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Stackelberg-Nash null controllability of heat equation with general dynamic boundary conditions

This paper deals with the hierarchical control of the anisotropic heat equation with dynamic boundary conditions and drift terms. We use the Stackelberg-Nash strategy with one leader and two followers. To each fixed leader, we find a Nash equilibrium corresponding to a bi-objective optimal control problem for the followers. Then, by some new Carleman estimates, we prove a null controllability result.


Introduction.
A control problem consists of finding a control that steers the system under consideration from an initial state to a fixed target. In many situations, the resolution process leads to a single-objective optimal control problem, and generally, under some appropriate conditions, the existence and uniqueness of optimal control problems can be proved.
The modeling of many industrial and economic complex problems reveals several criteria to optimize simultaneously. For example, in the heat transfer, in a room, it is meaningful to try to guide the temperature to be close as much as possible to a fixed target at the end of the day and, additionally, keep the temperature not too far from a prescribed value at some regions. This can be done by applying several controls at different locations of the room. This problem can be seen as a game with controls as players.
To solve a multi-objective optimal control problem, there are several strategies to choose the best controls, depending on the character of the problem. Among these strategies, let us mention the cooperative strategy proposed by Pareto [40], the Nash non-cooperative strategy [44], and the Stackelberg hierarchical strategy [41].
In the context of the control of evolution equations, a relevant question is whether one can steer the system to the desired state exactly or approximately by controls Section 5, we prove some suitable Carleman estimates and deduce our null controllability result in the linear case. The semilinear case is considered in Section 6.
Throughout this paper, we assume that the matrices A and AΓ satisfy the following assumptions.
This paper deals with Stackelberg-Nash strategy for the null controllability of the above heat equation with dynamic boundary conditions. To be more specific, we introduce, for i = 1, 2, the non-empty open sets ω i,d ⊂ Ω, representing the observation domains of the followers, and the fixed target functions y i,d ∈ L 2 (0, T ; L 2 (ω i,d )). Let us define the following main cost functional and the secondary cost functional where αi, µi are positive constants and Since the functionals Ji (i = 1, 2), are differentiable and convex, then, by well-known characterization results, see, e.g. Theorem 3.8 of [30] or [16], the pair Our goal is to prove that, for any initial data Y0 = (y0, y0,Γ) ∈ L 2 = L 2 (Ω)×L 2 (Γ), there exist a control f ∈ L 2 (0, T ; L 2 (ω)) with minimal norm (called leader) and an associated Nash To do this, we shall follow the Stackelberg-Nash strategy: for each choice of the leader f , we look for a Nash equilibrium pair for the cost functionals Ji (i = 1, 2); which means finding the controls v ⋆ 1 (f ) ∈ L 2 (0, T ; L 2 (ω1)) and v ⋆ 2 (f ) ∈ L 2 (0, T ; L 2 (ω2)), depending on f , satisfying (7). Once the Nash equilibrium has been identified and fixed for each f , we look for a controlf such that subject to the null controllability constraint Assume that the control regions satisfy the following assumption The main result in this paper is the following.
STACKELBERG-NASH NULL CONTROLLABILITY 5 Theorem 2.1. Let us assume that (10) holds and µi > 0, i = 1, 2, are sufficiently large. There exists a positive weight function ρ = ρ(t) blowing up at t = T such that for every y i,d ∈ L 2 (0, T ; L 2 (ωi)) satisfying and every Y0 = (y0, yΓ,0) ∈ L 2 , there exist a controlf ∈ L 2 (0, T ; L 2 (ω)) with minimal norm and an associated Nash equilibrium Remark 1. (a) As mentioned by several authors, the assumption (11) seems to be natural. It means that the follower targets y i,d , i = 1, 2, approach to 0 as T −→ 0, and with this the leader finds no obstruction to control the system. It remains an open problem to verify if this condition is necessary.
(b) In [4], the authors have considered some weak conditions than (10), for instance ω 1,d ∩ ω = ω 2,d ∩ ω, in the context of the heat equation with Dirichlet conditions. This generalization will be treated in a forthcoming paper for dynamic boundary conditions. (c) The hierarchical control is motivated by applications where more than one objective is desirable. For example, we can think of y and yΓ the concentration of some chemical product in Ω and on the boundary Γ, respectively. The process is to guide the system under consideration to 0 by means of an optimal control f acting on the domain ω, and without, in the course of the action, going too far from y d in a small subdomain w d .

