ψ (p * ) = - h (p * )ψ(p * ) + 2h (p * )ψ (p * ) h(p * ) = - h (p * )ψ(p * ) h(p * ) ≥ 0.
Thus, the inequality ψ (p * ) ≥ 0 may be viewed as a necessary condition for a solution of the free-boundary problem to be optimal in the investment timing problem. The inverse relation between solutions can be stated as follows.

Statement 1. If p * is the unique solution to the free-boundary problem (16)-(17), and ψ (p * )>0, then p * is an optimal threshold in the problem (7) for all x∈I.

Proof. Since h (p * ) = 0 and ψ (p * ) > 0 then h (p * ) = -h(p * )ψ (p * )/ψ(p * )<0. Therefore, h (p) strictly decreases at some neighborhood of p * . Then, it is easy to see that h (p) > 0 for p < p * and h (p) < 0 for p > p * . Otherwise h (q) = 0 for some q = p * , that contradicts the uniqueness of the solution to the free-boundary problem ( 16)-(17). Therefore, conditions (8)-( 9) hold and Theorem 1 gives the optimality of threshold p * .

The following result concerns the general case when the free-boundary problem has several solutions. Statement 2. Let p * and p be two solutions to the free-boundary problem (16)-(17), such that ψ (p * ) > 0 and (x -I)/ψ(x) ≤ (p * -I)/ψ(p * ) for l<x<p * . If p > p * is such that ψ (k) (p) = 0, k = 2, ..., n -1 and ψ (n) (p) > 0 for some n > 2, then p * is an optimal threshold in the problem (7) for all x ∈ I.

Proof. Let us prove that h (p) ≤ 0 for all p > p * . The inequality ψ (p * ) > 0 implies (as above) that h (p * ) < 0, and, therefore, h (p) < 0 for all p * < p < p 1 with some p 1 . If we suppose that h (p 2 ) > 0 for some p 2 > p * , then there exists p 0 ∈(p 1 , p 2 ) such that h (p 0 ) = 0 and h (p) > 0 for all p 0 < p < p 2 . Therefore, p 0 would be another solution to the free-boundary problem ( 16)-( 17). The conditions of the Statement imply that h (k) (p 0 ) = 0, k = 2, ..., n -1, h (n) (p 0 )<0 for some n > 2, which contradicts the positivity of h (p) for p 0 <p<p 2 .

Hence, h (p) ≤ 0 for all p > p * and conditions ( 8)-( 9) hold. Thus, according to Theorem 1, p * is an optimal threshold in the problem [START_REF] Dixit | A Markup Interpretation of Optimal Investment Rules[END_REF].

Optimal Strategies in the Investment Timing Problem

Now, let us return to the 'general' investment timing problem [START_REF] Arkin | Threshold Strategies in Optimal Stopping Problem for One-Dimensional Diffusion Processes[END_REF].

A specific version of the investment timing problem (2) over the class M 0 of stopping times τ such that τ <τ (0) = inf{t≥0 : X t ≤ 0} was considered by Alvarez [START_REF] Alvarez | Reward functionals, salvage values, and optimal stopping[END_REF]. He derived sufficient conditions under which an optimal investment time in [START_REF] Arkin | Threshold Strategies in Optimal Stopping Problem for One-Dimensional Diffusion Processes[END_REF] over the class M 0 will be a threshold stopping time. However these conditions are not necessary.

In this section we give necessary and sufficient conditions (criterion) for optimality of the threshold stopping time in the investment timing problem (2) over the class of all stopping times.

To reduce some technical difficulties we assume below that the drift a(x) and the diffusion σ(x) of the underlying process X t are continuous functions.

Here ψ(x) is an increasing solution to the ODE (6) and a(p) is the drift coefficient of the process X t .

Proof. Define the value function for the problem (2) over the class M of all stopping times as follows:

V (x) = sup τ ∈M E x (X τ -I) e -ρτ 1 {τ <∞} . i) Sufficiency. Let conditions (19)-(21) hold. Take the function Φ(x) = V (p * ; x) =    p * -I ψ(p * ) ψ(x), for x < p * , x -I, for x ≥ p * .
Obviously, Φ(x) > 0 (due to condition (20)) and V (x) ≥ Φ(x) for all x∈ I. On the other hand, condition (19) implies

p * -I ψ(p * ) ψ(x) ≥ x -I ψ(x) ψ(x) = x -I.
Therefore Φ(x) ≥ x -I for all x ∈ (l, r), i.e. Φ(x) is a majorant of the specific payoff function x -I.

For any stopping time τ ∈ M and a real number N > 0 put τ = τ ∧N . From Itô-Tanaka-Meyer formula (see, e.g. [START_REF] Karatzas | Brownian Motion and Stochastic Calculus[END_REF]) we have:

E x Φ(X τ )e -ρτ = Φ(x) + E x τ 0 (LΦ -ρΦ)(X t )e -ρt dt + 1 2 σ 2 (p * )[Φ (p * +0) -Φ (p * -0)]E x τ 0 e -ρt dL t (p * ), (22)
where L t (p * ) is the local time of the process X t at the point p * . By definition and in view of condition (20) we have

Φ (p * +0) -Φ (p * -0) = 1 - p * -I ψ(p * ) ψ (p * ) = 0. Denote T 1 ={t : 0 ≤ t ≤ τ , X t < p * }, T 2 ={t : 0 ≤ t ≤ τ , X t > p * }. We have: LΦ(X t ) -ρΦ(X t ) = p * -I ψ(p * ) Lψ(X t ) -ρψ(X t ) = 0 for t ∈ T 1 , LΦ(X t ) -ρΦ(X t ) = a(X t ) -ρ(X t -I) ≤ 0 for t ∈ T 2 .
These relations follow from the definition of the function ψ(x) and in view of condition (21). Then Since Φ(X τ )e -ρτ a.s.

-→ Φ(X τ )e -ρτ 1 {τ <∞} when N → ∞, then due to Fatou's Lemma : E x Φ(X τ )e -ρτ 1 {τ <∞} ≤ Φ(x) for all τ ∈ M and x ∈ I. Therefore, Φ(x) is a ρ-excessive function, which majorates the payoff function x -I. Since, by Dynkin's characterization, the value function V (x) is the least ρ-excessive majorant, then V (x) ≤ Φ(x).

