Bi-objective optimal control of some PDEs: Nash equilibria and quasi-equilibria

ESAIM: COCV 27 (2021) 50 ESAIM: Control, Optimisation and Calculus of Variations https://doi.org/10.1051/cocv/2021050 www.esaim-cocv.org BI-OBJECTIVE OPTIMAL CONTROL OF SOME PDES: NASH EQUILIBRIA AND QUASI-EQUILIBRIA∗ E. Fern´ andez-Cara1,** and I. Mar´ ın-Gayte2 Abstract. This paper deals with the solution of some multi-objective optimal control problems for several PDEs: linear and semilinear elliptic equations and stationary Navier-Stokes systems. Specifically, we look for Nash equilibria associated with standard cost functionals. For linear and semilinear elliptic equations, we prove the existence of equilibria and we deduce related optimality systems. For stationary Navier-Stokes equations, we prove the existence of Nash quasi-equilibria, i.e. solutions to the optimality system. In all cases, we present some iterative algorithms and, in some of them, we establish convergence results. For the existence and characterization of Nash quasi-equilibria in the Navier-Stokes case, we use the formalism of Dubovitskii and Milyutin. In this context, we also present a finite element approximation and we illustrate the techniques with numerical experiments. Mathematics Subject Classification. 35Q30, 76D05, 76N10. Received April 24, 2020. Accepted May 5, 2021. 1. Introduction We consider bi-objective optimal control problems for various PDEs and systems. First, an introductory problem corresponding to a linear elliptic PDE is analyzed with detail. Then, we deal with a similar semilinear elliptic PDE. Finally, we deal with the stationary Navier-Stokes system, that is, the equations satisfied by the time-independent velocity field and pressure of an incompressible viscous fluid flow. Our aims are to prove existence, characterize efficiently the equilibria and, also, compute numerical solutions to these multi-objective control problems. They are very important from the mathematical viewpoint and appear frequently in the applications; for some previous works on the subject, see for instance [2]. In classical control theory, we usually find a state equation or system and one control, with the task of achieving a predetermined goal. Frequently (but not always), the goal is to minimize a cost functional within a prescribed family of admissible controls; see for instance [10,16]. A different and 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 that an average provides a reasonable criterion. Also, it can 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 ∗Partially financed by MINECO, Grant MTM2016-76990-P. Keywords and phrases: Elliptic PDEs, Navier-Stokes equations, optimal control, bi-objective problems, Nash equilibria, Dubovitskii-Milyutin formalism. 1Dpto. EDAN e IMUS, Universidad de Sevilla, Aptdo. 1160, 41080 Sevilla, Spain. 2Departamento EDAN, Universidad de Sevilla, Campus Reina Mercedes, 41012 Sevilla, Spain. ** Corresponding author: [email protected] c The authors. Published by EDP Sciences, SMAI 2021 This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 2E. FERN´ ANDEZ-CARA AND I. MAR´ IN-GAYTE choice of good or the best controls can appear, depending on many circumstances. Moreover, these strategies can be cooperative or noncooperative (depending on whether or not several controls mutually cooperate in order to achieve prescribed goals). There exist several concepts of equilibrium for multi-objective problems, with origin in game theory. Each of them can be identified with a strategy. Thus, let us mention the non-cooperative optimization strategy proposed by Nash [18], the Pareto cooperative strategy [19] and the Stackelberg hierarchical-cooperative strategy [24]. In the context of the control of PDEs, up to date, there have been some works on the subject like the seminal papers by Lions [15,17] and other more recent contributions, like [6,21,22]. In this paper, we will be concerned with Nash equilibria and quasi-equilibria associated