Optimal control of the FitzHugh-Nagumo stochastic model with nonlinear diffusion

We consider the existence and first order conditions of optimality for a stochastic optimal control problem inspired by the celebrated FitzHugh-Nagumo model, with nonlinear diffusion term, perturbed by a linear multiplicative Brownian-type noise. The main novelty of the present paper relies on the application of the {\it rescaling method} which allows us to reduce the original problem to a random optimal one.

tion defined on a filtered probability space Ω, F , {F t } t≥0 , P , while {e n } ≥1 is an orthonormal basis in H −1 and µ n ∈ R.
Since the Laplacian operator ∆ ξ is a linear operator in L 2 (O), and −∆ ξ is self-adjoint, then there exists a complete orthonormal system {ē k } k≥1 in L 2 (O) of eigenfunctions of −∆ ξ , and we shall indicate the corresponding sequence of eigenvalues denoted by {λ k } k≥1 . Therefore, we have Also, we set and note that G is monotonically nondecreasing. The present paper addresses the problem of existence and uniqueness of a strong solution, in a sense to be better specified in a while, to equation (1). We stress that this is not a trivial problem as the nonlinear operator ∆γ is naturally defined on the space H −1 whereas the nonlinear polynomial perturbation I ion is not m−accreative on the same space. In order to solve above problem we will transform the original equation, via a rescaling transformation, to a random PDE. It turns out that the existence and uniqueness of transformed random PDE can be treated by the theory of nonlinear semigroup in L 1 .
We will further consider the problem of existence of an optimal control for the nonlinear FHN equation. Again, in order to solve the problem we will apply a rescaling transformation to obtain a corresponding random PDE. As already emerged in [5,21], the nonlinear polynomial term implies that standard minimization argument does not apply. Therefore, existence of an optimal control is achieved using Ekelands variational principle. First order conditions of optimality are given in terms of dual stochastic backward equation, see, e.g, [5,16], whereas, due to the applied rescaling transformation are expressed in terms of a random backward dual equation which allows to simplify the setting also giving more insights on the derived optimal controller.
The present paper is structured as follows: Section 1.1 introduces main notation used thorough the paper. Section 2 addresses the problem of proving existence and uniqueness for the state equation whereas in Section 3 the problem of the existence for an optimal control is considered.

Main notations
In what follows we will denote by | · |, resp. ·, · , the norm, resp. scalar product, R d . Also, L p (O) =: L p , for 1 ≤ p ≤ ∞, is the standard space of p−Lebesgue measurable function over the domain O ⊂ R d , with corresponding norm defined as | · | p . For the case p = 2, we will further denote by ·, · 2 the scalar product in L 2 . The space The dual of the space H 1 will be denoted as H −1 equipped with corresponding norm | · | −1 .
Similarly, we will denote by W n,p (O) =: W n,p , n ∈ N, 1 ≤ p ≤ ∞, the standard Sobolev space of p−integrable functions with p−integrable n−order derivatives. Coherently, W 1,p ([0, T ]; H −1 ) will be the space of absolutely continuous function u : In an analogous manner L 2 Above definition are still in place if instead of H −1 we consider a general Hilbert space H. It is also known that there is a natural embedding of We therefore can rewrite equation (1) as We will assume the following to hold.
Then, we can state the notion of solution to equation (3) that we will consider in subsequent analysis. Definition 1.2.1. Let x ∈ H −1 , we say that the process 2 Existence for the state equation The main problem in proving existence and uniqueness for a solution to equation (3) is that the operator G in not m-accretive on the space H −1 and so basic existence results in [6,7] are not applicable in the present case. It turns out that the proper space one has to consider to successfully treat equation (3) is the space L 1 , which, in turn, is not the proper one if one has to deal with SPDEs such as (3).
To overcome such a stalemate, we follow [8,9]. In particular, we apply the transformation X = e W y, which allows to reduce the stochastic equation (3) to a random PDE that can be treated with analytical techniques. In fact, the random equation can be successfully solved by exploiting the theory of nonlinear semigroup in L 1 . As noted in [8], we have still to face the problem that, because of the non regularity of the term W , the general theory cannot be applied straightforward to the resulting random PDE. Therefore, for ǫ > 0, we shall consider a suitable sequence of regular approximations W ǫ of W , to first establish a priori estimates for solutions y ǫ of the associated W ǫ −approximating problem, and then to show that, in the limit ǫ → 0, we obtain both existence and uniqueness of the solution for the original equation.
The following theorem constitutes the main result of this section.
Theorem 2.1. There is a unique strong solution to equation (3) X = e W y which satisfies In order to prove Theorem 2.1 we need some auxiliary lemmas. In particular, let us then introduce the transformation so that by an application of the Itô formula we obtain the random equation see, e.g. [8,9,12]. Following [8], we prove the existence of a unique strong solution to equation (5) by first considering an approximating problem. In particular, let us denote by β ǫ (t) : we thus have that For each ǫ > 0, let us thus consider the approximating equation associated to (5) where G ǫ is the Yosida approximation of G, that is Note that, G ǫ is monotoniccaly nondecreasing, Lipschitzian and lim ǫ→0 G ǫ (z) = G(z) , ∀z ∈ R uniformly on compacts. Defining z ǫ := e Wǫ y ǫ , equation (6) becomes where F ǫ := e −Wǫ F .
has a unique solution such that Proof. Let us first prove existence and uniqueness of a solution to equation (8) in the space H −1 . For a fixed ǫ > 0, let us define the operator A : We equip the space H −1 with the scalar product is m−accreative in the space H −1 , see, e.g., [3, p. 68], we have that, for a suitable α = α ǫ , it holds Moreover, for λ > 0 sufficiently large, we also have R ((λ + α)I + A) = H −1 , so that A is quasi-m-accretive. In other words, for f ∈ H −1 the equation has a unique solution in z ∈ L 2 . Indeed, introducing the operators we see that equation (10) can be rewritten as Since B is m-accretive and Γ is m-accretive and continuous in L 2 , it follows, see, e.g., [3, p.104], that R(ǫI + B + Γ) = L 2 , so that equation (11) admits a unique solution z in L 2 . Moreover, since γ(z) + ǫz ∈ H 1 and the inverse map of z → γ(z) + ǫz is Lipschitz, then z ∈ D(A and consequently z ǫ ∈ L ∞ (0, T ; L 2 ). Moreover we have that which implies that γ(z ǫ )+ǫz ǫ ∈ L ∞ (0, T ; H 1 0 ) and consequently z ǫ ∈ L ∞ (0, T ; H 1 0 ).
Proof. Let α ∈ C 1 ([0, T ]), such that α(0) = 0 and α ′ ≥ 0. Then, defining then, denoting by Moreover, by hypothesis 1.2, it follows that G ǫ is monotone, so that Proof. In what follows we will use the following with j(r) = r 0 γ(s)ds, r ∈ R + . Thus, multiplying equation (6) by γ(y ǫ ) and integrating over ( Concerning the last integral in the right hand side of equation (14), we have recalling that G ǫ is the Yosida approximant of G and using the monotonicity of γ and G ǫ that Lemma 2.3, while the other terms in equation (14) can be studied as done in [8,Lemma 3.3], so that the claim follows by Lemma 2.3.

