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On an optimal control problem of time-fractional advection-diffusion equation

We consider an optimal control problem of an advection-diffusion equation with Caputo time-fractional derivative. By convex duality method we obtain as optimality condition a forward-backward coupled system. We then prove the existence of a solution to this coupled system using Schauder fixed point theorem. The uniqueness of the solution is also established under certain monotonicity condition on the cost functional.

1. Introduction. In this paper we study the following time-fractional advectiondiffusion (AD) equation Throughout this paper, we always assume that β ∈ (0, 1]. We consider optimal control problem with the cost functional: As optimality condition we obtain the backward-forward system of coupled equations, F is the primitive of f with respect to m: Here ∂ β [t,T ) and ∂ β (0,t] denote the backward and forward Caputo fractional derivatives. To avoid both compactness and boundary conditions issues, instead working on a domain or R d , we will consider the case of the flat torus T d := R d /Z d . This choice is standard for works in forward-backward systems, e.g. [14]. The term f associates to a probability density m a real valued function f (x, m). v = v(t, x) is time dependent. ∇ and ∆ denote the gradient and Laplacian operators.
In [30], Li and Liu studied the existence, uniqueness and regularity of solution of fractional advection-diffusion equation (1) as a preliminary step to the study of a time-fractional Keller-Segel system in domain R 2 . Recently there has been growing interest in studying the non-local in time equations with similar structure. When the drift term v does not depend on time, for example when v(x) = −∇V (x) with V (x) being a potential, the equation is called a time-fractional Fokker Planck equation (FP equation), or Kolmogorov forward equation. Equation (3) may be used to describe the flow in time of probability density in subdiffusion regimes, i.e. particles are subjected to long-tailed random waiting time. Kemppainen and Zacher [24] studied the long-time behavior of a time-fractional FP equation via the entropy method. Hahn, Kobayashi and Umarov [20] derived this equation from a model of stochastic differential equations driven by time-changed Lévy processes. m(t, x) in equation (3) is the transition probability density for the stochastic differential equation: where E t is an inverse α-stable subordinator. B Et denotes the time-changed Brownian motion. For more general stochastic differential equation with drift v(t, X t ) the corresponding Fokker-Planck equation should be Simple calculation via fractional calculus can show that when v = v(x) the equations (6) and (1) are equivalent. But that is not the case when v = v(t, x). Under the premises of sufficient regularity on the velocity field v, we may rewrite (1) as This type of time-fractional parabolic equations has been intensively investigated in recent works, e.g. [30,2,37,38,27]. Optimal control problems of time-fractional diffusion type equations and numerical schemes have been considered by Antil et al. in [4] and by Annunziato et al. in [3]. Subdiffusion processes deviate from classical Gaussian process in that the motion of particles are interrupted by long sojourns, possibly due to trapping effects. It has very important applications in physics and finance, we refer to [8,33] for general theory of anomalous diffusion.
The adjoint equation we obtained in the optimality system (2) is a backward Hamilton-Jacobi equation with Caputo derivative: We note that by change of variable (from t to T − t), it may be transformed into In the case that f is sufficiently regular and does not depend on density m, i.e.
f (x, m) = f (t, x) ∈ W 1,∞ (0, T ; W 1,∞ (T d )), the classical solution to the forward fractional HJ equation has been obtained in [25,26,22] by Kolokoltsov et al. The viscosity solution theory for time-fractional PDEs was also developed in some recent works, e.g. [18,36]. In [10], Camilli, Maio and Iacomini introduced a Hopf-Lax formula for the solution of a fractional Hamilton Jacobi equation. When β = 1 the system (2) becomes the classical Lasry-Lions mean field game system: The theory of Mean Field Games (MFG, in short) is a branch of Dynamic Games which aims at modeling and analyzing complex decision processes involving a large number of indistinguishable rational agents who have individually a very small influence on the overall system and are, on the other hand, influenced by the distribution of the other agents (Achdou et al. [1]). The theory of Mean Field Games originated from the works of Lasry and Lions [28,29] and independently started by Huang, Caines and Malhamé [23]. For a general introduction of MFG we refer to [12,17].