with standard cost functionals. To be more precise, let us give some details in the case of a linear elliptic equation. Thus, let the fluid domain be a bounded open set Ω ⊂RN, with N≥1. Let us introduce four nonempty open subsets, O1,O2,ω1and ω2and let us assume that a function uid defined on Oiis given for i= 1,2. We would like to find a couple (f1, f2)∈L2(ω1)×L2(ω2) (the control pair) with the following property: the associated state u, that is, the solution to the Dirirchlet problem −ν∆u=f11ω1+f21ω2in Ω, u= 0 on ∂Ω,(1.1) is such that (f1, f2) is a Nash equilibrium for the functionals Ji(f1, f2) := a 2ZOi |u−uid|2+µ 2Zωi |fi|2, i = 1,2,(1.2) where a, µ > 0 (see Sect. 2.1). Our first main goal will be to find conditions under which at least one Nash equilibrium exists. The second one will be to characterize these equilibria in terms of first order optimality conditions, i.e. to derive a system that the optimal solution, together with some associated adjoint states must satisfy. The third one will be to show how Nash equilibria can be computed and present related convergent algorithms. These tasks will be successfully accomplished below. Then, we will deal with some more complex equations and systems. In particular, in what concerns optimality conditions, we will try to extend to this context the techniques usually employed for distributed control problems (see for instance [1,10,16]). At some point, we will apply the so called formalism of Dubovitskii and Milyutin. This approach was introduced in the context of mathematical programming and has been succesfully applied to the solution of many optimal control problems for ODEs since the 70’s. A good presentation of its applications to these areas can be found in Girsanov [11]; see also Flett [9]. Later, these techniques have been applied successfully to some distributed control problems; see [3,4]. In the present context this method is appropriate; note that this would not be so clear if state constraints were imposed. The plan of this paper is the following. In Section 2, we consider the relatively simple problem given by (1.1)–(1.2). We prove the existence of Nash equilibria, we provide an optimality system and we present some iterative algorithms. In Section 3, a more complicated problem is analyzed: a semilinear elliptic PDE together with functionals of the same kind. Here, in view of the nonlinearity, we must work with Nash equilibria and Nash quasi-equilibria, that is solutions to the optimality system. Again, existence, characterization and computation-oriented results are established. Section 4deals with the stationary Navier-Stokes system. New difficulties are found: nonlinearity, lack of uniqueness, lack of regularity of the functionals, etc. We also discuss the existence and optimality characterization of Nash equilibria and quasi-equilibria. Additionally, we present some iterative algorithms and the results of several numerical experiments. Finally, some additional comments and open questions are presented in Section 5. BI-OBJECTIVE OPTIMAL CONTROL OF SOME PDES 3 2. Introductory problem: a linear elliptic PDE In the sequel, we denote by k·kand (·,·) the usual L2norm and inner product, respectively. The symbol 1D will be used to denote the characteristic function of the set D. Also, Cwill stand for a generic positive constant. For simplicity, we will assume that only two controls act on the system and two functionals are taken into account but very similar considerations hold for systems with a larger number of controls and functionals. 2.1. Definition of Nash equilibria Let Ω ⊂RNbe a nonempty bounded connected open set with regular boundary ∂Ω and let us assume that ω1 and ω2are nonempty disjoint open subsets of Ω. We will consider the problems −∆u=f11ω1+f21ω2in Ω, u= 0 on ∂Ω,(2.1) where the fi∈L2(ωi) are the controls and uis the state. Let O1an O2be open sets, representing prescribed observation domains and let the Jibe given by Ji(f1, f2) := a 2ZOi |u−uid|2+µ 2Zωi |fi|2, i = 1,2,(2.2) where the uid ∈L2(Oi) are given functions and aand µare positive constants. The first bi-objective control problem considered in this paper is the following: Find a Nash equilibria associated with (2.1) and (2.2), that is, a pair ( ˆ f1,ˆ f2)∈L2(ω1)×L2(ω2) such that J1(ˆ f1,ˆ f2)≤J1(f1,ˆ f2)∀f1∈L2(ω1), J2(ˆ f1,ˆ f2)≤J2(ˆ f1, f2)∀f2∈L2(ω2).