Preliminaries.
Let Ω ⊂ R N be a bounded open set with smooth boundary Γ := ∂Ω.
Following [38], we introduce the product space defined by Here, we have considered the Lebesgue measure dx on Ω and the natural surface measure dσ on Γ. Equipped by the scalar product L 2 is a Hilbert space. Recall that H 1 (Γ) and H 2 (Γ) are real Hilbert spaces endowed with the respective norms We point out that the operator ∆Γ can be considered as an unbounded linear operator from L 2 (Γ) in L 2 (Γ), with domain and it is known that −∆Γ is a self-adjoint and nonnegative operator on L 2 (Γ). This implies that −∆Γ generates an analytic C0−semigroup (e t∆ Γ ) t≥0 on L 2 (Γ). If Γ is smooth, then one can show that D(∆Γ) = H 2 (Γ), and u → u L 2 (Γ) + ∆Γu L 2 (Γ) defines an equivalent norm on H 2 (Γ), see [38] and the references therein for more details. As in [38], we denote viewed as a subspace of H k (Ω) × H k (Γ) with the natural topology inherited by H k (Ω) × H k (Γ) and E(t0, t1) = H 2 (t0, t1; L 2 ) ∩ L 2 (t0, t1; H 2 ) for t1 > t0, and E1 = E(0, T ).
We denote (H k (Ω)) ′ , H −k (Γ) and H −k the topological dual of H k (Ω), H k (Γ) and H k , respectively, k = 1, 2, and Now we shall recall some results on the well-posedness of the nonhomogeneous forward system in ΩT (12) and the nonhomogeneous backward one in Ω × Γ. (13) Remark first that the system (12) can be rewritten as the following abstract Cauchy problem where we denoted Y = (y, yΓ), F = (f, g), Following [38], we can show that the operator A satisfies the following important property.

4.
Existence, uniqueness and characterization of Nash-equilibrium. In this section, using the same ideas as in [5], we shall prove the existence and provide a characterization, in term of an adjoint system, of the Nash-equilibrium in the sense of (7). In the sequel, if X is a Hilbert space, ·, · X stands for the scalar product of X.
Using some ideas of [5], we shall prove the existence and uniqueness of Nash-equilibrium. More precisely, we have the following result.
Proof. For i = 1, 2 and a fixed f , we have . We define the operator R = (R1, R2) ∈ L(H) as follows Let us prove that R is invertible. We have Choosing the parameters µ1 and µ2 such that there exists a constant C1 > 0, such that Since R is continuous, we deduce, from Lax-Milgramm theorem, that R invertible. Furthermore, we have This achieves the proof. Now, we shall characterize the Nash-equilibrium in term of the solution to an adjoint system. To this end, let us introduce the following adjoint systems and integrating by parts, we find We deduce that, Let us collect all the previous results in a same system. We obtain the following forward-backward system, called optimality system in Ω × Γ, In Section 5, we will show that the null controllability for the system (25) is equivalent to a suitable observability inequality for the following adjoint system 5. Carleman estimates and null controllability.