Therefore, V (x) = Φ(x) = V (p * ; x), i.e. τ p * is the optimal stopping time in problem (2) for all x.

ii) Necessity. Now, let τ p * be an optimal stopping time in the problem (2). Note, that τ p * will be also an optimal stopping time in the problem [START_REF] Dixit | A Markup Interpretation of Optimal Investment Rules[END_REF]. Therefore, Theorem 1 implies conditions (19) and (20), since p * is a point of maximum for the function (x -I)/ψ(x).

Further, assume that inequality (21) is not true at some point p 0 > p * , i.e. a(p) > ρ(p -I) in some interval J ⊂ (p * , r) (by continuity). For some x ∈ J define τ = inf{t ≥ 0 : X t / ∈ J}, where the process X t starts from the point x. Then for any N > 0 from Dynkin's formula E x(X τ ∧N -I)e -ρ(τ ∧N ) = x -I + E x τ ∧N 0 [a(X t ) -ρ(X t -I)]e -ρt dt > x -I. Therefore, V (x) > x -I which contradicts the relation V (x) = V (p * ; x) = g(x), since x > p * .

Example 1. Let the process X t be the geometric Brownian motion [START_REF] Peskir | Optimal Stopping and Free-Boundary Problems[END_REF]. Then Theorem 2 implies that the threshold stopping time τ p * will be optimal in the investment timing problem (2) over all investment times if and only if p * = Iβ/(β -1), where β is the positive root of the equation 1 2 σ 2 β(β -1)+αβ -ρ = 0.

Theorem 2 .

 2 The threshold stopping time τ p * , p * ∈I, is optimal in the investment timing problem (2) for all x∈I if and only if the following conditions hold: (p -I)ψ(p * ) ≤ (p * -I)ψ(p) for p < p * ; (19) ψ(p * ) = (p * -I)ψ (p * ); (20) a(p) ≤ ρ(p -I) for p > p * .

ET1(

  x Φ(X τ )e -ρτ ≤ Φ(x) + E x   LΦ-ρΦ)(X t )e -ρt dt + T2 (LΦ-ρΦ)(X t )e -ρt dt   ≤ Φ(x).

We consider stopping times which can take infinite values (with positive probability).

Acknowledgments. The work was supported by Russian Foundation for Basic Researches (project 15-06-03723) and Russian Foundation for Humanities (project 14-02-00036).


George Baravdish
email: george.baravdish@liu.se

Yuanji Cheng
email: yuanji.cheng@mah.se

Olof Svensson
email: olof.svensson@liu.se

Freddie Åström
email: freddie.astroem@iwr.uni-heidelberg.debuild

Extension of p-Laplace Operator for Image Denoising

Keywords: p-Laplace operator, parabolic equations, image denoising, anisotropic diffusion, inverse problems

In this work we introduce a novel operator ∆ (p,q) as an extended family of operators that generalize the p-Laplace operator. The operator is derived with an emphasis on image processing applications, and particularly, with a focus on image denoising applications. We propose a non-linear transition function, coupling p and q, which yields a non-linear filtering scheme analogous to adaptive spatially dependent total variation and linear filtering. Well-posedness of the final parabolic PDE is established via pertubation theory and connection to classical results in functional analysis. Numerical results demonstrates the applicability of the novel operator ∆ (p,q) .

Introduction

A well known inverse problem in image processing is image denoising [?]. In the last decades the energy functional approach together with its corresponding Euler-Lagrange (E-L) equation has attracted great attention in solving inverse problems applied to image reconstruction. One important case of E-L equations is the one which involves the p-Laplace operator

∆ p u = div(|∇u| p-2 ∇u), p ≥ 1, (1)
associated with the evolution equation of p-Laplacian

   ∂ t u -∆ p u = 0, in Ω × (0, T ) u(0) = u 0 , in Ω ∂ n u = 0, on ∂Ω × (0, T ) (2)
where Ω is a bounded domain in R 2 and u 0 : Ω → R is a given degraded image [?], [?], [?] and ∇u is the gradient. The degenerate parabolic equation in (??) has been studied by many authors and we limit ourselves here to refer the reader to [?], It is well known that the case p = 2 gives the linear Gaussian filter, which however, impose strong spatial regularity and therefore image details such as lines and edges are oversmoothed. p = 1 is often refereed to as the method of total variation [?] and p = 0 is an instance of the so called balanced forward backward evolution [?].

In this work we study a decoupled form of the p-Laplace operator, expressed as a non-linear combination of the ∆ 1 and ∆ ∞ operators, introduced below. We call our new operator ∆ (p,q) . Via established existence theory we show that the corresponding perturbed parabolic equation is well-posed and close to the original operator.

In Section 2 we review some of the properties of the p-Laplace operator applied in image denoising and compare it with Perona-Malik models. Our main contribution is in section 3 where we extend the p-Laplace operator to a new operator ∆ (p,q) with focus on image denoising. We consider the corresponding variable version ∆ (p(x),q(x)) in section 4. Finally, in section 5 we demonstrates the applicability of ∆ (p,q) by numerical results.

p-Laplacian for Image Denosing

An important feature in any evolution process for image denoising is preservation of certain geometrical features of the underlying image. In the case of image restoration these features include edges and corners. It is straight-forward to express the p-Laplace operator (??) as

∆ p u = |∇u| p-1 ∆ 1 u + (p -1)|∇u| p-2 ∆ ∞ u, (3)
where

∆ 1 u = div ∇u |∇u| , ∆ ∞ u = ∇u |∇u| • (D 2 u) ∇u |∇u|
and D 2 u is the Hessian of u. However, an intuitive way to represent ∆ p , giving direct interpretation of the diffusivity directions is to express ∆ p by using Gauge coordinates (x, y) → (η, ξ):

∆ p u = |∇u| p-2 (u ξξ + (p -1)u ηη ) (4)
where

η = ∇u |∇u| , ξ = ∇ ⊥ u |∇u| , (5)
and

u ξξ = |∇u|∆ 1 u, u ηη = ∆ ∞ u. (6)
From (??) it is now clear that ∆ p imposes the same diffusivity strength in both directions ξ and η independent of the magnitude of the gradient. In an attempt to resolve this drawback, Perona and Malik (P-M) [?] proposed to replace |∇u| p-2 with g(|∇u| 2 ) in (??). The idea is that the weight should satisfy g(s) → 0, s → ∞ Noisy ∆ (p(x),q(x)) TV PSNR: 22.2 36.2 28.3 SSIM: 0.30 0.96 0.94 ∆ (p(x),q(x)) TV Fig. 1: Synthetic test image with 20 standard deviations of noise (left) and the obtained results for TV (green/thick) and the ∆ (p(x),q(x)) -operator (red/dashed). We see that both error measures improve with our operator and we get less staircasing artifacts while preserving corner points and edges as shown in the detailed images. and g(s) → 1, s → 0 + . P-M studied weights like g = k/(k + s) and g = e -ks and demonstrated the advantages of these weight functions for edge preservations.