(2.3) In this case, since the control-to-state mapping is well-defined, linear and continuous and the cost functionals Jiare quadratic, strictly convex and C1, it is immediate to deduce that ( ˆ f1,ˆ f2) is a Nash equilibria if and only if ∂J1 ∂f1 (ˆ f1,ˆ f2)=0,∂J2 ∂f2 (ˆ f1,ˆ f2)=0.(2.4) 2.2. Existence and characterization of Nash equilibria For future purposes, note that (∂Ji ∂fi (f1, f2), gi) = Zωi (aϕi+µfi)gi∀fi, gi∈L2(ωi),(2.5a) where ϕiis the i-th adjoint state associated with (f1, f2), i.e. the solution to −∆ϕi= (u−uid)1Oiin Ω, ϕi= 0 on ∂Ω,(2.5b) where uis the state associated with (f1, f2). We can now present the main result of the section. It deals with the existence and characterization of Nash equilibria. 4E. FERN´ ANDEZ-CARA AND I. MAR´ IN-GAYTE Theorem 2.1. Let us assume that ( ˆ f1,ˆ f2)∈L2(ω1)×L2(ω2). Then 1. ( ˆ f1,ˆ f2) is a Nash equilibrium if only if there exist ˆu, ˆϕ1and ˆϕ2such that              −∆ˆu=ˆ f11ω1+ˆ f21ω2in Ω, ˆu= 0,on ∂Ω, −∆ ˆϕi= (ˆu−uid)1Oiin Ω, ˆϕi= 0 on ∂Ω, ˆ fi=−a µˆϕi|ωi, i = 1,2. (2.6) 2. There exists χ=χ(Ω,O1,O2, ω1, ω2) such that, if a/µ ≤χ, (2.6) possesses exactly one solution. Consequently, if a > 0, µ > 0 and a/µ is sufficiently small, there exists a unique Nash equilibrium for J1 and J2. 3. Let us assume that O1=O2. Then, for any a > 0 and µ > 0, there exists exactly one solution to (2.6), that is, a unique Nash equilibrium associated with (2.1) and (2.2). Proof. 1. Let us first assume that ( ˆ f1,ˆ f2) is a Nash equilibrium. Then, (2.4) holds. In view of (2.5a)–(2.5b), one must have ˆ fi=−a µˆϕi|ωifor i= 1,2 (here, ˆϕisolves (2.5b) for u= ˆu). Hence, (2.6) is satisfied. Conversely, if ( ˆ f1,ˆ f2),ˆuand the ˆϕisatisfy (2.6), then we see from (2.5a) that (2.4) holds and ( ˆ f1,ˆ f2) is a Nash equilibrium. This proves part 1. 2. Note that (2.6) can be written in the form Id +a µΛ0(ˆ f1,ˆ f2) = a µ(z1, z2),(ˆ f1,ˆ f2)∈L2(ω1)×L2(ω2),(2.7) where Λ0:L2(ω1)×L2(ω2)7→ L2(ω1)×L2(ω2) is given by Λ0(f1, f2) := (ψ1|ω1, ψ2|ω2), ψi= (−∆)−1((−∆)−1(f11ω1+f21ω2))1Oi) and (z1, z2) := ((−∆)−1(u1d1O)|ω1,(−∆)−1(u2d1O)|ω2). It is easy to check that Λ0is a linear compact operator. Consequently, if a/µ is sufficiently small, Id +a µΛ0 is an isomorphism and there exists exactly one solution to (2.7). This proves part 2. 3. Suppose now that O1=O2, Then Λ0is self-adjoint and positive, that is, (Λ0(f1, f2),(f1, f2)) ≥0∀(f1, f2)∈L2(ω1)×L2(ω2). Therefore, for any a, µ > 0, (2.7) is uniquely solvable and the solution is moreover the unique minimizer of the functional K:L2(ω1)×L2(ω2)7→ R, given by K(f1, f2) := 1 2((Id +a µΛ0)(f1, f2),(f1, f2)) + a µ((z1, z2),(f1, f2)).(2.8) Thus, the proof of part 3 is also done. BI-OBJECTIVE OPTIMAL CONTROL OF SOME PDES 5 Note that, when O16=O2, the operator Λ0is not necessarily self-adjoint. As a consequence, for large a/µ there can be no solution to (2.7). However, since Λ0is compact, we can make use of Fredholm’s Alternative and affirm that there exist at most β1, β2, . . . with βn→+∞such that, if a/µ 6=βnfor all n≥1, a unique Nash equilibrium exists. 2.3. Algorithms and convergence We present in this section several algorithms that can be used for the computation of Nash equilibria. The first one relies on a fixed-point reformulation of (2.6): ALG 1: (a) Choose f0 i∈L2(ωi), i= 1,2. (b) Then, for given n≥0 and fn i∈L2(ωi), compute the solution unto −∆un=fn 11ω1+fn 21ω2in Ω, un= 0 on ∂Ω(2.9) and the solutions ϕn ito the systems −∆ϕn i= (un−uid)1Oiin Ω, ϕn i= 0 on ∂Ω(2.10) and, finally, take fn+1 i=−a µϕn i|ωi, i = 1,2.