Stackelberg-Nash exact controllability for linear and semilinear parabolic equations

ESAIM: COCV 21 (2015) 835–856 ESAIM: Control, Optimisation and Calculus of Variations DOI: 10.1051/cocv/2014052 www.esaim-cocv.org STACKELBERG–NASH EXACT CONTROLLABILITY FOR LINEAR AND SEMILINEAR PARABOLIC EQUATIONS ∗,∗∗ F.D. Araruna1,E.Fern ´ andez-Cara2and M.C. Santos1,3 Abstract. This paper deals with the application of Stackelberg–Nash strategies to the control of parabolic equations. We assume that we can act on the system through a hierarchy of controls. A first control (the leader) is assumed to choose the policy. Then, a Nash equilibrium pair (corresponding to a noncooperative multiple-objective optimization strategy) is found; this governs the action of the other controls (the followers). The main novelty in this paper is that, this way, we can obtain the exact controllability to a prescribed (but arbitrary) trajectory. We study linear and semilinear problems and, also, problems with pointwise constraints on the followers. Mathematics Subject Classification. 34K35, 49J20, 35K10. Received June 27, 2014. Revised September 27, 2014 Published online May 20, 2015. 1. Introduction In classical control theory, we usually find a state equation or system and one control with the mission of achieving a predetermined goal. Frequently (but not always), the goal is to minimize a cost functional in a prescribed family of admissible controls. A more interesting situation arises when several (in general, conflictive or contradictory) objectives are considered. This may happen, for example, if the cost function is the sum of several terms and it is not clear how to average. It can also be expectable to have more than one control acting on the equation. In these cases, we are led to consider multi-objective control problems. In contrast with the mono-objective case, various strategies for the choice of good controls can appear, depending of the characteristics of the problem. Moreover, these strategies can be cooperative (when the controls mutually cooperate in order to achieve some goals) or noncooperative. Keywords and phrases. Controllability, Stackelberg–Nash strategies, Carleman inequalities. ∗Partially supported by INCTMat, CAPES, CNPq (Brasil) and MathAmSud COSIP. ∗∗ Partially supported by grant MTM2010-15592 (DGI-MICINN, Spain) and CAPES (Brasil). 1Dpto. de Matem´atica, Universidade Federal da Para´ıba, 58051-900 Jo˜ao Pessoa – PB, Brasil. [email protected] 2Dpto. EDAN and IMUS, University of Sevilla, Aptdo. 1160, 41080 Sevilla, Spain. [email protected] 3Dpto. de Matem´atica, Universidade Federal de Pernambuco, 50740-540 Recife-PE, Brasil. [email protected] Article published by EDP Sciences c EDP Sciences, SMAI 2015 836 F.D. ARARUNA ET AL. There exist several equilibrium concepts for multi-objective problems, with origin in game theory, mainly motivated by economics. Each of them determines a strategy. Thus, let us mention the noncooperative optimization strategy proposed by Nash [16], the Pareto cooperative strategy [17] and the Stackelberg hierarchical-cooperative strategy [21]. In the context of the control of PDEs, a relevant question is whether one is able to steer the system to a desired state (exactly or approximately) by applying controls that correspond to one of these strategies. Up to date, there has been some work on the subject: •The papers by Lions [14,15], where the author gives some results concerning Pareto and Stackelberg strategies, respectively. •The paper by D´ıaz and Lions [4], where the approximate controllability of a system is established following a Stackelberg–Nash strategy and the extension in D´ıaz [3], that provides a characterization of the solution by means of Fenchel–Rockafellar duality theory. •The papers [18,19], where Ramos et al. study Nash equilibria from the theoretical and numerical viewpoints for linear parabolic PDEs and for the Burgers equation. •Finally, let us mention that the Stackelberg–Nash strategy for the Stokes systems has been studied by Guill´en-Gonz´alez et al. in [11]. The controllability issues considered in these works only provide answers at the approximate level. This means that the main results assert that one can lead the system to a state that is arbitrarily close (but not identical) to a desired target. The main novelty of the present paper is to extend the analysis and the results to an exact controllability framework. 