Rewriting the operator given by the P-M method in Gauge coordinates, we obtain

div g(|∇u| 2 )∇u = g(|∇u| 2 )u ξξ + φ(|∇u| 2 )u ηη , (7)
where φ(s) = (sg(s)) . Thus the diffusion in the direction η differs from the diffusion in the direction ξ. Since φ(s) is negative for large s, the evolution will be of backward diffusion effect near edges. This backward evolution cause problem for the well-posedness of the model and could also lead to staircasing problem [?].

The Perona-Malik PDE is a forward-backward type equation, and the diffusion is forward in the region {|∇u| < k} and is backward and hence ill-posed in the region {|∇u| > k}. From evolution point of view, the forward-backward type equation may have infinitely many solutions and from variational minimization perspective the energy functional may have infinitely many minima [?]. To overcome the ill-posedness in the P-M model the authors in [?] introduced a regularization and proposed the following model

∂ t u = div(g(|∇G σ * u|)∇u)),
where G σ is a Gaussian function with standard deviation σ. A similar but time dependent variance σ(t) was used in [?]. However, it's rather tricky to choose σ(t) since it should neither decay too fast nor too slow during the evolution.

3 Extending the p-Laplace Operator

Anisotropic decomposition via constant coefficients

The p-Laplace operator in (??), is an isotropic operator and expanding the divergence we obtain the equivalent form

∆ p u = ∂ x ([(∂ x u) 2 + (∂ y u) 2 ] p-2 2 ∂ x u) + ∂ y ([(∂ x u) 2 + (∂ y u) 2 ] p-2 2 ∂ y u). (8)
An anisotropic behavior of the ∆ p -operator is induced by suppressing mixed derivatives, i.e., we define

L (p,p) u = ∂ x (|∂ x u| p-2 ∂ x u) + ∂ y (|∂ y u| p-2 ∂ y u), 1 ≤ p < ∞. (9)
In the case p = 1, (??) is known as isotropic TV and (??) is anisotropic TV [?].

By decoupling the exponents in (??) one obtains the operator L (p1,p2)

L (p1,p2) u = ∂ x (|∂ x u| p1-2 ∂ x u) + ∂ y (|∂ y u| p2-2 ∂ y u), 1 ≤ p 1 , p 2 < ∞. ( 10
)
which has previously appeared in fluid mechanics and we refer to [?], [?].

Next, to see how the diffusion appears in the operator (??) we reformulate it in Gauge coordinates (??) by making the following definition.

Definition 1. The L (p1,p2) -operator is given by

L (p1,p2) u = ∂ ξ (|∂ ξ u| p1-2 ∂ ξ u) + ∂ η (|∂ η u| p2-2 ∂ η u), (11)
where

1 ≤ p 1 , p 2 < ∞.
The above operator L (p1,p2) is in fact a generalization of several known operators. We have 1. The case p 1 = p 2 = 2. The operator L (2,2) in (??) is the Laplacian, by now well studied. Due to the Laplacian's rotation invariance property we get

L (2,2) u = ∂ ξ (∂ ξ u) + ∂ η (∂ η u) = ∆u = u xx + u yy . (12)
2. The case p 1 = 2 and p 2 = 1. The operator L (p1,p2) in (??) is then given by

L (2,1) u = ∂ ξ (∂ ξ u) + ∂ η (|∂ η u| -1 ∂ η u). (13)
Since we have |∂ η u| -1 ∂ η u = 1, it follows that L (2,1) u = u ξξ . In Cartesian coordinates this corresponds to

L (2,1) u = |∇u|∆ 1 u, (14)
i.e. the mean curvature equation (see e.g., [?]). The corresponding regularized (weighted) mean curvature equation is given by

∂ t u = g(|∇G σ * u|)|∇u|∆ 1 u (15)
previously studied in the context of image analysis, see e.g., [?].

3. The case p 1 = 2 and p 2 = p ∈ (1, 2). It follows from (??) that

L (2,p) u = ∂ ξξ u + ∂ η (|∂ η u| p-2 ∂ η u) = ∂ ξξ u + (p -1)u p-2 η u ηη (16a) = |∇u|∆ 1 u + (p -1)|∇u| p-2 ∆ ∞ u (16b)
i.e. a mean curvature operator (??) with a second order term corresponding to (??). Note that the second order term induce invariant smoothing of the image data. Since (??) is merely a special case of (??), we further relax the mean curvature term next to better reflect the trade-off between edgepreservation and obtained smoothness. Although this modification appears straight-forward, its implications are non-trivial, however.

The anisotropic L (2,p) operator is given by Definition 2. The operator ∆ (p,q) is

∆ (p,q) u = |∇u| q ∆ 1 u + (p -1)|∇u| p-2 ∆ ∞ u, p ∈ [1, 2], q ≥ 0, (17)
Remark 1. Definition ?? is a straight-forward relaxation of the exponents in (??), motivated by the discussion in above point 3. The introduction of q in (??), defines an additional degree of freedom, allowing us to control the trade-off between ∆ 1 and ∆ ∞ , i.e., a trade-off between edge preservation and smoothness.

The corresponding evolution problem of the operator ∆ (p,q) is Definition 3. The evolution problem of ∆ (p,q) is given by

   ∂ t u -∆ (p,q) u = 0, p ∈ [1, 2], q ≥ 0 u(0) = u 0 ∂ n u = 0 (18)
We point out that q = 0, p = 1 yields the familiar isotropic TV regularizer and q = 1, p = 2 results in the heat equation, i.e., isotropic filtering. In the next section, we propose to couple p and q via a smooth non-linear transition function such that ∆ (p,q) can be thought of a spatially variant TV and isotropic regularizer.