(2.11) The following convergence result holds: Theorem 2.2. There exists χ0∈(0, χ], such that, if a/µ ≤χ0, the controls provided by ALG 1 satisfy (fn 1, fn 2)→(ˆ f1,ˆ f2) strongly in L2(ω1)×L2(ω2), where ( ˆ f1,ˆ f2) is the unique Nash equilibrium associated with J1and J2. Furthermore, the speed of convergence is at least linear. The proof is immediate: it suffices to rewrite (2.6) in the form (2.7) and note that ALG 1 is the usual fixed-point iteration method for the mapping Λ : L2(ω1)×L2(ω2)7→ L2(ω1)×L2(ω2), where Λ(f1, f2) := −a µΛ0(f1, f2) + a µ(z1, z2). The following two algorithms are inspired by the classical optimal step gradient and conjugate gradient methods. ALG 2: (a) Choose f0 i∈L2(ωi), i= 1,2. (b) Then, for given n≥0 and fn i∈L2(ωi), compute the solution unto (2.9) and the solution ϕn ito (2.10) and take fn+1 i=fn i−ρn ign i, i = 1,2,(2.12) 6E. FERN´ ANDEZ-CARA AND I. MAR´ IN-GAYTE where gn i=a ϕn i|ωi+µfn i(2.13) and ρn 1= argmin ρ≥0 J1(fn 1−ρgn 1, fn 2), ρn 2= argmin ρ≥0 J2(fn 1, fn 2−ρgn 2).(2.14) ALG 3: (a) Choose f0 i∈L2(ωi), i= 1,2. (b) For n= 0, perform one step of ALG 2 and take d0 i=g0 i,i= 1,2. (c) Then, for given n≥1 and fn i∈L2(ωi), compute the solution unto to (2.9) and the solution ϕn ito (2.10) and take fn+1 i=fn i−ρn idn i,(2.15) where      dn i=gn i+γn idn−1 i, γn i=kgn ik2 L2(ωi) kgn−1 ik2 L2(ωi) , gn i=a ϕn i|ωi+µfn i (2.16) and ρn 1= argmin ρ≥0 J1(fn 1−ρdn 1, fn 2), ρn 2= argmin ρ≥0 J2(fn 1, fn 2−ρdn 2).(2.17) Note that, in these iterates, the functions ρ7→ J1(fn 1−ρgn 1, fn 2), ρ7→ J2(fn 1, fn 2−ρgn 2), etc. are quadratic and strictly convex. Consequently, the ρn ican be computed explicitly. To the best of our knowledge, the convergence of ALG 2 and ALG 3 cannot be guaranteed. Note however that we advance in the directions indicated by the partial derivatives of the Jiand, consequently, it is reasonable to expect that they do converge. Now, let us assume as before that O1=O2=O. Then, the Nash equilibrium is the unique minimizer of the functional Kin (2.8) and it makes sense to consider the following algorithms: ALG 2’: Optimal Step Gradient Method for K. (a) Solve the systems: −∆Zi=uid1Oin Ω, Zi= 0 on ∂Ω for i= 1 and 2. (b) Choose f0 i∈L2(ωi), i= 1,2. (c) Then, for given n≥0 and fn i∈L2(ωi), compute the solution unto (2.9) and the solution ψnto −∆ψn=un1O,in Ω, ψn= 0 on ∂Ω.(2.18) BI-OBJECTIVE OPTIMAL CONTROL OF SOME PDES 7 and take fn+1 i=fn i−ρndn ii= 1,2,(2.19) where dn i=fn i+a µψn|ωi+a µZi(2.20) and ρn= argmin ρ≥0 K(fn 1−ρdn 1, fn 2−ρdn 2),(2.21) ALG 3’: Optimal Step Conjugate Gradient Method for K. (a) Solve the systems: −∆Zi=uid1Oin Ω, Zi= 0 on ∂Ω for i= 1 and 2. (b) Choose f0 i∈L2(ωi), i= 1,2. (c) For n= 0, perform one step of ALG 2’ and take d0 i,i= 1,2. (d) Then, for given n≥1 and fn i∈L2(ωi), compute the solution unto to (2.9) and the solution ψnto (2.18) and take fn+1 i=fn i−ρndn i,(2.22) where        dn i=gn i+γndn−1 i, γn=k(gn 1, gn 2)k2 L2(ω1)×L2(ω2) k(gn−1 1, gn−1 2)k2 L2(ω1)×L2(ω2) , gn i=fn i+a µψn|ωi+a µZi (2.23) and ρn= argmin ρ≥0 K(fn 1−ρdn 1, fn 2−ρdn 2).(2.24) Again, the functions ρ7→ K(f1 n−ρd1 n, f2 n−ρd2 n) are quadratic and strictly convex, whence the ρncan be easily found. The following result holds: Theorem 2.3. Assume that O1=O2=O. Then, the controls furnished by ALG 2’ and ALG 3’ satisfy (fn 1, fn 2)→(ˆ f1,ˆ f2) strongly in L2(ω1)×L2(ω2), where ( ˆ f1,ˆ f2) is the unique Nash equilibrium associated with J1and J2. The proof is immediate in view of the properties of the linear operator Λ0. It suffices to directly apply well known classical results; see for instance [7,20]. It is remarkable that, in this particular case, ALG 2 and ALG 2’ and ALG 3 and ALG 3’ only differ in the definitions of the step parameters ρn iand ρn. 8E. FERN´ ANDEZ-CARA AND I. MAR´ IN-GAYTE 3. The case of a semilinear elliptic PDE 3.1. Nash equilibria and quasi-equilibria This section is devoted to introduce Nash optima in the semilinear case. Now, we must distinguish equilibria from quasi-equilibria and take into account the particularities of each of them. Let us assume that φ:R7→ Ris of class C1, 0≤φ0(s)≤C∀s∈R.