1.1. The problems and their motivations Let Ω⊂RNbe a bounded domain whose boundary Γis regular enough. Let T>0 be given and let us consider the cylinder Q=Ω×(0,T), with lateral boundary Σ=Γ×(0,T).In the sequel, we will denote by Ca generic positive constant. Sometimes, we will indicate the data on which Cdepends by writing C(Ω), C(Ω,T), etc. The usual norm and scalar product in L2(Ω) will be respectively denoted by ·and (·,·). We are interested in the proof of the exact controllability to the trajectories of a multi-objective parabolic PDE problem in Q, where we apply a Stackelberg–Nash strategy. For simplicity, we will assume that only three controls are applied (one leader and two followers), but very similar considerations hold for systems with a higher number of controls. We will consider systems of the form ⎧ ⎨ ⎩ yt−Δy +a(x, t)y=F(y)+f1O+v11O1+v21O2in Q, y=0 onΣ, y(·,0) = y0in Ω, (1.1) where y=y(x, t)isthestate,a∈L∞(Q), Fis a locally Lipschitz-continuous function and y0is prescribed. In (1.1), the set O⊂Ωis the main control domain and O1,O2⊂Ωare the secondary control domains (all them are supposed to be small); 1O,1 O1and 1O2are the characteristic functions of O,O1and O2, respectively; the controls are f,v1and v2,wherefis the leader and v1and v2are the followers. Let O1,d,O2,d ⊂Ωbe open sets, representing observation domains for the followers. We will consider the (secondary) functionals Ji(f;v1,v2):=αi 2Oi,d×(0,T ) |y−yi,d|2dxdt+μi 2Oi×(0,T ) |vi|2dxdt, i =1,2 (1.2) and the main functional J(f):=1 2O×(0,T) |f|2dxdt, (1.3) where the αi>0, μi>0 are constants and the yi,d =yi,d(x, t) are given functions. STACKELBERG–NASH EXACT CONTROLLABILITY FOR LINEAR AND SEMILINEAR PARABOLIC EQUATIONS 837 The control process can be described as follows: 1. The followers v1and v2assume that the leader fhas made a choice and intend to be a Nash equilibrium for the costs Ji(i=1,2). Thus, once fhas been fixed, we look for controls vi∈L2(Oi×(0,T)) that satisfy J1(f;v1,v2)= min ˆv1J1(f;ˆv1,v2),J 2f;v1,v2=min ˆv2J2(f;v1,ˆv2).(1.4) Any pair (v1,v2) satisfying (1.4) is called a Nash equilibrium for J1and J2. Note that, if the functionals Ji(i=1,2) are convex, then (v1,v2) is a Nash equilibrium if and only if J 1(f;v1,v2)(ˆv1,0) = 0,∀ˆv1∈L2(O1×(0,T)) ,v 1∈L2(O1×(0,T)) (1.5) and J 2(f;v1,v2)(0,ˆv2)=0,∀ˆv2∈L2(O2×(0,T)) ,v 2∈L2(O2×(0,T)).(1.6) 2. Let us fix an uncontrolled trajectory of (1.1), that is, a sufficiently regular solution to the system ⎧ ⎨ ⎩ yt−Δy +a(x, t)¯y=F(y)in Q, y=0 onΣ, y(·,0) = y0in Ω. (1.7) Once the Nash equilibrium has been identified and fixed for each f, we look for a control ˆ f∈L2(O×(0,T)) such that J(ˆ f)= min fJ(f),(1.8) subject to the restriction of exact controllability y(·,T)=y(·,T)inΩ. (1.9) Several motivations can be found for control problems of this kind: •If y=y(x, t) is viewed as a temperature distribution in a body, we interpret that our intention is to drive y to a desired yat time Tby heating and cooling (acting only on the small subdomains O,O1and O2), trying at the same time to keep reasonable temperatures in O1,d and O2,d during the whole time interval (0,T). •The same control strategy makes sense in the context of fluid mechanics. Thus, we can replace (1.1)and(1.7) by similar Stokes and/or Navier–Stokes systems and we can look for controls ¯ fand associated Nash equilibria (v1,v 2) satisfying (1.8)–(1.9). In this case, it is assumed that we act on the system through mechanical forces applied on O,O1and O2and the goal is to reach yat time Tkeeping the velocity field ynot too far from yi,d in Oi,d ×(0,T)(i=1,2). •In the framework of mathematical finance, this can also be an interesting question. For instance, it is well known that the price of an European call option is governed by a backward PDE similar to (1.1). Now, the independent variable xmust be interpreted as the stock price and tis in fact the reverse of time (we fix a situation at t=Tand we want to know what to do in order to arrive at this situation from a well chosen state). In this regard, it can be interesting to control the solution of the system with the composed action of several agents, each of them corresponding to a different range of values of x. For further information on the modeling and control of phenomena of this kind, see for instance [2,20,22]. 