4 Variable coefficients

Coefficient coupling

In the previous section we motivated the ∆ (p,q) operator and advocated to introduce variable coefficients, depending on the image data. The behavior of the operator ∆ (p(x),q(x)) that we seek, is the edge-preserving effect of TV leading to ∆ (p(x),q(x)) → ∆ (1,0) (the case of TV) as |∇u| → ∞. In regions with small gradients we define the (p(x), q(x))-coefficients such that the operator is a linear filter. leading to ∆ (p(x),q(x)) → ∆ (2,1) = ∆ (the case of isotropic diffusion) as |∇u| → 0. To see how the diffusion appears in ∆ (p(x),q(x)) we rewrite the operator in Gauge coordinates Lemma 1. The operator ∆ (p,q) in (??) can be written as

∆ (p,q) u = |∇u| q-1 ξ (D 2 u)ξ + (p -1)|∇u| p-q-1 η (D 2 u)η . ( 19
)
Proof. Using the relations in (??) and by rotation invariance of Laplacian, we obtain

∆u = u ξξ + u ηη = ∆ ∞ u + |∇u|∆ 1 u. ( 20
)
Now we observe that the operator ∆ 1 u can be expanded as

∆ 1 u = u 2 y u xx -2u x u y u xy + u 2 x u yy |∇u| 3 = (∇ ⊥ u) (D 2 u)∇ ⊥ u |∇u| 3 , ( 21
)
and using this in (??) gives

∆ ∞ u = |∇u| 2 ∆u -(∇ ⊥ u) (D 2 u)∇ ⊥ u = ∇ u(D 2 u)∇u. ( 22
)
Thus, the ∆ (p,q) operator in (??) reformulates to

∆ (p,q) u = |∇u| q-3 (∇ ⊥ u) (D 2 u)∇ ⊥ u + (p -1)|∇u| p-4 ∇ u(D 2 u)∇u = |∇u| q-1 ξ (D 2 u)ξ + (p -1)|∇u| p-2 η (D 2 u)η,
which shows the result.

Next, we couple p and q via the relation

p(|∇u|) = 1 + q(|∇u|) (23)
and from which we derive the following result Lemma 2. If p(x) = 1 + q(x), then

∆ (1+q(x),q(x)) u = |∇u| q(x)-1 ξ (D 2 u)ξ + q(x)η (D 2 u)η . ( 24
)
Proof. The proof follows immediately from Lemma ??.

In this study, we couple p and q via the negative exponential function (although other selections are possible), i.e., we set

q(|∇u|) = k 2 exp (-|∇u|/k 1 ) , (25a)
p(|∇u|) = 1 + q(|∇u|), (25b)
where k 1 > 0 and 0 < k 2 < 1. Under this selection we form the following parabolic PDE

∂ t u -|∇u| q(|∇u|)-1 ξ (D 2 u)ξ + q(|∇u|)η (D 2 u)η = 0. ( 26
)
One easily checks that (??) describes a smooth transition between total variation and linear filtering for the selection of p, q in (??). By using |∇u|∆ 1 u = ∆u -∆ ∞ u, the operator ∆ (1+q,q) becomes

∆ (1+q,q) u = |∇u| q(|∇u|)-1 T r(D 2 u) + q(|∇u|) -1 |∇u| 2 ∇u (D 2 u)∇u . (27)
Remark 2. The operator ∆ (1+q,q) is non-linear and have unbounded coefficients.

In this first study, we perturb the ∆ (1+q,q) -operator to obtain a regularized version of the evolution problem (??). This regularization enables us pose the necessary conditions for well-posedness, introduced next.

Regularity of solutions

In order to set up the framework for numerical calculation, we define Definition 4. Let 0 < ε, δ < 1 and q as in (??). Then

∆ ε,δ (1+q,q) u = (|∇u| 2 + ε 2 ) (q(|∇u|)-1)/2 (1 + δ)T r(D 2 u) + q(|∇u|) -1 |∇u| 2 + ε 2 ∇u (D 2 u)∇u . ( 28
)
Thus we study the regularized evolution equation

u t = ∆ ε,δ (1+q,q) u = i,j=1 a ε,δ ij (∇u)u ij (29)
where

a ε,δ ij (ζ) = (|ζ| 2 + ε 2 ) (q(|ζ|)-1)/2 (1 + δ)δ ij + q(|ζ|) -1 |ζ| 2 + ε 2 ζ i ζ j , (30)
and ζ = ∇u. Given u (k) , we obtain the next update u (k+1) by solving the following, equivalent (see [?]), initial value problem iteratively

     u (k+1) t = a ε,δ ij (∇u (k) )u (k+1) ij , in Ω × (0, T ) u (k+1) (0) = u 0 , in Ω ∂ n u (k+1) = 0, on ∂Ω × (0, T ) (31)
Proposition 1. If the initial data u (0) = 0 then the solution u (k) to (??) exists and is in C ∞ (Ω × (0, T )).

Proof. If the initial guess u (0) = 0, then a ε,δ ij (0)u ij = (1+δ)ε q(0)-1 ∆u and we deduce Thm 4.31 in [?] that u [START_REF] Alvarez | Image selective smoothing and edge detection by nonlinear diffusion[END_REF] exists and is C ∞ (Ω × (0, T )). Given u (k) ∈ C ∞ , then ||u (k) || C 1 (Ω×(0,T )) is bounded, (|ζ| 2 + ε 2 ) (q(|ζ|)-1)/2 is bounded from below and a ε,δ ij are also C ∞ (Ω × (0, T )). Hence, there are constants c = c(ε, δ, k) and

C = C(ε, δ, k) > 0, depending only on ε, δ, ||u (k) || C 1 , such that c|ζ| 2 ≤ a ε,δ ij (∇u (k) )ζ i ζ j ≤ C|ζ| 2 .
It follows once again from Thm 4.31 in [?] that u (k+1) exists and also is C ∞ (Ω × (0, T )). -operator where we stopped the filtering process at the maximum SSIM value. "TV"-was obtained using the Split Bregman method [?]. We set k 1 = k 2 = 0.1 in ∆ (p(x),q(x)) . See text for details.