(3.1) The state equation reads as follows: −∆u+φ(u) = f11ω1+f21ω2in Ω, u= 0 on ∂Ω.(3.2) It is then well known that, for each (f1, f2)∈L2(ω1)×L2(ω2), there exists exactly one (strong) solution u to (3.2). As in Section 2, we will consider the cost functionals Jiin (2.2), where the uid ∈L2(Oi) and a, µ > 0. Again, it will be said that ( ˆ f1,ˆ f2) is a Nash equilibrium for (3.2) and (2.2) if (2.3) is satisfied. From standard optimal control theory, it is known that, under the previous assumptions on φ, the cost functionals Jiare again C1and satisfy (∂Ji ∂fi (f1, f2), gi) = Zωi (aϕi+µfi)gi∀fi, gi∈L2(ωi), where ϕiis the unique solution to −∆ϕi+φ0(u)ϕi= (u−uid)1Oiin Ω, ϕi= 0,on ∂Ω(3.3) (that is, ϕiis the i-th adjoint state corresponding to f1and f2) and uis the solution to (3.2); see for instance [10,16]. Definition 3.1. It will be said that ( ˆ f1,ˆ f2) is a Nash quasi-equilibrium for (3.2) and (2.2) if ( ˆ f1,ˆ f2) satisfies (2.4), that is, ( ˆ f1,ˆ f2) solves, together with ˆuand the ˆϕi, the optimality system              −∆ˆu+φ(ˆu) = ˆ f11ω1+ˆ f21ω2in Ω, ˆu= 0 on ∂Ω, −∆ ˆϕi+φ0(ˆu) ˆϕi= (ˆu−uid)1Oiin Ω, ˆϕi= 0 on ∂Ω, ˆ fi=−a µˆϕi|ωi, i = 1,2. (3.4) Note that, if ( ˆ f1,ˆ f2) is a Nash equilibrium, then ( ˆ f1,ˆ f2) is also a Nash quasi-equilibrium. However, the converse is not necessarily true. Theorem 3.2. Let us assume that N≤8 and, besides (3.1), φis of class C2and |φ0|+|φ00| ≤ Mfor some M > 0. Then, there exists η > 0, only depending on Ω, u1d, u2dand M, such that, if a/µ ≤η, the following assertions are equivalent: (a) ( ˆ f1,ˆ f2) is a Nash equilibrium for (3.2) and (2.2). (b) ( ˆ f1,ˆ f2) is a Nash quasi-equilibrium for (3.2) and (2.2). BI-OBJECTIVE OPTIMAL CONTROL OF SOME PDES 9 Proof. We have to prove that, under the previous conditions, any quasi-equilibrium satisfies (2.3). Thus, let us assume that (2.4) holds and let the functionals ˜ Jibe given as follows: ˜ J1(f1) := J1(f1,ˆ f2) and ˜ J2(f2) := J2(ˆ f1, f2). Let us first prove that there exists η0>0 and C0>0, only depending on Ω, the uid and M, such that, if a/µ ≤η0, any quasi-equilibrium ( ˆ f1,ˆ f2) satisfies kˆ fikL2(ωi)≤C0, i = 1,2.(3.5) Indeed, if we take the PDEs for uand ϕi, we respectively multiply by uand ϕiand we integrate in Ω, we get k∇uk2+ZΩ φ(u)u+a µZω1 uϕ1+Zω2 uϕ2= 0 and k∇ϕik2+ZΩ φ0(u)|ϕi|2−ZOi uϕi=−ZOi uidϕi. Now, from the properties of φand H¨older and Young inequalities, we see that k∇uk2≤Ca µk∇ϕ1k2+k∇ϕ2k2 and k∇ϕik2≤C+Ca µk∇ϕ1k2+k∇ϕ2k2. Consequently, if a/µ is sufficiently small one has (3.5) for some C0. Observe that, in view of the properties of φ, the ˜ Jiare twice continuosly differentiable. Let us see that, if a/µ is sufficiently small, ˜ J00 i(fi;gi, gi)>0 for all fi∈L2(ωi), all nonzero gi∈L2(ωi) and i= 1,2. For instance, let us take i= 1. Then, (˜ J0 1(f1), g1)L2(ω1)=Zω1aϕ1+µf1g1dx ∀f1, g1∈L2(ω1),(3.6) where ϕ1is, together with u, the solution to        −∆u+φ(u) = f11ω1+ˆ f21ω2in Ω, u= 0 on ∂Ω, −∆ϕ1+φ0(u)ϕ1= (u−u1d)1O1in Ω, ϕ1= 0 on ∂Ω. (3.7a) For any small ε > 0 and any g1∈L2(ω1)N, let us introduce uεand ϕε 1, with        −∆uε+φ(uε)=(f1+εg1)1ω1+ˆ f21ω2in Ω, uε= 0 on ∂Ω, −∆ϕε 1+φ0(uε)ϕε 1= (uε−u1d)1O1in Ω, ϕε 1= 0 on ∂Ω. (3.7b) 16 E. FERN´ ANDEZ-CARA AND I. MAR´ IN-GAYTE Note that there exist bilinear continuous mappings Band ˆ Bwith hB(u, v), wi=b(u, v, w) and hˆ B(u, v), wi=ˆ b(u, v, w)∀u, v, w ∈V. The existence of a quasi-equilibrium for (4.1) and (4.2) is guaranteed by the following result: Theorem 4.3. There exists β > 0, only depending on Ω, ν, the ωi, the Oiand the uid, such that, if a/µ ≤β, the optimality system (4.4) possesses at least one solution. Proof. The proof is not difficult. Thus, let us rewrite (4.4) in the form   u ϕ1 ϕ2  =Z  u ϕ1 ϕ2  , where Z:V37→ V3is the following continuous and compact mapping: Z  u ϕ1 ϕ2  := 1 ν  A−1−(u· ∇)u−a µ(ϕ11ω1+ϕ21ω2 A−1−(u· ∇)ϕ1−(∇u)tϕ1+ (u−u1d)1O1 A−1−(u· ∇)ϕ2−(∇u)tϕ2+ (u−u2d)1O2  . Now, we want to find a fixed-point of Z. It is easy