838 F.D. ARARUNA ET AL. 1.2. The main results We will have to impose the following assumption: O1,d =O2,d.(1.10) Accordingly, we will denote these sets by Od; see below, in Section 5, some comments on the necessity of the hypothesis (1.10). In the linear case (F≡0), the exact controllability to the trajectories is equivalent to the null controllability property. The following result holds: Theorem 1.1. Let us assume that F≡0,Od∩O=∅and the μi>0(i=1,2) are sufficiently large. Then, there exists a positive function ˆρ=ˆρ(t)blowing up at t=Twith the following property: if yis the unique solution to (1.7)with (F≡0) associated to the initial state y0∈L2(Ω)and the yi,d are such that Od×(0,T) ˆρ2|y−yi,d|2dxdt<+∞,i=1,2,(1.11) for any y0∈L2(Ω), there exist controls f∈L2(O×(0,T)) and associated Nash equilibria (v1,v2)such that the corresponding solutions to (1.1)satisfy (1.9). Roughly speaking, the assumption on the μimeans that the followers must have moderate L2norms. On the other hand, the assumption (1.11) means that both y1,d and y2,d approach yas t→T. In the semilinear case, with Fbeing a locally Lipschitz-continuous function, we can consider the same controllability questions. However, it is important to note that, in this case, we lose the convexity of the functionals Jiand the Nash equilibrium condition (1.4) is not necessarily equivalent to (1.5)and(1.6). For this reason, it is convenient to weaken the definition of equilibrium as follows: Definition 1.2. Let f∈L2(O×(0,T)) be given. The pair (v1,v2) is called a Nash quasi-equilibrium of (1.1)–(1.2) associated to fif the conditions (1.5)and(1.6) are satisfied. For the semilinear case, we have the following result: Theorem 1.3. Let us assume that F∈W1,∞(R),Od∩O =∅and the μi>0(i=1,2) are sufficiently large. Let ybetheuniquesolutionto(1.7)associated to the initial state y0∈L2(Ω)and let us assume that (1.11)holds, where ˆρis the weight furnished by Theorem 1.1. Then, for each y0∈L2(Ω), there exist controls f∈L2(O×(0,T)) and associated Nash quasi-equilibria (v1,v2)such that the corresponding solutions to (1.1)satisfy (1.9). A natural question is whether there are semilinear systems for which the concepts of Nash equilibrium and Nash quasi-equilibrium are equivalent. An answer is given by the following result: Proposition 1.4. Let us assume that F∈W2,∞(R)and yi,d ∈L∞(Oi,d ×(0,T)) (i=1,2). Suppose that y0∈ H1 0(Ω)(resp. y0∈L2(Ω))andN≤14 (resp. N≤12). Then, there exists C>0such that, if f∈L2(O×(0,T)) and the μisatisfy μi≥C(1 + fL2(O×(0,T))), the conditions (1.4)and (1.5)–(1.6)are equivalent. In this paper, we also analyze if a result like Theorem 1.1 holds true when the followers are constrained to belong to appropriate convex sets Ui⊂L2(Oi×(0,T)). Thus, let I1and I2be two nonempty closed intervals with 0 ∈I1∩I2,letustake Ui={v∈L2(Oi×(0,T)) : v(x, t)∈Iia.e. },i=1,2,(1.12) and let us suppose that the minimization of J1and J2in (1.4) is subject to the restrictions ˆv1∈U 1and ˆv2∈U 2. STACKELBERG–NASH EXACT CONTROLLABILITY FOR LINEAR AND SEMILINEAR PARABOLIC EQUATIONS 839 The controllability result is the following: Theorem 1.5. Let us assume that F≡0,Od∩O=∅and the μi>0(i=1,2) are sufficiently large. Let ybetheuniquesolutionto(1.7)associated to the initial state y0∈L2(Ω). Then, for each y0∈L2(Ω),there exist controls f∈L2(O×(0,T)) and associated Nash equilibria (v1,v2)∈U 1×U 2such that the corresponding solutions to (1.1)satisfy (1.9). As mentioned above, the main novelty of this paper is that we deal with exact and not approximate controllability. There are other points that distinguish our contribution as well. Thus, contrarily to what was imposed in other previous papers (see for instance [11]), we do not make any assumption on the open sets Oi.Inparticular, the Oican be disjoint of O, which is obviously the most interesting situation. On the other hand, the analysis and results also hold, after appropriate modifications, for mfollowers with m>2. The rest of the paper is organized as follows. In Section 2we prove Theorem 1.1, which concerns the linear case. This result will be strongly used in the other sections. In Section 3we prove Theorem 1.3 and Proposition 1.4. As a consequence, we see that the Stackelberg–Nash strategy can be applied to nonlinear problems and, also, that under adequate hypotheses on F, we still obtain a Nash equilibrium. Section 4deals with the proof of Theorem 1.5. Finally, we present some additional comments and questions in Section 5. 