Evaluation

Implementation

The parabolic PDE is discretized by using finite differences and a simple forward Euler scheme. We used a state of the art Split Bregman (SB) implementation of total variation. For details on SB see [?]. The regularization parameter for the SB was optimized in the range [0.01, 1] in increasing steps of 0.11. The regularization parameter of the Bregman-variables of SB was set to 1 and the scheme was terminated as 10 -3 > ||u (k) -u (k-1) || 2 /||u (k-1) || 2 and we choose the regularization parameter that produces highest SSIM value [?]. We also report the peak signal-to-noise value (PSNR). For the (p(x), q(x))-operator we found that k 1 = k 2 = 0.1 and the update stepsize as α = 10 -5 and τ = 0.5 works well for the considered noise level of 20 standard deviations of additive Gaussian noise. These values are ad-hoc and future work include methods for parameter estimation.

Results

First we test our algorithm on a synthetic test image seen in figure ??. We see that the result of the proposed operator appears smoother than the result of Original Noisy (P: 22.1, S: 0.54) ∆ (p(x),q(x)) TV PSNR (P): 25.9 24.0 SSIM (S): 0.81 0.74 ∆ (p(x),q(x)) TV ∆ (p(x),q(x)) TV Fig. 3: Example denoising a grayscale image with 20 standard deviations of noise.

In the close-up images to the right it can be seen that TV (thick/green) produces the characteristic staircasing effect while the operator ∆ (p,q) (dashed/red) shows good visual similarity with the noise free patch. TV shows good result in the sky (up right), but oversmooths, e.g., the window tiles seen in the close-up down right.

Conclusion

In this paper we introduced a new family of operators, ∆ (p,q) . Preliminary numerical results indicate that there could be a relationship between p(x) and q(x) that further improves the restoration effect. In forthcoming works we will investigate the operator ∆ (p,q) further regarding both regularity and different areas of applications.

Fig. 2 :

 2 Fig.2: Example results for the ∆ (p,q) -operator where we stopped the filtering process at the maximum SSIM value. "TV"-was obtained using the Split Bregman method [?]. We set k 1 = k 2 = 0.1 in ∆ (p(x),q(x)) . See text for details.

Acknowledgment Support by the German Science Foundation and the Research Training Group (GRK 1653) is gratefully acknowledged.


M Hassan

Noriaki Yamazaki

Optimal control of doubly nonlinear evolution equations governed by subdifferentials without uniqueness of solutions

Keywords: Optimal control, doubly nonlinear evolution equations, subdifferentials, without uniqueness

In this paper we study an optimal control problem for a doubly nonlinear evolution equation governed by time-dependent subdifferentials. We prove the existence of solutions to our equation. Also, we consider an optimal control problem without uniqueness of solutions to the state system. Then, we prove the existence of an optimal control which minimizes the nonlinear cost functional. Moreover, we apply our general result to some model problem.

Introduction

The present paper is concerned with an optimal control problem without uniqueness of solutions to a doubly nonlinear evolution equation governed by timedependent subdifferentials in a real Hilbert space H.

In our optimal control problem, for each control f , the state system (P; f ) is as follows: State system (P; f ):

(P; f ) ∂ψ t (u (t)) + ∂ϕ(u(t)) + g(u(t)) f (t) in H for a.e. t ∈ (0, T ), u(0) = u 0 in H, (1.1)
where 0 < T < ∞, u = du/dt in H, ψ t : H → R ∪ {∞} is a time-dependent proper, l.s.c. (lower semi-continuous), convex function for each t ∈ [0, T ], ϕ : H → R ∪ {∞} is a time-independent proper, l.s.c., convex function, ∂ψ t and ∂ϕ are their subdifferential in H, g(•) is a single-valued Lipschitz operator in H, f is a given H-valued control function and u 0 ∈ H is a given initial data.

In this present paper, we consider the optimal control problem without uniqueness of solutions to the state system (P; f ). To this end, let V be a real Hilbert space such that the embedding V → H is dense and compact. Then, we study the following optimal control problem without uniqueness of solutions to (P; f ): Problem (OP): Find the optimal control f * ∈ F such that

J(f * ) = inf f ∈F J(f ).
Here F := W 1,2 (0, T ; H) ∩ L 2 (0, T ; V ) is the control space and J(f ) is the cost functional defined by

J(f ) := inf u∈S(f ) π f (u), (1.2)
where f ∈ F is the control, S(f ) is the set of all solutions to (P; f ) with the control function f . Also, u is a solution to the state system (P; f ) and π f (u) is its functional defined by

π f (u) := 1 2 T 0 |u(t) -u ad | 2 H dt + 1 2 T 0 |f (t)| 2 V dt + 1 2 T 0 |f t (t)| 2 H dt, (1.3)
where

u ad ∈ L 2 (0, T ; H) is a given target profile and | • | H (resp. | • | V ) is the norm of H (resp. V ).
There is vast literature on optimal control problems to (parabolic or elliptic) variational inequalities. For instance, we refer to [START_REF] Barbu | Optimal control of variational inequalities[END_REF][START_REF] Hoffmann | Optimal control problems for ellipticparabolic variational inequalities with time-dependent constraints[END_REF][START_REF] Hu | Time-dependent subdifferential evolution inclusions and optimal control[END_REF][START_REF] Lions | Optimal control of systems governed by partial differential equations[END_REF][START_REF] Lions | Contrôle des systèmes distribués singuliers[END_REF][START_REF] Neittaanmäki | Optimization of Elliptic Systems: Theory and Applications[END_REF][START_REF] Yamazaki | Convergence and optimal control problems of nonlinear evolution equations governed by time-dependent operator[END_REF]. In particular, Lions [START_REF] Lions | Contrôle des systèmes distribués singuliers[END_REF] and Neittaanmäki et al. [START_REF] Neittaanmäki | Optimization of Elliptic Systems: Theory and Applications[END_REF]Section 3.1.3.1] discussed the singular control problems, which is the class of control problems characterized by not well-posed state systems.