to see that, if a/µ is sufficiently small, the fixed-points of Zare uniformly bounded. Consequently, form the Leray-Schauder Principle, we deduce that, for small a/µ, (4.4) is solvable. In this case, we cannot ensure the existence of Nash equilibria. This is due to the lack of uniqueness of the control-to-state mapping and the lack of convexity of the related functionals. However, the following holds: Theorem 4.4. Let ( ˆ f1,ˆ f2) be a Nash equilibrium for (4.1) and (4.2). Then, ( ˆ f1,ˆ f2) is a Nash quasi-equilibrium. We will give a proof of this result relying on the Dubovitsky-Milyutin formalism (see [11]). To this purpose, we must recall some technical results. Lemma 4.5. Let K1, . . . , Knbe convex cones in a Banach space Xwith apex at 0. For each i, we assume that either Kiis open or it is a closed subspace. Then the following conditions are equivalent: • n \ i=1 Ki=∅. •There exist linear functionals fi∈K∗ iwith i= 1, . . . , n, not all zero, such that f1+f2+· · · +fn= 0. Here, for any i, we have denoted by K∗ ithe dual cone to Ki, that is, K∗ i:= {f∈X0:f(e)≥0∀e∈Ki}. For the proof, see for instance Lemma 5.11 of [11]. Note that the adjoint system    −ν∆ϕi+ (u· ∇)ϕi+ (∇u)tϕi+∇qi= (u−uid)1Oiin Ω, ∇ · ϕi= 0 in Ω, ϕi= 0 on ∂Ω, can be equivalently rewritten in the form    −ν∆ϕi−Dϕi·u+∇˜qi= (u−uid)1Oiin Ω, ∇ · ϕi= 0 in Ω, ϕi= 0 on ∂Ω, BI-OBJECTIVE OPTIMAL CONTROL OF SOME PDES 17 where Dϕi=1 2(∇ϕi+(∇ϕi)t) and the pressure has been redefined. This observation will be used in the following lemma and the following section. Lemma 4.6. Let (f1, f2)∈L2(ω1)N×L2(ω2)Nbe given and let u∈H2(Ω)N∩Vbe (together with p) an associated solution to (4.1). Let R:V7→ Vbe the linear mapping defined by Rϕ := (νA)−1(−ˆ B(u, ϕ)), where Ais the (extended) Stokes operator in V. Then, for any non-empty open set ω⊂Ω, ϕω:= kϕ|ωkL2(ω)(4.5) is a norm in N(Id +R). Proof. From the usual alliptic estimates for the solutions to stationary Navier-Stokes systems, one has u∈ H2(Ω)N,∇u∈L6(Ω)N×Nand u∈L∞(Ω)N, see for instance [23]. Consequently, Ris well defined and compact. We only have to prove that, for every ϕ∈N(Id +R) with ϕ|ω= 0, one has ϕ≡0. Thus, let us assume that ϕ∈Vand ϕ+ (νA)−1(−ˆ B(u, ϕ)) = 0, that is,    −ν∆ϕ−Dϕ ·u+∇˜q= 0 in Ω, ∇ · ϕ= 0 in Ω, ϕ= 0 on ∂Ω. Assume that ϕ= 0 a.e. in ω. Then, we can use the unique continuation property of the Stokes system with coefficients in L∞(see [8]) and deduce that, certainly, ϕvanishes identically. Lemma 4.7. Let (f1, f2), (u, p) and ωbe as in Lemma 4.6. Let the (ϕn, ψn) be given in V×Vwith ϕn|ω→ϕ|ωin L2(ω), ψn:= ϕn+Rϕnand ψn→ψ∈V. Then ϕnis uniformly bounded in V. Proof. First, note that, since Ris a compact operator, dim(N(Id +R)) <+∞and R(Id +R) is closed, in view of Fredholm’s Alternative Theorem (see for instance [5]). Now, let ˜ϕnbe, for each n, the unique function in N(Id +R) satisfying kϕn−˜ϕnkH1 0= inf ˜ϕ∈N(Id+R)kϕn−˜ϕkH1 0. Then, ψn= (ϕn−˜ϕn) + R(ϕn−˜ϕn). Also, kϕn−˜ϕnkH1 0is bounded by a constant C. Indeed, if this were not the case, we could assume that kϕn−˜ϕnkV= dist(ϕn, N(Id +R)) →+∞. Let us introduce ζn:= ϕn−˜ϕn kϕn−˜ϕnkH1 0 . Then ζn+Rζn=ψn kϕn−˜ϕnkH1 0 .(4.6) 18 E. FERN´ ANDEZ-CARA AND I. MAR´ IN-GAYTE Since the kζnkV= 1 and ψn→ψin V, we see from (4.6) that ζn→ζfor some ζ∈N(Id +R). So, kϕn−˜ϕn+kϕn−˜ϕnkH1 0kH1 0=kϕn−˜ϕnkH1 0kζn−ζkH1 0≥ kϕn−˜ϕnkH1 0 for all n. But this is an absurd, since ζn→ζin V. Finally, let us deduce that kϕnkH1 0≤C. We can write that ϕn= ˜ϕn+ηn, with • kηnkH1 0=kϕn−˜ϕnkH1 0≤C. • k ˜ϕnkH1 0≤C[ ˜ϕn]ω( note that the ˜ϕnbelong to a finite dimensional space) and [ ˜ϕn]ω≤[ϕn]ω+ [ηn]ω≤[ϕn]ω+CkηnkH1 0≤C. This ends the proof. Now, we can prove the main result in this section. Thus, let ( ˆ f1,ˆ f2) be a Nash equilibrium for (4.1) and (4.2). Let us introduce the sets F1={(f1, u)∈L2(ω1)N×D(A) : (f1,ˆ f2, u) solves (4.1) (together with some pfor f2=ˆ f2} and F2={(f2, u)∈L2(ω2)N×D(A):(ˆ f1, f2, u) solves (4.1) (together with some pfor f1=ˆ f1}. By assumption, there exists a strong solution (ˆu, ˆp) to (4.1) such that ( ˆ f1,ˆu) and ( ˆ f2,ˆu) respectively solve the following extremal problems: Minimize ˆ J1(f1, u) = J1(f1,ˆ f2, u) Subject to (f1, u)∈ F1 (4.7a) and Minimize ˆ J2(f2, u) = J1(ˆ f1, f2, u) Subject to (f2, u)∈ F2.