2. The linear case In this section we prove Theorem 1.1. The proof is long and, for clarity, has been decomposed in two parts. In Section 2.1 we will recall the existence, uniqueness and characterization of a Nash equilibrium (for fixed but arbitrary f); then, in Section 2.2, we will prove the desired controllability result. Thanks to the linearity of the problem, we may reduce the exact controllability to the trajectories to a null controllability property. In fact, after the change of variable y=z+y, it is immediate to see from (1.1)and(1.7), with F≡0, that zis the solution to the problem ⎧ ⎨ ⎩ zt−Δz +a(x, t)z=f1O+v11O1+v21O2in Q, z=0 onΣ, z(·,0) = z0in Ω, (2.1) where z0=y0−y0. It is clear that the condition (1.9)isequivalentto z(x, T )=0 in Ω. (2.2) Also, we can write the functionals Jiin (1.2)intermsofz,whichgives Ji(f;v1,v2)=αi 2Oi,d×(0,T ) |z−zi,d|2dxdt+μi 2Oi×(0,T) |vi|2dxdt, i =1,2, where zi,d := yi,d −y(i=1,2). 2.1. Nash equilibrium In this subsection, we will recall an existence/uniqueness result concerning a Nash equilibrium, in the sense of (1.4), for any f∈L2(O×(0,T)). We will also recall a result which characterizes this Nash equilibrium in terms of the solution to an adjoint system. These results are due to D´ıaz and Lions (see [3,4,15]). For the moment, we do not have to impose the assumption (1.10). This requirement only appears later, in Section 2.2, when the choice of fhas to be made. Accordingly, in this section we keep the notation Oi,d (i=1,2). 840 F.D. ARARUNA ET AL. 2.1.1. Existence and uniqueness Let us introduce the spaces Hi:= L2(Oi×(0,T)) and H:= H1×H 2and let us consider the operators Li∈L(Hi;L2(Q)) with Livi=zi,whereziis the solution to the system ⎧ ⎨ ⎩ zi t−Δzi+a(x, t)z=vi1Oiin Q, zi=0 onΣ, zi(·,0) = 0 in Ω. By definition, for any control f, the pair (v1,v2) is a Nash equilibrium if and only if it satisfies (1.5)and(1.6), that is to say, αiOi,d×(0,T) (z−zi,d)widxdt+μiOi×(0,T ) viˆvidxdt=0,∀ˆvi∈H i,(2.3) where wiis the derivative of zwith respect to viin the direction ˆvi.Notethat ⎧ ⎨ ⎩ wi t−Δwi+a(x, t)wi=ˆvi1Oiin Q, wi=0 onΣ, wi(·,0) = 0 in Ω. Consequently, Liˆvi=wi.Wealsohavez=L1v1+L2v2+u,where ⎧ ⎨ ⎩ ut−Δu +a(x, t)u=f1Oin Q, u=0 onΣ, u(·,0) = z0in Ω. Therefore, we may rewrite (2.3)intheform αiOi,d×(0,T )L1v1+L2v2−(zi,d −u)Liˆvidxdt +μiOi×(0,T ) viˆvidxdt=0,∀ˆvi∈H i or Oi×(0,T) αiL∗ iL1v1+L2v2−(zi,d −u)1Oi,d +μiviˆvidxdt=0,∀ˆvi∈H i, where L∗ i∈L(L2(Q); Hi)istheadjointofLi.Inotherwords,(v1,v2) is a Nash equilibrium if and only if αiL∗ iL1v1+L2v21Oi,d +μivi=αiL∗ i((zi,d −u)1Oi,d )inHi,i=1,2. Let us introduce the operator L∈L(H;H), given by Lv1,v2=α1L∗ 1L1v1+L2v21O1,d +μ1v1,α 2L∗ 2((L1v1+L2v2)1O2,d )+μ2v2,(2.4) for all (v1,v2)∈H. Then, the task is to prove the existence and uniqueness of a solution for the equation Lv1,v2=Ψ, v1,v2∈H,(2.5) where Ψ=(α1L∗ 1((z1,d −u)1O1,d ),α 2L∗ 2((z2,d −u)1O2,d )).(2.6) STACKELBERG–NASH EXACT CONTROLLABILITY FOR LINEAR AND SEMILINEAR PARABOLIC EQUATIONS 841 In this direction, the following holds: Proposition 2.1. Let us assume that α11O1,d L2(1) <4μ2and α21O2,d L1(2) <4μ1,(2.7) where ·(i)denotes the norm in the space L(H3−i;L2(Oi,d ×(0,T))).ThenLis an isomorphism. In particular, for each f∈L2(O×(0,T)), there exists exactly one Nash equilibrium (v1(f),v2(f)) in the sense of (1.4). Proof. From (2.4) and Young’s inequality, we observe that Lv1,v2,v1,v2H= 2  i=1 μivi2 Hi+ 2  i,j=1 αi(Ljvj,L ivi)L2(Oi,d×(0,T)) ≥ 2  i=1 μivi2 Hi+αiLivi2 L2(Oi,d×(0,T )) − 2  i=1 αiLivi2 L2(Oi,d×(0,T )) +1 4L3−iv3−i2 L2(Oi,d×(0,T)) ≥ 2  i=1 μi−α3−i 41O3−i,d Li2 (3−i)vi2 Hi. Therefore, Lv1,v2,v1,v2H≥γv1,v22 H,∀v1,v2∈H,(2.8) where γ=min i{μi−α3−i1O3−i,d Li2 (3−i)}>0, see (2.7). Now, let us introduce the bilinear form a:H×H→R,with av1,v2,ˆv1,ˆv2:= Lv1,v2,ˆv1,ˆv2H. From the definition of the operator Land the inequality (2.8), we readily see that a(·,·) is continuous and coercive on H. Consequently, the Lax–Milgram’s Theorem implies that, for any Φ∈H , there exists exactly one (v1,v2)∈Hsatisfying a(v1,v2,(ˆv1,ˆv2)) = Φ, (ˆv1,ˆv2)H×H ∀(ˆv1,ˆv2)∈H;v1,v2∈H. In particular, we get (2.5) and the proof is done.  