The system (10) may be derived via a convex optimization problem of the Fokker-Planck equation, this is known as the theory of variational mean field games [13,14]. This approach is based on the dynamic formulation of the Monge-Kantorovich optimal transport problem proposed by Benamou and Brenier [5]. Recent advances in this direction include [6,32,7]. An application of variational MFG to nonlinear mobilities in pedestrian dynamics can be found in [9]. Camilli and Maio [11] considered a time-fractional MFG system: where D 1−β [t,T ) and D 1−β (0,t] denote the backward and forward Riemann-Liouville fractional derivatives. Later, Tang and Camilli in [35] have derived the system (11) from a variational MFG approach. It is important to note the relation and differences between the two systems (2) and (11). The system (2) is the variant of the classical MFG system with Caputo time derivatives. It has the advantage of being more trackable via PDE methods, and may be naturely extended to include space non-local diffusion (by using the fractional Laplacian instead of Laplacian). The positivity and mass preservation of solutions to equation (1) has been proven in [30]. This equation itself is also instrumental in modeling of anomalous diffusion and aggregation processes, e.g. chemotaxis of bacteria. However, it has the drawback that the game theory interpretation is not clear, as the fractional HJ equation has not been derived via dynamical programming from a single agent point of view, but only as the adjoint equation in a PDE control problem. In comparison, in system (11), Camilli and Maio in [11] have proved it to be a bona fide MFG system. Therefore we believe the two fractional parabolic systems are of independent interests and both worthy of study.
The following assumptions and notations are supposed to hold throughout the rest of the paper. (H1) The coupling f : T d × [0, +∞) → R is continuous in both variables. Moreover the following normalization condition holds: f (x, 0) = 0, ∀x ∈ T d .
Plan of the paper. The paper is organized as follows. In Section 2, we review some basic facts about the fractional calculus, introduce the notion of weak solution to the time-fractional advection-diffusion equation and prove the existence and uniqueness of the solution. In Section 3, we derive the coupled fractional forward-backward system (2) as the optimality condition of control problems driven by the fractional advection-diffusion equation and the equivalence with a control problem of the fractional HJ equation via Fenchel-Rockerfeller duality theorem. We also give verification results regarding the equivalence between solutions of coupled fractional system and the two optimization problems. Finally, we discuss some interesting directions of investigation for future work.

Fractional calculus and advection-diffusion equation.
2.1. Fractional calculus. The idea of defining a derivative of fractional order ( 1 2 for example) dates back to Leibniz. This problem has also been considered by Riemann and Liouville among others in 19th century. The nonlocal operators under the name Riemann-Liouville derivative and integral are the most important definitions in this subject to this day. In the first half of 20th century advances has been made by Hardy and Littlewood [21]. We refer to Samko, Kilbas and Marichev [34] for a comprehensive account of the theory. We start with a brief introduction to some definitions and basic results in fractional calculus. We start from a formal level and assume functions φ(t), κ(t) ∈ C 1 (R). The forward and backward Riemann-Liouville fractional integrals are defined by The forward Riemann-Liouville and Caputo derivatives are defined by while the backward Riemann-Liouville and Caputo derivatives are defined by For β → 1 the forward Riemann-Liouville and Caputo derivatives of φ converge to the classical derivative dφ dt , meanwhile, the backward derivatives converge to − dφ dt . It should be noticed that the Riemann-Liouville derivative of a constant is not 0: .
We note the following relations between fractional integral and Caputo derivative, which may be regarded as the fractional version of fundamental theorem of calculus: The following fractional version to the classic Gronwall lemma will be used in priori estimates for solution to fractional PDEs.
Proof. This lemma is just a particular case of Proposition 6.6 of [27], using (12) and the fact that Moreover, we have the following result regarding the fractional integration by parts with Caputo derivative ( [4], Lemma 3): In order to study partial differential equations with fractional time derivatives, one needs to construct the fractional calculus for functions valued in general Banach spaces. We refer readers to [30] by Li and Liu for this theory. Denote by B a Banach space, we introduce the following sets:

QING TANG
We are motivated by the fractional integration by parts Lemma (2.2) to define the weak Caputo derivative in the sense of distributions: The following result ( [30], Lemma 3.1) is due to Hardy and Littlewood (Theorem 12 of [21]). An alternative proof has recently been obtained by Kubica and Yamamoto ([27], Proposition 6.7).

Time-fractional advection diffusion equation.
We define the weak formulation to the fractional advection-diffusion equation: Definition 2.5. Let β ∈ (0, 1). We say that m is a weak solution to the fractional advection-diffusion equation (1), if for any test function We also note the notion of mild solution to (1). Introduce the Mittag-Leffler function: by taking the Laplace transform one can obtain the fractional (non-Markovian) version of Duhamel's principle for solution to the equation (1): The mild solution is a strong solution in C([0, T ]; L 2 (T d )).