Lemma 2.5.
There is a unique solution to equation (5) with Moreover, the process y is (F t ) t≥0 −adapted.
Then, since it holds then, for ǫ → 0, we get Thus, again from the fact that G : R → R is maximal monotone it follows that it is also closed and therefore we have that ζ = G(e W y).
Therefore, by letting ǫ → 0, from equation (6) we obtain Then, by the uniqueness result already proved, we also have that the sequence (y ǫ ) is independent of ω ∈ Ω, implying that y is (F t ) −adapted, ending the proof.
We can finally prove that it exists a unique strong solution X to equation (3) which satisfies Proof of Theorem 2.1. Using [9, Lemma 8.1] we have the equivalence between the stochastic PDE 3 and the random PDE 5 via the rescaling transformation 4, so that existence and uniqueness of a solution X in the sense of Definition 1.2.1 follows by Lemma 2.5.

The optimal control problem
In this section we will focus the attention to a controlled version of equation (1). We denote by X = L 2 ad ((0, T ) × O) the space of all F t −adapted processes u : [0, T ] → R d , and we consider the following optimal control problem Here F 0 (Ω) and α > 0 are given. In what follows we are going to treat the problem (P) by a rescaling procedure which allows us to reduce it to a random optimal control problem.
Theorem 3.1. Let hypothesis 1.2 holds, then, for T sufficiently small, there exists at least one optimal pair (u * , v * ) solution to problem (P).
Proof. As in Section 2,we will apply the rescaling transformation y := e −W v so that the optimal control problem (P) reads subject to Existence and uniqueness for a solution to equation (23) follows from Lemma 2.5.
By equation (26) we have P − a.s., hence, multiplying equation (27) by (−∆) −1 (y ǫ − y λ ) and integrating over O, it holds where η is defined as in (19). While, the first four integrals in the right hand side of equation (28) can be bounded similarly as done proving Lemma 2.5, see (19), the last two terms can be treated exploiting the Young inequality as to obtain Applying the Gronwall lemma and taking the mean value, we have Regarding the second equation in (26), we obtain Then, multiplying equation (32) by (p ǫ − p λ ) and integrating over O, we obtain 2(e W (y ǫ − y λ ))(p ǫ (s) − p λ (s))dξds .
Rearranging terms above, we further have Therefore, by using the Young inequality, we have where we have used C to denote possibly different constants as to simplify notation. Taking the expectation in equation (34) and combining it with equation (31), we thus have so that, if T is small enough, we can infer that implying that (y ǫ , p ǫ ) is a Cauchy sequence, therefore ,along a subsequence still denoted by {ǫ} → 0 for the sake of clarity, we have Letting then ǫ → 0 in the first equation in (26), we have hence, since Ψ is lower-semicontinuous, previous computations give us: and the claimed existence result follows.
Theorem 3.2 (Necessary condition of optimality). Let be (v * , u * ) an optimal pair for problem (P), then if α > 0 it holds where p is the solution to the dual backward equation (41) and Remark 3.3. We would like to underline that in literature about stochastic control problem, the first order conditions of optimality (the Pontryagin maximum principle) are expressed in terms of dual stochastic backward equation, see, e.g, [5,16]. Here, instead, optimality conditions are given in terms of a random backward dual equation which allows to simplify the setting also giving more insights on the derived optimal controller. Proof. We provide the result exploiting the rescaling transformation y := e −W X, hence proving necessary condition for the problem (P2).
Theorem 3.4 (The bang-bang principle). Let be (v * , u * ) an optimal pair for problem (P) and let α = 0, then it holds where p is the solution to the dual backward equation (41).
Proof. Proceeding as in Theorem 3.2 with obtain the equivalent of equation (42)to be which yields equation (43), and the claim follows.