From the proof, it becomes clear that, under the assumptions of Proposition 2.1, for any f∈L2(O×(0,T)) the associated Nash equilibrium (v1(f),v2(f)) satisfies  v1(f),v2(f) H≤C1+fL2(O×(0,T)),(2.9) where the constant Cdepends on Ω,O,T,Oi,Oi,d,α i,μ i,z0and aL∞(Q). These estimates will be used below. Notice that, in view of (2.9), the state zassociated to fand (v1(f),v2(f)) satisfies zL2(0,T;H1 0(Ω)) +ztL2(0,T;H−1(Ω)) ≤C(1 + fL2(O×(0,T ))),(2.10) where Cis as above. 842 F.D. ARARUNA ET AL. 2.1.2. Characterization of the Nash equilibrium We will express the followers v1(f)andv2(f) in terms of a new adjoint variable. Let f∈L2(O×(0,T)) be given. For any (v1,v2)∈H, let us consider the associated state z(the solution for (2.1)). In view of (2.3), it is very natural to introduce the adjoint states φi(i=1,2), with ⎧ ⎨ ⎩ −φi t−Δφi+a(x, t)φi=αi(z−zi,d)1Oi,d in Q, φi=0 onΣ, φi(·,T)=0 in Ω. Using integration by parts, we see that (v1,v2) is a Nash equilibrium if and only if Oi×(0,T )φi+μiviˆvidxdt=0,∀ˆvi∈H i,v i∈H i. This directly implies that vi=−1 μi φiOi×(0,T),i=1,2. Let us gather all these informations in the same system. We obtain the following: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ zt−Δz +a(x, t)z=f1O− 2  i=1 1 μi φi1Oiin Q, −φi t−Δφi+a(x, t)φi=αi(z−zi,d)1Oi,d in Q, z=0,φ i=0 onΣ, z(·,0) = z0,φ i(·,T)=0 in Ω. (2.11) Recall that our main objective is to prove the null controllability of zat time t=T. Therefore, the task is to find a distributed control f∈L2(O×(0,T)) such that the solution to (2.11)satisfies(2.2). 2.2. Null controllability In this subsection, we will achieve the proof of Theorem 1.1. We will establish an observability inequality for the system ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ −ψt−Δψ +a(x, t)ψ= 2  i=1 αiγi1Oi,d in Q, γi t−Δγi+a(x, t)γi=−1 μi ψ1Oiin Q, ψ=0,γ i=0 onΣ, ψ(·,T)=ψT,γ i(·,0) = 0 in Ω, (2.12) which can be viewed as the adjoint of (2.11). This will suffice. This observability estimate is given in the following result: Proposition 2.2. Assume that (1.10)holds, Od∩O=∅and the μiare sufficiently large. There exist C>0, only depending on Ω,O,T,Oi,Od,α i,μ iand aL∞(Q)and a weight function ˆρ=ˆρ(t)blowing up at t=T, only depending on Ω,O,Od,Tand aL∞(Q), such that, for any ψT∈L2(Ω), the following inequality holds true for the solution (ψ,γi)of (2.12): Ω |ψ(x, 0)|2dx+ 2  i=1 Q ˆρ−2|γi|2dxdt≤CO×(0,T ) |ψ|2dxdt. (2.13) STACKELBERG–NASH EXACT CONTROLLABILITY FOR LINEAR AND SEMILINEAR PARABOLIC EQUATIONS 843 Let us assume for a moment that Proposition 2.2 holds and let us prove the controllability result in Theorem 1.1. From a well known duality argument, we have that, for any z0∈L2(Ω)andanyψT∈L2(Ω), Ωz(x, T )ψT(x)−z0(x)ψ(x, 0)dx=O×(0,T) fψdxdt− 2  i=1 αiOd×(0,T) zi,dγidxdt, (2.14) where (z,φ1,φ 2)and(ψ,γ1,γ2) are the solutions to (2.11)and(2.12), respectively associated to z0and ψT. Thus, to prove the null controllability property is equivalent to find, for each z0∈L2(Ω), a control fsuch that, for any ψT∈L2(Ω), one has O×(0,T ) fψdxdt=−Ω z0(x)ψ(x, 0) dx+ 2  i=1 αiOd×(0,T ) zi,dγidxdt. There are several ways to show that (2.13) implies the existence of such a control. They rely on well known arguments. For completeness, let us sketch one of them. For each >0, let us consider the following functional: F(ψT):=1 2O×(0,T ) |ψ|2dxdt+ψT+Ω z0(x)ψ(x, 0) dx − 2  i=1 αiOd×(0,T ) zi,dγidxdt, ∀ψT∈L2(Ω). It is then clear that F:L2(Ω)→Ris continuous and strictly convex. Moreover, F(ψT)≥1 4O×(0,T ) |ψ|2dxdt −CΩ |z0|2dx+ 2  i=1 α2 iOd×(0,T) ˆρ2|zi,d|2dxdt +ψT, where Cand ˆρare furnished by Proposition 2.2.Consequently,Fis also coercive in L2(Ω). Note that, here, we have used the assumption (1.11)onzi,d =yi,d −¯y. Let ψT be the unique minimizer of F. Then, either ψT =0or F (ψT ),ψT=0,∀ψT∈L2(Ω). Suppose that ψT = 0. In this case, we have O×(0,T) ψψdxdt+ψT  ψT ,ψT+Ω z0(x)ψ(x, 0) dx − 2  i=1 αiOd×(0,T) zi,dγidxdt=0,∀ψT∈L2(Ω), (2.15) where we have denoted by (ψ,γ1 ,γ2 ) the solution to (2.12) corresponding to ψT=ψT .Takingf=f:= ψ1O×(0,T)in (2.14), denoting by zthe associated stateandcomparingto(2.15), we see that Ωz(x, T )− ψT ψT ψT(x)dx=0,∀ψT∈L2(Ω), 850 F.D. ARARUNA ET AL. In particular, for all w1∈L2(O1×(0,T)), one has D2 1J1(f;v1,v2),(w1,w1)=μ1O1×(0,T) |w1|2dxdt+O1×(0,T ) ηw1dxdt. (3.14) Let M>0 be such that |F(s)|≤Ma.e. in R. Let us show that, for some Conly depending on Ω,O,T, Oi,Od,αi,M,K,aL∞(Ω)and y0,wehave O1×(0,T) ηw1dxdt ≤C(1 + fL2(O×(0,T)))w1H1,∀w1∈L2(O1×(0,T)).