Remark 1. The Lemma 2.7 was proven in [30] for space domain R 2 , but the proof can be repeated for T d without essential changes.
Throughout the paper we denote by · p the norm of Lebesgue space L p (T d ) with p ∈ [1, +∞).
Moreover, ∀t ∈ [0, T ], we have positivity and mass conservation of the solution, i.e. m(t, x) ≥ 0 and . Proof. We start with some basic a priori estimates for equation (1). We will find a constant c 0 > 0 such that uniformly ∀t ∈ [0, T ]: From Poincare inequality there exists a constant c 2 > 0 such that m(t, ·) 2 ≤ c 2 ∇m(t, ·) 2 . Then ∀t ∈ (0, T ], The first inequality is due to the Proposition 2.2 of [30]. We then apply Lemma 2.1 and obtain Thus we may conclude that m ∈ L ∞ ([0, T ]; L 2 (T d )). Moreover from the calculations above we have Integrate both sides of this inequality on [0, T ] we obtain: Thus We now turn to the estimates on ∂ β (0,t] m. For all test function We now consider the mollified equation Using Lemma 2.7 we see that there exists a unique mild solution m such that We note that the previous a priori estimates results we obtained for m holds for m uniformly in such that Passing to the limit → 0, since L 2 ([0, T ]; H 1 (T d )) is a reflexive Banach spaces (so is its dual space L 2 ([0, T ]; H −1 (T d )), of course), we may construct subsequence m k such that m k converges in L ∞ ([0, T ]; L 2 (T d )) in the weak* topology, m k converges weakly in L 2 ([0, T ]; H 1 (T d )) and ∂ β (0,t] m k converges weakly in L 2 ([0, T ]; H −1 (T d )). More prescisely, ∀ϕ ∈ Ξ * (and therefore ∀ϕ(t, Since v convergent strongly to v, using classical diagonal argument we can subtract a further subsequence (m κ , v κ ) such that, for κ → 0: Uniqueness follows analogously via a priori estimates. Suppose there exist two solutions m 1 and m 2 such that m 1 (0, x) = m 2 (0, x). Due to the linear structure of the equation it is easy to obtain from the above calculations Hence the solution is unique in Ξ.  3. The coupled time-fractional forward-backward system.

3.1.
A formal derivation of the coupled system. In this section we introduce the coupled fractional forward-backward system and the corresponding variational interpretation. Consider the following optimal control problem inf v,m with m and v constrained by the advection-diffusion equation (1) in the weak sense.
We can reformulate this problem as a saddle point problem: As in [6], we first show at a formal level the connection between the coupled forward-backward system and the control problem of advection-diffusion equation via Lagrange multiplier methods, assuming that the optimal control problem is convex.
we claim that (u, m) is a solution to the coupled system (10) with v = −Du. Indeed, via integration by parts, we have: By convexity of the Lagrangian and Legendre-Fenchel theory, Given a variation δv with initial condition δv(0, x) = 0, consider first variation, We have obtained that optimal control is v = −∇u and we have 1 Take a variation δm with boundary condition δm(0, x) = 0 and recall that ∂F (x,m) We conclude that (u, m) is solution to the HJ equation 3.2. Duality of two optimal control problems. We now show the duality between two optimal control problems constrained, respectively, by a fractional HJ equation and a fractional advection-diffusion equation. We start from introducing a control problem for HJ equation. Denote by K 0 the set of maps u ∈ C 2 ([0, T ] × T d ) such that u(T, x) = u T (x) and define, on K 0 , the functional the Hamiltonian is defined as H(x, p) = 1 2 |p| 2 . Here F * defines the Legendre-Fenchel transform of F (x, m) such that ∀α: This is in fact a control problem of the Hamilton Jacobi equation: Next we reformulate the control problem J AD . To linearize the constraint , we use a standard trick in optimal transport theory and introduce the variable Then the advection-diffusion equation can be written as: where the solution is in the sense of distributions. The Legendre-Fenchel transform of Hamiltonian can be written as: otherwise.
Let us denote K 1 the set of pairs (m, w) On the set K 1 , define the following functional Since H * and F are bounded from below and m ≥ 0 a.e. (from Theorem 2.8), the first integral in B(m, w) is well defined in R ∪ {+∞}. We proceed to our main duality result. Proof.