The theory of nonlinear evolution equations are useful in the systematic study of variational inequalities. For instance, many mathematicians studied the nonlinear evolution equation of the form:

u (t) + ∂ϕ t (u(t)) f (t) in H for a.e. t ∈ (0, T ), (1.4)
where ϕ t (•) : H → R ∪ {∞} is a proper, l.s.c. and convex function. For various aspects of (1.4), we refer to [START_REF] Hu | Time-dependent subdifferential evolution inclusions and optimal control[END_REF][START_REF] Kenmochi | Solvability of nonlinear evolution equations with time-dependent constraints and applications[END_REF][START_REF] Ôtani | Nonlinear evolution equations with time-dependent constraints[END_REF][START_REF] Yamada | On evolution equations generated by subdifferential operators[END_REF]. In particular, Hu-Papageorgiou [START_REF] Hu | Time-dependent subdifferential evolution inclusions and optimal control[END_REF] studied the optimal control problems to (1.4). Also, doubly nonlinear evolution equations were studied. For instance, Kenmochi-Pawlow [START_REF] Kenmochi | A class of nonlinear elliptic-parabolic equations with time-dependent constraints[END_REF] studied the following type of doubly nonlinear evolution equations:

d dt ∂ψ(u(t)) + ∂ϕ t (u(t)) f (t) in H for a.e. t ∈ (0, T ), (1.5)
where ψ(•) : H → R ∪ {∞} is a proper, l.s.c. and convex function. The abstract results of doubly nonlinear evolution equations (1.5) can be applied to ellipticparabolic equations. Therefore, from the view point of (1.5), Hoffmann et al. [START_REF] Hoffmann | Optimal control problems for ellipticparabolic variational inequalities with time-dependent constraints[END_REF] studied optimal control problems for quasi-linear elliptic-parabolic variational inequalities with time-dependent constraints. Also, Kadoya-Kenmochi [START_REF] Kadoya | Optimal Shape Design in Elliptic-Parabolic Equations[END_REF] studied the optimal sharp design of elliptic-parabolic equations.

On the other hand, Akagi [START_REF] Akagi | Doubly nonlinear evolution equations with non-monotone perturbations in reflexive Banach spaces[END_REF], Arai [START_REF] Arai | On the existence of the solution for ∂ϕ(u (t)) + ∂ψ(u(t)) f (t)[END_REF], Aso et al. [START_REF] Aso | Phase change problems with temperature dependent constraints for the volume fraction velocities[END_REF][START_REF] Aso | Quasivariational evolution inequalities for a class of reaction-diffusion systems[END_REF], Colli [START_REF] Colli | On some doubly nonlinear evolution equations in Banach spaces[END_REF], Colli-Visintin [START_REF] Colli | On a class of doubly nonlinear evolution equations[END_REF], Senba [START_REF] Senba | On some nonlinear evolution equation[END_REF] investigated the following type of doubly nonlinear evolution equations (cf. (1.1)): ∂ψ t (u (t)) + ∂ϕ(u(t)) f (t) in H for a.e. t ∈ (0, T ).

(1.6)

However, there was no result of optimal control for (1.1) and (1.6) since (1.1) and (1.6) are not well-posed state systems, in general. Therefore, by arguments similar to Kadoya et al. [START_REF] Kadoya | A class of nonlinear parabolic systems with environmental constraints[END_REF], more precisely, using the cost functional defined by (1.2) and (1.3), we establish the abstract theory of the optimal control problem (OP) without uniqueness of solutions to the state system (1.1). The plan of this paper is as follows. In the next Section 2, we state the main result in this paper. In Section 3, we first give the sketch of the proof of solvability for (1.1). Also, we prove the convergence result (Proposition 3) of solutions to (P; f ). Moreover, we prove the main result (Theorem 1) on the existence of the optimal control to (OP). In the final Section 4, we apply our abstract result to a parabolic PDE with Neumann boundary condition.

Notations

Throughout this paper, let H be a real Hilbert space with the inner product (•, •) and norm | • | H , respectively. Also, let V be a real Hilbert space with the norm | • | V such that the embedding V → H is dense and compact.

Let us here prepare some notations and definitions of subdifferential of convex functions. To this end, let E be a real Hilbert space with the inner product (•, •) E . Then, for a proper (i.e., not identically equal to infinity), l.s.c. and convex function φ : E → R ∪ {∞}, the effective domain D(φ) is defined by

D(φ) := {z ∈ E; φ(z) < ∞}.
The subdifferential of φ is a possibly multi-valued operator in E and is defined by z * ∈ ∂φ(z) if and only if

z ∈ D(φ) and (z * , y -z) E ≤ φ(y) -φ(z) for all y ∈ E.
The next proposition is concerned with the closedness of maximal monotone operator ∂φ in E.

Proposition 1 (cf. [7, Lemma 1.2]). Let E be a real Hilbert space with the inner product (•, •) E . Let φ : E → R∪{∞} be a proper, l.s.c. and convex function.

Also, let [z n , z * n ] ∈ ∂φ and [z, z * ] ∈ E × E be such that z n → z weakly in E and z * n → z * weakly in E as n → ∞. Suppose that lim sup n→∞ (z n , z * n ) E ≤ (z, z * ) E . Then, it follows that [z, z * ] ∈ ∂φ and lim n→∞ (z n , z * n ) E = (z, z * ) E .
For various properties and related notions of the proper, l.s.c., convex function φ and its subdifferential ∂φ, we refer to a monograph by Brézis [START_REF] Brézis | Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert[END_REF].

Main Theorem

We begin by defining the notion of a solution to (P; f ).

Definition 1. Given f ∈ L 2 (0, T ; H) and u 0 ∈ H, the function u : [0, T ] → H is called a solution to (P;f ) on [0, T ],
if the following conditions are satisfied:

(i) u ∈ W 1,2 (0, T ; H). (ii) There exist functions ξ ∈ L 2 (0, T ; H) and ζ ∈ L 2 (0, T ; H) such that ξ(t) ∈ ∂ψ t (u (t)) in H for a.e. t ∈ (0, T ), ζ(t) ∈ ∂ϕ(u(t)) in H for a.e. t ∈ (0, T ) and ξ(t) + ζ(t) + g(u(t)) = f (t) in H for a.e. t ∈ (0, T ). (iii) u(0) = u 0 in H.
Now, we give the assumptions on ψ t , ϕ and g.