(4.7b) Let us introduce some cones associated with (4.7a) and (4.7b). To this end, we set M1(f1, u) := −νAu +B(u, u)−f11ω1−ˆ f21ω2∀(f1, u)∈L2(ω1)N×V and M2(f2, u) := −νAu +B(u, u)−ˆ f11ω1−f21ω2∀(f2, u)∈L2(ω2)N×V. It is then clear that M1and M2are continuosly differentiable mappings, with M0 i(fi, u)(hi, w) = νAw +B(u, w) + B(w, u)−hi1ωi∀(hi, w)∈L2(ωi)N×V. We also have M0 i(ˆ fi,ˆu)∗ϕi= (−ϕi|ωi, νAϕi−ˆ B(ˆu, ϕi)) ∀ϕi∈V. BI-OBJECTIVE OPTIMAL CONTROL OF SOME PDES 19 Then the cones of descent directions associated with (4.7a) and (4.7b) are the sets Di:= {(hi, v)∈L2(ωi)N×V:hˆ J0 i(ˆ fi,ˆu),(hi, v)i<0} and the spaces of tangent directions are the Ti:= {(hi, v)∈L2(ωi)N×V:M0 i(ˆ fi,ˆu)(hi, v)=0}=N(M0 i(ˆ fi,ˆu)). Let us prove that R(M0 i(f, u)∗) is closed in L2(ωi)N×Vfor each i= 1,2. Indeed, if (h, w)∈R(M0 i(fi, u)∗), there exists fields ϕn∈Vsuch that −ϕn|ωi→hstrongly in L2(ωi)Nand νAϕn−ˆ B(ˆu, ϕn)→wstrongly in V0. Let us introduce the linear mapping S=νA(Id +R), where Ris the linear operator in Lemma 4.6. From Fredholm’s Alternative, we have that R(S) is closed in V0and, from Lemma 4.7, we find that the ϕnare uniformly bounded in V. As an inmediate consequence, we see that, at least for a subsequence, ϕn→ϕweakly in V, with −ϕ|ωi=hand Sϕ =w. In other words, (h, w)∈R(M0 i(ˆ fi,ˆu)∗). At this moment, we can apply the Dubovitskiy-Milyutin formalism to (4.7a) and (4.7b) and deduce that the descent cone Diand the tangent space Tiare disjoint for i= 1,2 (see [11]): Di∩Ti=∅for i= 1,2. In view of Lemma 4.5, there exist (h0 i, w0 i)∈D∗ iand (h0 i, w0 i)∈T∗ i, not both zero, such that (h0 i, w0 i)+(h0 i, w0 i) = (0,0).(4.8) Taking into account the definitions of Diand Tiand the fact that R(M0 i(ˆ fi,ˆu)∗) is closed, we see at once that D∗ i={−λˆ J0 i(ˆ fi,ˆu) : λ≥0}and T∗ i= (N(M0 i(ˆ fiˆu)))⊥=R(M0 i(fi, u)∗). Hence, there exist ϕ1, ϕ2∈Vsuch that h0 i=−ϕi|ωi, w0 i=νAϕi−ˆ B(ˆu, ϕi) (4.9) and (4.8) can be rewritten in the form ZOi (ˆu−uid)wi+µZωi ˆ fihi=Zωi h0 ihi+hw0 i, wii ∀(hi, wi)∈L2(ωi)N×V. (4.10) Taking wi= 0, we find that h0 i=µfi. On the other hand, taking hi= 0 and recalling (4.9), we see that ϕi|ωi=−µˆ fifor each i= 1,2 and ZΩν∇ϕi· ∇wi−(ˆu· ∇)ϕi·wi+ (∇ˆu)tϕi·wi=aZOi (ˆu−uid)wi∀wi∈V. Therefore, ϕisolves, together with some qi∈L2(Ω), the following linear system for each i= 1,2    −ν∆ϕi−(ˆu· ∇)ϕi+ (∇ˆu)tϕi+∇qi=a(ˆu−uid)1Oiin Ω, ∇ · ϕi= 0 in Ω, ϕi= 0 on ∂Ω (4.11a) 20 E. FERN´ ANDEZ-CARA AND I. MAR´ IN-GAYTE Figure 1. Test 1 – The domain and a “rough” mesh; Ω is composed of the bands ω1and ω2, the large rectangle O1and the small rectangle O2. Number of nodes: 1519. Number of triangles: 2876. and, furthermore, ˆ fi=−1 µϕi|ωi.(4.11b) From (4.1), (4.11a) and (4.11b), we deduce that ( ˆ f1,ˆ f2) is a Nash quasi-equilibrium and the proof is done. 4.3. Algorithms This section is dedicated to the presentation of four iterative algorithms for the computation of a Nash quasi-equilibrium for (4.1) and (4.2). Three of them are similar to those in Sections 2.3 and 3.3; the fourth one is of the Newton kind. ALG 7: Fixed-Point-like Method (a) Choose (f0 1, f0 2)∈L2(ω1)N×L2(ω2)Nand u0∈V. (b) Then, for given n≥0, (fn 1, fn 2)∈L2(ω1)N×L2(ω2)Nand un∈V, compute a solution (un+1, pn+1) to    −ν∆un+1 + (un· ∇)un+1 +∇pn+1 =fn 11ω1+fn 21ω2in Ω, ∇ · un+1 = 0 in Ω, un+1 = 0 on ∂Ω, (4.12) a solution (ϕn+1 i, qn+1 i) to the system    −ν∆ϕn+1 i−Dϕn+1 i·un+1 +∇qn+1 i= (un+1 −uid)1Oiin Ω, ∇ · ϕn+1 i= 0 in Ω, ϕn+1 i= 0 on ∂Ω (4.13) BI-OBJECTIVE OPTIMAL CONTROL OF SOME PDES 21 and, finally, set fn+1 i=−a µϕn+1 iωi(4.14) for i= 1,2. ALG 8: Optimal-Step-Gradient-like Method (a) Choose (f0 1, f0 2)∈L2(ω1)N×L2(ω2)Nand u0∈V. (b) Then, for given n≥0, (fn 1, fn 2)∈L2(ω1)N×L2(ω2)Nand un∈V, compute the solution (un+1, pn+1) to (4.12) and the solution (ϕn+1 i, qn+1 i) to (4.13) and set fn+1 i=fn i−ρn ign+1 i,(4.15) where gn+1 i=a ϕn+1 iωi+µfn i(4.16) and ρn 1= argmin ρ≥0 Jn 1(fn 1−ρgn 1, fn 2), ρn 2= argmin ρ≥0 Jn 2(fn 1, fn 2−ρgn 2).