Define on E 0 the functional where χ S is the characteristic function of the set S = {u ∈ E 0 , u(T, ·) = u T }, i.e., χ S (u) = 0 if u ∈ S and +∞ otherwise. For (a, b) ∈ E 1 , we define: Let Λ : E 0 → E 1 be the bounded linear operator defined by: Then we obtain It follows by Fenchel-Rockafellar duality theorem that where E 1 is the dual space of E 1 , i.e., the set of vector valued Radon measures (m, w) is the dual operator of Λ and F * and G * are the convex conjugate of F and G respectively.
Moreover, for any (m, w) ∈ R × R d , Hence, for (−m) > 0, due to the convexity of F , we have Therefore, for m > 0, The minimum is taken over the L 1 maps (m, w) such that m(t, x) ≥ 0 a.e. and equation (22) is satisfied in the sense of distributions. Since T d m 0 dx = 1, by Theorem 2.8 we have T d m(t, x)dx = 1 for for all t ∈ [0, T ]. Hence the pair (m, w) belongs to the set K 1 and we have proved the duality of the optimal control problems.
3.3. Verification arguments. In the next theorem, we show the connection between the optimal control problems for the fractional advection-diffusion and HJ equations and the coupled system (2). Proof. The proof is by verification arguments. "i ⇒ ii": assume that (m,û) is solution to coupled system (2). Denote byα and α respectivelyα Thus we can write By definition of Legendre transform F * (x, α) we know it is convex with regard to α, hence Since, by definition, ∂ α F * (x,α) =m and |∇u| 2 − |∇û| 2 ≥ 2∇û(u −û), we obtain J HJ (α) As (m,û) is solution to coupled system (2) then is satisfied in the sense of distributions. Hence we get J HJ (α) ≥ J HJ (α). From the uniqueness of solution to the fractional HJ equation we conclude: A(u) ≥ A(û). m(t, x)). Take a smooth function δα ∈ C 1 ([0, T ] × T d ), denote by u h the solution to the equation We remind thatû is the solution to the system Thus (u h −û)/h converges to a smooth fucntion φ which is the solution to the linearized system: Calculate the first variation we can obtain Since by density argument this holds for any test function φ ∈ C 3 with φ(T, x) = 0, thus we obtainm is a weak solution to the advection-diffusion equation in the coupled system (2) withm(0, x) = m 0 (x).
"(i)⇒ (iii)" This can be proven by verification argument, completely analogous to the reasoning above, hence we omit it.
Proof. We obtain the existence of solution via Schauder fixed point theorem. Boundedness: Let C be a large constant to be chosen below and denote by C the set of maps µ ∈ L 2 ([0, T ]; L 2 (T d )) such that µ L 2 ([0,T ];L 2 (T d )) ≤ C, constant C is to be fixed later. Then C is a convex closed setset of L 2 ([0, T ]; L 2 (T d )). To any µ ∈ C we associate m = Ψ(µ). Let u be the solution to and µ ≤ c 1 . It has been proven that in this case there is a unique classical solution u. we denote with constant c 3 > 0 such that ∀t ∈ [0, T ], ∇u(t, ·) ≤ c 3 . Then m solves the advection-diffusion equation By a priori estimates we have m ∈ Ξ and the bound Therefore we can choose such that we have Ψ : µ → m maps from C into itself. Compactness: Consider the triplet (Gelfand triple): H 1 (T d ) is compactly embedded in L 2 (T d ), by using the generalized Aubin-Lions argument (Theorem 4.1 in [30]) we obtain that: Ξ is compactly embedded in L 2 ([0, T ]; L 2 (T d )).
Continuity: Consider a sequence µ n ∈ C and denote by (u n , m n ) and (u, m) the solution obtained via µ n and µ, respectively. It has been shown by Kolokolstov et al. ( [26], Theorem 11) that there exist uniform bound c 4 such that sup t∈[0,T ] u n C 2 (T d ) ≤ c 4 .
Hence we conclude that u n and ∇u n converges locally uniformly to u and ∇u, respectively. Moreover, from the estimates in the proof of Theorem 2.8 we have again a uniform bound for m n : ) m 0 (x) 2 2 .
We can then extract a weakly convergent subsequence of m n , the limit will be a weak solution of the advection-diffusion equation in (2). Since we have proved the weak solution is unique in Ξ for fixed ∇u, thus m is the limit and the mapping Ψ is continuous. As Ψ maps a Banach space continuously into a convex compact subset, we conclude with Schauder fixed point theorem it admits at least one fixed point Ψ(m) = m. It follows that the pair (u, m) is a solution to the system.