(A1) For each t ∈ [0, T ], ψ t (•) : H → R ∪ {∞}
ψ t (z) ≥ C 1 |z| 2 H , ∀t ∈ [0, T ], ∀z ∈ D(ψ t ).
(A3) There exists a positive constant C 2 > 0 such that

|z * | 2 H ≤ C 2 (ψ t (z) + 1), ∀[z, z * ] ∈ ∂ψ t , ∀t ∈ [0, T ].
(A4) There are functions α ∈ W 1,2 (0, T ) and β ∈ W 1,1 (0, T ) satisfying the following property: for any s, t ∈ [0, T ] with s ≤ t and z ∈ D(ψ s ), there exists z ∈ D(ψ t ) such that

|z -z| H ≤ |α(t) -α(s)| 1 + ψ s (z) 1 2 , ψ t (z) -ψ s (z) ≤ |β(t) -β(s)| (1 + ψ s (z)) .
(A5) There exists a positive constant

C 3 > 0 such that ϕ(z) ≥ C 3 |z| 2 H , ∀z ∈ D(ϕ). (A6) For each r > 0, the level set {z ∈ H; ϕ(z) ≤ r} is compact in H. (A7) g : H → H is a single-valued Lipschitz operator. Namely, there is a positive constant L g > 0 such that |g(z 1 ) -g(z 2 )| H ≤ L g |z 1 -z 2 | H , ∀z i ∈ H (i = 1, 2). Remark 1.
The assumption (A4) is the standard time-dependence condition of convex functions (cf. [START_REF] Kenmochi | Solvability of nonlinear evolution equations with time-dependent constraints and applications[END_REF][START_REF] Ôtani | Nonlinear evolution equations with time-dependent constraints[END_REF][START_REF] Yamada | On evolution equations generated by subdifferential operators[END_REF]).

By a slight modification of [START_REF] Akagi | Doubly nonlinear evolution equations with non-monotone perturbations in reflexive Banach spaces[END_REF][START_REF] Aso | Phase change problems with temperature dependent constraints for the volume fraction velocities[END_REF], we can prove the following existence result for problem (P;f ). We give a sketch of its proof in Section 3.

Proposition 2 (cf. [1, Theorem 3.2], [3, Theorem 2.1]). Assume (A1)-(A7). Then, for each u 0 ∈ D(ϕ) and f ∈ L 2 (0, T ; H), there exists at least one solution u to (P;f ) on [0, T ]. Moreover, there exists a positive constant N 0 > 0, independent of u 0 , such that Now, we state the main result of this paper, which is directed to the existence of an optimal control to (OP) without uniqueness of solutions to (P;f ). Theorem 1. Assume (A1)-(A7) and u 0 ∈ D(ϕ). Let u ad be an element in L 2 (0, T ; H). Then, (OP) has at least one optimal control f * ∈ F such that

T 0 ψ t (u (t))dt + sup t∈[0,T ] ϕ(u(t)) ≤ N 0 ϕ(u 0 ) + |f | 2 L 2 (0,T ;H) + 1 . ( 2
J(f * ) = inf f ∈F J(f ).

Proof of Main Theorem 1

In this section, we give the sketch of the proof of Proposition 2 by arguments similar to Akagi [START_REF] Akagi | Doubly nonlinear evolution equations with non-monotone perturbations in reflexive Banach spaces[END_REF] and Aso et al. [START_REF] Aso | Phase change problems with temperature dependent constraints for the volume fraction velocities[END_REF]. Moreover, we prove Theorem 1.

Throughout this section, we suppose that all the assumptions of Theorem 1 hold.

Sketch of the proof of Proposition 2.

By arguments similar to Akagi [START_REF] Akagi | Doubly nonlinear evolution equations with non-monotone perturbations in reflexive Banach spaces[END_REF] and Aso et al. [START_REF] Aso | Phase change problems with temperature dependent constraints for the volume fraction velocities[END_REF], we can prove Proposition 2. In fact, for each λ ∈ (0, 1], we consider the following approximate problem for (P;f ), denoted by (P;f ) λ :

(P; f ) λ    λu λ (t) + ∂ψ t (u λ (t)) + ∂ϕ λ (u λ (t)) + g(J ϕ λ u λ (t)) f (t) in H for a.e. t ∈ (0, T ), u λ (0) = u 0 in H,
where ∂ϕ λ and J ϕ λ := (I + λ∂ϕ) -1 denote the Yosida approximation and the resolvent of ∂ϕ , respectively.

By Cauchy-Lipschitz-Picard's existence theorem and the fixed point argument for compact operators (e.g. the Schauder's fixed point theorem), we can prove the existence of solutions u λ to (P; f ) λ on [0, T ].

From the standard calculation, we can establish a priori estimate (cf. (2.1)) of solutions u λ to (P; f ) λ with respect to λ ∈ (0, 1]. Therefore, by the limiting procedure of solutions u λ to (P; f ) λ as λ → 0, we can construct the solution to (P; f ) on [0, T ] satisfying the boundedness estimate (2.1). For a detailed argument, see [1, Sections 4 and 5] or [3, Sections 3 and 4], for instance.

Here, let us mention the result of the convergence of solutions to (P; f ), which is a key proposition to proving Theorem 1.

Proposition 3. Assume (A1)-(A7). Let {f n } ⊂ L 2 (0, T ; H), {u 0,n } ⊂ D(ϕ), f ∈ L 2 (0, T ; H) and u 0 ∈ D(ϕ). Assume that f n → f strongly in L 2 (0, T ; H), (3.1)
u 0,n → u 0 in H and ϕ(u 0,n ) → ϕ(u 0 ) (3.2)
as n → ∞. Let u n be a solution to (P;f n ) on [0, T ] with initial data u 0,n . Then, there exist a subsequence {n k } ⊂ {n} and a function u ∈ W 1,2 (0, T ; H) such that u is a solution to (P;f ) on [0, T ] with initial data u 0 and

u n k → u in C([0, T ]; H) as k → ∞. (3.3)
Proof. From the bounded estimate (2.1), (A2), (A5) and the level set compactness of ϕ (cf. (A6)), we derive that there are a subsequence {n k } of {n} and a function u ∈ W 1,2 (0, T ; H) such that n k → ∞,

u n k → u weakly in W 1,2 (0, T ; H), in C([0, T ]; H), weakly- * in L ∞ (0, T ; H)      (3.4)
as k → ∞. Hence, we observe from (3.2) and (3.4) that u(0) = u 0 in H. Now, let us show that u is a solution of (P; f ) on [0, T ] with initial data u 0 . Since u n k is a solution of (P; f n k ) on [0, T ] with initial data u 0,n k , there exist functions ξ n k ∈ L 2 (0, T ; H) and ζ n k ∈ L 2 (0, T ; H) such that