Carleman estimates.
In this section, we shall state and show some suitable Carleman estimates needed to prove our main result concerning null controllability. To this end, let us first introduce the following well-known Morse function, see [19] for the existence of such function. Let ω ′ be an open set of Ω such that for some constant c > 0. Introduce the following classical weight functions where x ∈ Ω, t ∈ (0, T ) and λ ≥ 1. The following lemma summarizes some important properties of the above functions. In what follows, C stands for a generic positive constant only depending on Ω and ω, whose value can change from line to line.
Proof. Let ω ′ 1 be open sets such that Let θ ∈ C ∞ (Ω) such that 0 ≤ θ ≤ 1, θ = 1 in ω ′ and supp(θ) ⊂ ω ′ 1 . Applying Carleman estimate given in the second point of Lemma 5.2 to Z with the observation region ω ′ instead of ω and F0 = h1ω d − az, F = Bz, FΓ,0 = −bzΓ, FΓ = BΓzΓ, we find Choosing s2 large enough, the previous inequality becomes Using the Carleman estimate given in the first point of Lemma 5.2 for H, and the fact that ξ ≥ 4 T 2 , we obtain By choosing s1 large enough, one has From (30) and (31)  Thanks to (10) and (29), we have Using (33) and the fact that supp(θ) ⊂ ω ′ 1 , we find By the symmetry of the matrix A, we obtain div(A∇(θξ 3 e −2sα h)) =div(A∇(θξ 3 e −2sα ))h + div(A∇h)θξ 3 e −2sα + 2∇(θξ 3 e −2sα ) · A∇h, ∇(θξ 3 e −2sα h) = h∇(θξ 3 e −2sα ) + θξ 3 e −2sα ∇h. Then Using the equation satisfied by h = α1ψ 1 + α2ψ 2 , we find By the properties of the functions α and ξ given in Lemma 5.1, we can estimate the the four last terms in the right hand side in the above inequality. To this end, fix ǫ small enough and using Young inequality, we deduce for the first term For the second one, we have The third one can be estimated as For the last one, we have Using  The proof is then finished.
To prove the needed observability inequality, we are going to improve the Carleman inequality (28). To this end, following [5], we modify the weight functions ξ and α. More precisely, we introduce the functions Point out that the functions ξ and α do not blow up at t = 0, this will be important to get an improved Carleman estimate. We also denote where s, λ > 1 and Z = (z, zΓ) . With these definitions and the same notations of Theorem 5.3, we have the following result. Proof. The proof relies on the Carleman estimate (28) and some energy estimates. Following the strategy in [5], let us introduce a function η ∈ C 1 ([0, T ]) such that Set P = ηZ. It is clear that P = (p, pΓ) satisfies in Ω × Γ. (41) From the energy estimate (18) for P , there exists a positive constant C such that Using the fact that the weight functions e −2sα and ξ (resp. e On the other hand, since α = α and ξ = ξ in (T /2, T ), we have From (42) and (43), we obtain Using again the energy estimate and the fact that the weight functions e −2sα and ξ are bounded in [0, 3T /4], we see that For µ1 and µ2 large enough, we deduce Then, we conclude that This finishes the proof.

5.2.
Observability and null controllability. In this section we will prove our main result of controllability given in Theorem 2.1. Let us first recall that the characterization (24) of the follower controls adds two additional equations to the system (1), and our problem is then reduced to look for a control f ∈ L 2 (0, T ; L 2 (ω)) such that the solution (Y, Φ 1 , Φ 2 ) to the optimality system (25) satisfies Y (T ) = 0. To this end, we shall prove the following observability inequality.
Proposition 5. Assume that (10) holds and µi > 0, i = 1, 2, are large enough. Then, there exist a constant C > 0 and a positive weight function ρ = ρ(t) blowing up at t = T , such that, for any ZT ∈ L 2 , the solution (Z, Ψ 1 , Ψ 2 ) to (26) satisfies the following inequality Proof. Fix s large enough and set ρ = e sα ⋆ . Thus ρ is a positive function blowing up at t = T . For i = 1, 2, using the equation satisfied by Ψ i in (25), we readily see that By Young inequality and for a positive λ, we have in (0, T ).
Using the ellipticity of A and AΓ and choosing λ large enough, we obtain 1 2 Thus, from Gronwall 's lemma and the fact that Ψ i (0) = 0, it follows that Using Lemma 5.4, we find From the inequalities (45) and (46) and the fact that the function e −2sα ξ 7 is bounded in ΩT , we obtain This ends the proof. Now, we prove the controllability result in Theorem 2.1. Since we have proved the existence and uniqueness of Nash equilibrium in Proposition 4, it remains to prove the following result. Proposition 6. Let ρ = ρ(t) the weight function given in Proposition 5. Then, for any y i,d ∈ H i,d , i = 1, 2, satisfying (11) and Y0 ∈ L 2 , there exists a control f ∈ L 2 (0, T ; L 2 (ω)) with minimal norm such that f L 2 (0,T ;L 2 (ω)) ≤ C Y0 L 2 + ρy 1,d L 2 (Ω T ) + ρy 2,d L 2 (Ω T ) , (47) and the associated state satisfies Y (T ) = 0, where (Y, Φ 1 , Φ 2 ) is the solution to (25) and C is a positive constant.
By energy estimate, we deduce Using (54) and (57), we deduce that Y (T ) = 0. This concludes the null controllability result.
As a consequence of the above HUM method, see for instance [46], the control we have constructed is one of the minimal norm, and it is characterized as follows.
7. Conclusion. In this paper we have studied a hierarchical control problem of heat equation with general dynamic boundary conditions. Following the Stackelberg-Nash strategy with one leader and two followers, we have proved, for each fixed leader, the existence and uniqueness of a Nash-equilibrium, and by means of an adjoint system, we have characterized the Nash-equilibrium, and we have deduced an optimality system. By suitable Carleman estimates, we have established an observability inequality which is the key to deduce our controllability result. The similar results are also obtained for the semilinear system.