(3.15) In fact, from standard energy estimates, since F∈L∞(Q), we have Ω |h(x, t)|2dx+Q |∇h|2dx≤CO1×(0,T ) |w1|2dxdt. Using the PDEs in (3.13), we also get the following: O1×(0,T ) ηw1dxdt=Q (ht−Δh +a(x, t)h−F(y)h)ηdxdt =Q h(−ηt−Δη +a(x, t)η−F(y)η)dxdt =Q (F(y)hφ +α1h1Od)hdxdt =Q (F(y)|h|2φ+α1|h|21Od)dxdt. (3.16) Letusfirstassumethaty0∈H1 0(Ω). The idea is to find rand ssuch that φ∈Lr(0,T;Ls(Ω)) and h∈L2r(0,T;L2s(Ω)),(3.17) where rand sare the conjugate of rand s, respectively. This will make possible to bound from above the last integral in (3.16). It is clear that h∈L2(0,T;H2(Ω)) ∩L∞(0,T;H1 0(Ω)). For this reason, it is natural to ask for which values of αand βthe following embedding holds: L2(0,T;H2(Ω)) ∩L∞(0,T;H1 0(Ω)) →Lα(0,T;Lβ(Ω)).(3.18) By interpolation, we have that, for each 0 <θ<1, (3.18)holdswhen 1 α=θ 2and 1 β=(N−4)θ 2N+(N−2)(1 −θ) 2N=α(N−2) −4 2αN · Taking α=2rand β=2s, we conclude that r=α/(α−2) and s=αN/2(α+2). Analogously, we have that y∈L2(0,T;H2(Ω))∩L∞(0,T;H1 0(Ω)) →La(0,T;Lb(Ω)), with b=2aN/(a(N− 2) −4). Using the regularity results of the heat equation and the fact that yi,d ∈L∞(Oi,d ×(0,T)), it follows that φ∈La(0,T;W2,b(Ω)) →La(0,T;LNb N−2b(Ω)) = La(0,T;L2aN aN−6a−4(Ω)). If a=r=α/(α−2), we get φ∈Lr(0,T;L2αN αN−10α+8 (Ω)). To finish, we must have L2αN αN−10α+8 (Ω)→Ls(Ω), which is equivalent to αN 2(α+2) ≤2αN α(N−10) + 8· Thus, we see that this inequality holds true if and only if N≤14. STACKELBERG–NASH EXACT CONTROLLABILITY FOR LINEAR AND SEMILINEAR PARABOLIC EQUATIONS 851 From (3.2), (3.3), (3.10)fors=0,(3.12)fors=0,(3.16) and the estimates at Section 3.2,weseethat, if y0∈H1 0(Ω)andN≤14, O1×(0,T) ηw1dxdt ≤Mh2 L2r(0,T;L2s(Ω))φLr(0,T ;Ls(Ω)) +α1h2 L2(Od×(0,T)) ≤C(φLr(0,T;Ls(Ω)) +1)w12 H1 ≤C(yL2(Q)+1)w12 H1 ≤C2  i=1 1 μi φiHi+f+y0+1 w12 H1 ≤C(1 + f)w12 H1. This proves (3.15)inthiscase. Now, let us assume that we only have y0∈L2(Ω). As in the first situation, the idea is to find rand ssuch that (3.17) holds. Since the regularity of ηdoes not depend on the data y0, we still have η∈L2(0,T;H2(Ω)) ∩ L∞(0,T;H1 0(Ω)) and, therefore, η∈Lα(0,T;Lβ(Ω)), where αand βare as above. In this case, we have by a interpolation argument that y∈L2(0,T;H1 0(Ω)) ∩L∞(0,T;L2(Ω)) →L¯a(0,T;L¯ b(Ω)), where ¯a≥2and ¯ b=2N¯a/(¯aN −4). Using again parabolic regularity, we get φ∈L¯a(0,T;W2,¯ b(Ω)) →L¯a(0,T;LN¯ b N−2¯ b(Ω)) = L¯a(L2¯aN ¯a(N−4)−4(Ω)). If ¯a=r=α/(α−2), we have φ∈Lr(L2αN α(N−8)+8 (Ω)). Now, to finish the proof, we must have L2αN α(N−8)+8 (Ω)→ Ls(Ω), which is equivalent to αN 2(α+2) ≤2αN α(N−8) + 8· Since this holds if and only if N≤12, the estimate (3.15)isalsoprovedinthiscase. Taking into account (3.14)and(3.15), we see that D2 1J1(f;v1,v2),(w1,w1)≥μ1−C(1 + fL2(O×(0,T )))w12 H1dxdt. Note that the previous constant Ccan be chosen independent of μ1and μ2. In a similar way, it can be shown that, under the previous assumptions on y0and N, D2 2J2(f;v1,v2),(w2,w2)≥μ2−C(1 + fL2(O×(0,T )))w22 H2dxdt. for another constant Cindependent of μ1and μ2. It is now clear that, for sufficiently large μ1and μ2, the couple (v1,v2) is a Nash equilibrium in the sense of (1.4). 4. The case with restrictions In this section, we will prove Theorem 1.5. We return to the Stackelberg–Nash null controllability problem for a linear parabolic PDE, but we impose some restrictions: the followers (v1,v2) are supposed to minimize the functionals (1.2) subject to the convex constraints vi∈U i(i=1,2), where the Uiare given by (1.12). This is a more difficult problem. The search of a pair (v1,v2) satisfying (1.4), where the minimizations are performed in U1,d and U2,d,isequivalenttothe following: D1J1(f;v1,v2)(ˆv1−v1,0) ≥0,∀ˆv1∈U 1,d,v 1∈U 1,d (4.1) 852 F.D. ARARUNA ET AL. and D2J2(f;v1,v2)(0,ˆv2−v2)≥0,∀ˆv2∈U 2,d,v 2∈U 2,d.(4.2) As in Section 2, with the change of variable z=y−¯y, we are led to a null controllability problem. Then, we see that (4.1)–(4.2)isequivalentto ⎧ ⎨ ⎩ αiOi,d×(0,T) (z−zi,d)widxdt+μiOi×(0,T ) vi(ˆvi−vi)dxdt≥0, ∀ˆvi∈U i,d,v i∈U i,d, (4.3) where wiis the derivative of zwith respect to ˆviin the direction vi,thatistosay,thesolutionto ⎧ ⎨ ⎩ wi t−Δwi+a(x, t)wi=vi1Oiin Q, wi=0 onΣ, wi(·,0) = 0 in Ω. (4.4) The adjoint system associated to (4.4)isgivenby ⎧ ⎨ ⎩ −φi t−Δφi+a(x, t)φi=αi(z−zi,d)1Oi,d in Q, φ=0 onΣ, φ(·,T)=0 in Ω. Replacing the equation satisfied by φiin (4.3), we obtain Oi×(0,T )φi+μivi(ˆvi−vi)dxdt≥0,∀ˆvi∈U i,d,v i∈U i,d,i=1,2.