ξ n k (t) ∈ ∂ψ t (u n k (t)) in H for a.e. t ∈ (0, T ), (3.5)
ζ n k (t) ∈ ∂ϕ(u n k (t)) in H for a.e. t ∈ (0, T ), (3.6)
ξ n k (t) + ζ n k (t) + g(u n k (t)) = f n k (t) in H for a.e. t ∈ (0, T ). (3.7)
Then, it follows from (2.1) and (A3) that

{ξ n k } is bounded in L 2 (0, T ; H). (3.8)
Therefore, taking a subsequence if necessary (still denote it by {n k }), we observe that:

ξ n k → ξ weakly in L 2 (0, T ; H) for some ξ ∈ L 2 (0, T ; H) as k → ∞. (3.9) Also, it follows from (A7) and (3.4) and that

g(u n k ) → g(u) in C([0, T ]; H) as k → ∞. ( 3
{ζ n k } is bounded in L 2 (0, T ; H).
Hence, taking a subsequence if necessary (still denote it by {n k }), we observe that:

ζ n k → ζ weakly in L 2 (0, T ; H) for some ζ ∈ L 2 (0, T ; H) as k → ∞. (3.11)
Thus, we infer from (3.1), (3.7), (3.9), (3.10) and (3.11) that:

ξ + ζ + g(u) = f in L 2 (0, T ; H). (3.12)
Also, from (3.4), (3.6), (3.11) and the demi-closedness of maximal monotone operator ∂ϕ (cf. Proposition 1), we infer that

ζ ∈ ∂ϕ(u) in L 2 (0, T ; H), (3.13
(f n k (t) -ζ n k (t) -g(u n k (t)), u n k (t))dt = lim sup k→∞ T 0 (f n k (t) -g(u n k (t)), u n k (t))dt - T 0 d ds ϕ(u n k (s))ds ≤ T 0 (f (t) -g(u(t)), u (t))dt + lim sup k→∞ (-ϕ(u n k (T )) + ϕ(u 0,n k )) ≤ T 0 (f (t) -g(u(t)), u (t))dt -ϕ(u(T )) + ϕ(u 0 ) = T 0 (f (t) -g(u(t)) -ζ(t), u (t))dt = T 0 (ξ(t), u (t))dt,
thus, we observe from Proposition 1, namely, the closedness of maximal monotone operator ∂ψ t that ξ ∈ ∂ψ t (u ) in L 2 (0, T ; H), which implies that (3.14) holds. Therefore, we observe that u is a solution of (P; f ) on [0, T ] with initial data u 0 . Thus, the proof of this proposition has been completed. Now, let us prove the main Theorem 1 in our paper, which is the existence of an optimal control to (OP). Proof of Theorem 1. Note that we show the existence of an optimal control to (OP) without uniqueness of solutions to state problem (P; f ) Also note from (1.2) and (1.3) that J(f ) ≥ 0 for all f ∈ F. Let {f n } ⊂ F be a minimizing sequence such that

d * := inf f ∈F J(f ) = lim n→∞ J(f n ).
Then, we observe that {J(f n )} is bounded. Therefore, by the definition (1.2) of J(f n ), for each n there is a solution u n ∈ S(f n ) such that

π fn (u n ) < J(f n ) + 1 n .
Hence, we observe that {π fn (u n )} is bounded. Thus, by the definition of π fn (u n ) (cf.

d * = inf f ∈F J(f ) ≤ J(f * ) = inf u∈S(f * ) π f * (u) ≤ π f * (u * ) = 1 2 T 0 |u * (t) -u ad | 2 H dt + 1 2 T 0 |f * (t)| 2 V dt + 1 2 T 0 |f * t (t)| 2 H dt ≤ lim inf k→∞ π fn k (u n k ) ≤ lim inf k→∞ J(f n k ) + 1 n k = lim k→∞ J(f n k ) = d * .

  is a proper, l.s.c. and convex function. Also, ϕ(•) : H → R ∪ {∞} is a proper, l.s.c. and convex function. (A2) There exists a positive constant C 1 > 0 such that

. 10 )

 10 Therefore, we infer from (3.1), (3.7), (3.8) and (3.10) that

  (1.3)) and by the Aubin's compactness theorem (cf.[START_REF] Lions | Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires[END_REF] Chapter1, Section 5]), there are a subsequence {n k } ⊂ {n} and a function f * ∈ F such thatf n k → f * weakly in W 1,2 (0, T ; H),weakly in L 2 (0, T ; V ), in L 2 (0, T ; H) Now, taking a subsequence if necessary, we infer from Proposition 3 that there is a solution u * to (P;f * ) on [0, T ] with initial data u 0 satisfyingu n k → u * in C([0, T ]; H) as k → ∞. (3.16) Therefore, it follows from (3.15)-(3.16), u * ∈ S(f * ) and the weak lower semicontinuity of L 2 -norm that

This work was supported by Grant-in-Aid for Scientific Research (C) No. 26400164 and No. 26400179, JSPS

Hence, we have d * = inf f ∈F J(f ) = J(f * ), which implies that f * ∈ F is an optimal control to (OP). Thus, the proof of Theorem 1 has been completed.

Application

In this section, we apply Theorem 1 to the simple model problem as follows:

ν is an outward normal vector on Γ and u 0 is a given initial data. Also, A(t, •) is the given time-dependent function defined by

To apply the abstract result to (P;f ), we put H := L 2 (Ω) and V := H 1 (Ω) with usual real Hilbert space structures. Define a function ϕ on H by

Also, for each t ∈ [0, T ], define a function ψ t on H by

where Â(t, •) is the primitive of A(t, •) such that Â(t, •) ≥ 0 for all t ∈ [0, T ]. It is not difficult to show that the assumptions (A1)-(A7) are satisfied. For instance, put z = z -c(s) + c(t), α(t) := t 0 |c (τ )|dτ and β(t) ≡ 0 for (A4) (cf. [START_REF] Kenmochi | Solvability of nonlinear evolution equations with time-dependent constraints and applications[END_REF]Chapter 3]). Therefore, by applying Theorem 1, we can consider the control problem (OP) without uniqueness of solutins to (SMP) p .