(4.5) Now, by introducing the projectors PUi,d :L2(Oi×(0,T)) →U i,d,weseethat(4.5) can be rewritten equivalently in the form vi=PUi,d −1 μi φiOi×(0,T),i=1,2. We may group all this information to get the following system: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ zt−Δz +a(x, t)z=f1O+ 2  i=1 PUi,d −1 μi φiOi×(0,T)in Q, −φi t−Δφi+a(x, t)φi=αi(z−zi,d)1Oi,d in Q, z=0,φ i=0 onΣ, z(·,0) = z0,φ i(·,T)=0 in Ω. (4.6) Let us prove that, under the assumptions (2.7), for each f∈L2(O×(0,T)) there exists exactly one solution to (4.6), i.e. there exists a unique Nash equilibrium (v1,v2)inU1,d ×U 2,d.Indeed,noticethat(4.5) can also be rewrittenintheform Lv1,v2,(ˆv1,ˆv2)−v1,v2≥Ψ,(ˆv1,ˆv2)−v1,v2H ∀v1,v2∈U 1,d ×U 2,d,(ˆv1,ˆv2)∈U 1,d ×U 2,d,(4.7) where Land Ψare respectively given by (2.4)and(2.6). If μ1and μ2satisfy (2.7), Lis a coercive continuous bilinear form on H, whence (4.7) is uniquely solvable. Furthermore, it is clear that the couple (v1,v2)andthe associated state zsatisfy (again) the estimates (2.9)and(2.10). As in the semilinear case, we will analyze and solve the null controllability problem for (4.6) by a fixed-point method. To this end, note that the projectors PUi,d are given as follows: PUi,d (k)(x, t)=k(x, t)ifk(x, t)∈Ii, Pi(k(x, t)) otherwise, STACKELBERG–NASH EXACT CONTROLLABILITY FOR LINEAR AND SEMILINEAR PARABOLIC EQUATIONS 853 for (x, t)a.e. in Oi×(0,T), where Pi:R→Iiis the usual projector on the interval Ii. Also, note that, for every k∈H i,PUi,d canbewrittenintheformPUi,d (k)=qi(k)k, where the function k→ qi(k) is continuous on Hiand qi(k)∞≤C, ∀k∈H i. Therefore, the controllability problem is reduced to find f∈L2(O×(0,T)) such that the solution to ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ zt−Δz +a(x, t)z=f1O− 2  i=1 ˜qi(φi)φi1Oiin Q, −φi t−Δφi+a(x, t)ψ=αi(z−zi,d)1Oi,d in Q, z=0,φ i=0 onΣ, z(·,0) = z0,φ i(·,T)=0 in Ω, (4.8) where ˜qi(φi) stands for the function ˜qi(φi)=qi(−1 μiφiOi×(0,T )), satisfies (2.2). But this can be done easily. Indeed, for each couple (˜ φ1,˜ φ2)∈[L2(Q)]2we can consider the system ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ zt−Δz +a(x, t)z=f1O− 2  i=1 ˜qi(˜ φi)φi1Oiin Q, −φi t−Δφi+a(x, t)ψ=αi(z−zi,d)1Oi,d in Q, z=0 φi=0 onΣ, z(·,0) = z0,φ i(·,T)=0 in Ω. (4.9) The arguments in Sections 2.2 and 3.2 can be applied again to (4.9). The main consequence is that there exists exactly one minimal L2norm null control ffor this system with fL2(O×(0,T)) ≤C(4.10) and, also, z,φ1and φ2uniformly bounded in L2(0,T;H1 0(Ω)) ∩L∞(0,T;L2(Ω)) and zt,φ1 tand φ2 tuniformly bounded (at least) in L2(0,T;H−1(Ω)). Hence, it is not difficult to deduce that the mapping (˜ φ1,˜ φ2)→ (φ1,φ 2) possesses at least one fixed-point. Such a fixed-point satisfies, together with some fand some z,(4.8)and(2.2). This concludes the proof of Theorem 1.5. 5. Some additional comments and questions 5.1. On the assumption O1,d=O2,d The assumption (1.10)isusedin(2.20) and only there. Indeed, in combination with (2.21)and(2.22), (2.20) yields (2.23). At present, we do not know whether an estimate like (2.13) remains true for O1,d =O2,d. However, this is the case if we modify appropriately the secondary functionals Ji.Infact,letρ∗=ρ∗(x, t)beaweight(a positive continuous function on Ω×(0,T)) such that ρ∗≥esσ/2,see(2.18). We assume now that the followers produce a Nash equilibrium with respect to the functionals ˜ Ji(f;v1,v2):=αi 2Oi,d×(0,T ) |y−yi,d|2dxdt+μi 2Oi×(0,T) ρ2 ∗|vi|2dxdt, i =1,2. With computations similar to those in Section 2.1, we obtain the following optimality system: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ zt−Δz +a(x, t)z=f1O− 2  i=1 1 μi ρ−2 ∗φi1Oiin Q, −φi t−Δφi+a(x, t)φi=αi(z−zi,d)1Oi,d in Q, y=0,φ i=0 onΣ, y(·,0) = y0,φ i(·,T)=0 in Ω. 854 F.D. ARARUNA ET AL. The associated adjoint system is given by ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ −ψt−Δψ +a(x, t)ψ= 2  i=1 αiγi1Oi,d in Q, γi t−Δγi+a(x, t)γi=−1 μi ρ−2 ∗ψ1Oiin Q, ψ=0,γ i=0 onΣ, ψ(·,T)=ψT,γ i(·,0) = 0 in Ω and the main task is to prove an estimate like (2.13) for the solutions (ψ,γ1,γ2). In this situation, we have an useful energy inequality for the γi: γi(·,τ)2+τ 0 ∇γi(·,t)2dt≤C μ2 iQ ρ−4 ∗|ψ|2dxdt. (5.1) Using (5.1) in the right-hand side of (2.20), since the μiare sufficiently large, we get I3(ψ)≤Cs3λ4ω×(0,T ) e−2sαξ3|ψ|2dxdt. (5.2) Combining (5.2)and(2.26), we arrive at (2.13). This shows that if we replace Jiby ˜ Ji(i=1,2), the claims in Theorem 1.1 to 1.5 remain true. In fact, this is not surprising: if we impose ˜ Ji<+∞,thenweforcethe controls vito vanish exponentially as t→T−and the leader ffinds no obstruction to control the system. As mentioned above, it is unknown whether (2.13) continues to be true in the original framework (1.2)when O1,d =O2,d. 5.2. Stackelberg–Nash controllability and Stokes and Navier–Stokes systems It makes complete sense to consider the Stokes-like system ⎧ ⎪ ⎨ ⎪ ⎩ yt−Δy +(w·∇)y+∇p=f1O+v11O1+v21O2in Q, ∇·y=0 in Q, y=0 onΣ, y(·,0) = y0in Ω, (5.3) where Ω,T,Oand the Oiare as above, y0belongs to the Hilbert space H:= {z∈L2(Ω)N:∇·z=0 in Ω, z ·n=0 on Γ}, the field wbelongs to L∞(0,T;H) and the controls fand visatisfy f∈L2(O×(0,T))N,v i∈L2(Oi×(0,T))N. With functionals Jand Jisimilar to those in the previous sections, we can formulate again the Stackelberg– Nash null controllability problem for (5.3). Results of the same kind can be obtained easily by adapting the arguments in Sections 2to 4. The situation is obviously more difficult to analyze when we consider the Navier–Stokes system ⎧ ⎪ ⎨ ⎪ ⎩ yt−Δy +(y·∇)y+∇p=f1O+v11O1+v21O2in Q, ∇·y=0 in Q, y=0 onΣ, y(·,0) = y0in Ω. Now, the existence of Nash equilibria or quasi-equilibria for each fand, of course, whether or not there exist null controls and associated Nash equilibrium pairs are open problems. For other controllability results for Stokes and Navier–Stokes systems, see [6,8–10,12]. STACKELBERG–NASH EXACT CONTROLLABILITY FOR LINEAR AND SEMILINEAR PARABOLIC EQUATIONS 855 5.3. Other Stackelberg strategies It is possible to introduce other strategies to control systems of the kind (1.1). One of them is the so called Stackelberg–Pareto method. For each f∈L2(O×(0,T)), we can associate one or several Pareto equilibrium pairs (u1(f),u 2(f)) ∈H.By definition, this means that there is no (ˆu1,ˆu2)∈Hsatisfying Ji(ˆu1,ˆu2)≤Ji(u1(f),u 2(f)),i=1,2, one of these inequalities at least being strict. Then, we search for fsuch that the states yassociated to fand the (u1(f),u 2(f)) satisfy (1.9), where y=y(x, t) is a prescribed uncontrolled solution to (1.1). The analysis of Stackelberg–Pareto controllability will be the goal of a forthcoming paper. 5.4. The boundary case It is natural to try to prove results similar to Theorems 1.1,1.3 and 1.5 with boundary controls. For instance, let us consider the system ⎧ ⎪ ⎨ ⎪ ⎩ zt−Δz +a(x, t)z=0 in Q, z=f1S+v11S1+v21S2on Σ, z(·,0) = z0in Ω, where S,S1,S2⊂Γare non-empty closed sets and let us introduce the functionals Li(f;v1,v2):=αi 2Oi,d×(0,T) |z−zi,d|2dxdt+μi 2Si×(0,T ) |vi|2dΓdt, i =1,2.(5.4) Now, the problem is to find for each fa Nash equilibrium (v1(f),v2(f)) associated to the functionals Liand, then, choose fin a appropriate way such that z(x, T )≡0. We can try to solve this problem as before. However, we find some technical difficulties, as shown below. Arguing as in Section 2, we see that the optimality system for (v1(f),v2(f)) is the following: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ zt−Δz +a(x, t)z=0 Q, −φi t−Δφi+a(x, t)φi=αi(z−zi,d)1Oi,d in Q, z=f1S+1 μ1 ∂φ1 ∂n 1S1+1 μ2 ∂φ2 ∂n 1S2,φ i=0onΣ, z(·,0) = z0,φ i(·,T)=0 in Ω, (5.5) where n=n(x) is the outward unit normal to Ωat the point x∈Γ. The corresponding adjoint is given by ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ −ψt−Δψ +a(x, t)ψ= 2  i=1 αiγi1Oi,d in Q, γi t−Δγi+a(x, t)γi=0 in Q, ψ=0,γ i=1 μi ψ1Sion Σ, ψ(·,T)=ψT,γ i(·,0) = 0 in Ω. (5.6) Thus, if we try to adapt the proof of Proposition 2.2, we see at once that the following conditions are required: O1,d =O2,d =Odand Od∩S=∅.(5.7) The main difficulty in this case is that we have to combine a boundary Carleman inequality for ψand a distributed Carleman inequality for h=α1γ1+α2γ2for functions satisfying nonhomogeneous Dirichlet boundary conditions on Σ. This interesting situation will be also analyzed in a forthcoming paper. 856 F.D. ARARUNA ET AL. 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