(4.17) Here, Jn 1and Jn 2are the quadratic functionals respectively similar to J1and J1obtained when we replace (4.1) by the n-th linear system    −ν∆u+ (un· ∇)u+∇p=f11ω1+f21ω2in Ω, ∇ · u= 0 in Ω, u= 0 on ∂Ω. ALG 9: Optimal-Step-Conjugate-Gradient-like Method (a) Choose (f0 1, f0 2)∈L2(ω1)N×L2(ω2)Nand u0∈V. (b) For n= 0, perform one step of ALG 8 and set d0 i=g0 i. (c) Then, for given n≥1, (fn 1, fn 2)∈L2(ω1)N×L2(ω2)Nand un∈V, compute the solution (un+1, pn+1) to (4.12), the solution (ϕn+1 i, qn+1 i) to (4.13) and then set fn+1 i=fn i−ρn idn i,(4.18) where    dn i=gn i+γn idn−1 i, γn i=(gn i−gn−1 i, gn i) kgn−1 ik2, gn i=a ϕn i|ωi+µfn i (4.19) and ρn 1= argmin ρ≥0 Jn 1(fn 1−ρdn 1, fn 2), ρn 2= argmin ρ≥0 Jn 2(fn 1, fn 2−ρdn 2).(4.20) 22 E. FERN´ ANDEZ-CARA AND I. MAR´ IN-GAYTE As in Section 3, we find in ALG 8 and ALG 9 some 1D extremal problems. Again, the functions to minimize are not quadratic and a numerical method is needed. This will be explained below. Before presenting the results of some simulations, let us describe another algorithm. It is based on Newton’s iterates and, like ALG 7, aims to compute directly a solution to the optimality system (4.4). In practice, this method is much faster but, as is usual for Newton methods and variants, needs a nontrivial starting process (see below). ALG 10: Newton Method We want to solve (4.4) with ν= ˜ν. We fix a decreasing factor ˜a∈(0,1) and we do as follows. (a) Choose (f0 1, f0 2)∈L2(ω1)N×L2(ω2)Nand ν0∈R+and compute a solution (u0, p0) to the Stokes problem    −ν0∆u0+∇p0=f0 11ω1+f0 21ω2in Ω, ∇ · u0= 0 in Ω, u0= 0 on ∂Ω (4.21) and a solution (ϕ0 i, q0 i) to    −ν0∆ϕ0 i+∇q0 i= (u0−uid)1Oiin Ω, ∇ · ϕ0 i= 0 in Ω, ϕ0 i= 0 on ∂Ω (4.22) and take f0 i=−a µϕ0 iωifor i= 1,2 and then ν1= max{˜ν, ˜aν0}. (b) Then, for given n≥0, νn, (fn 1, fn 2)∈L2(ω1)N×L2(ω2)N, (un, pn) and (ϕn i, qn i), do the following: (b.1) Take fn,0 i=−a µϕn i|ωi, un,0=un, ϕn,0 i=ϕn iand νn+1 = max(˜aνn,˜ν). (b.2) For given k≥0, fn,k i, un,k and ϕn,k i, set until convergence Fn,k := −νn+1∆un,k + (un,k · ∇)un,k −fn,k 11ω1−fn,k 21ω2 and Gn,k i:= −νn+1∆ϕn,k i−(un,k · ∇)ϕn,k i+ (∇un,k)tϕn,k i−(un,k −uid)1Oi, compute the solution (vk, hk, ψk i, ηk i) to                    −νn+1∆vk+ (un,k · ∇)vk+ (vk· ∇)un,k +∇hk=Fn,k,in Ω, ∇ · vk= 0 in Ω, vk= 0 on ∂Ω, −νn+1∆ψk i−(un,k ·∇)ψk i−(vk·∇)ϕn,k i + (∇un,k)tψk i+ (∇vk)tϕn,k i+∇ηk i=Gn,k iin Ω, ∇ · ψk i= 0 in Ω, ψk i= 0 on ∂Ω (4.23) and take un,k+1 =un,k −vk, ϕn,k+1 i=ϕn,k i−ψk i, νn+1 = max(˜ν, ˜aνn).(4.24) BI-OBJECTIVE OPTIMAL CONTROL OF SOME PDES 23 Figure 2. Test 1 – The function u1d. Note that ALG 7 and ALG 10 are designed to compute a solution to the optimality system (4.4) (a quasiequilibrium) that, possibly, is not be a Nash equilibrium. Thus, from the viewpoint of the Calculus of Variations, ALG 7 and ALG 10 can be seen as “indirect methods”. Contrarily, ALG 8 and ALG 9 can be viewed as realizations of the “direct method” of Calculus of Variations. 4.4. Numerical experiments In order to illustrate the behavior of the previous algorithms, we will present in this section the results of some numerical experiments. The computations have been performed with the FreeFem++ package (see [14]). In our experiments, we try to solve numerically the optimality system (4.4), where ais chosen with 0 <a<2 and µ= 2 −a. In the tests, several values of aand the Reynolds number Re := UL/ν have been chosen. Here, the characteristic speed and length are respectively given by U= max(ku1dk,ku2dk) and L= maxx,x0∈Ω|x−x0|. 4.4.1. Test 1 In this first test, our domain is composed of two rectangles O1and O2and we assume that the controls act on the narrow bands ω1and ω2. In order to solve numerically the systems (4.12), (4.13), (4.21), (4.22) and (4.23), we have used meshes like the one in Figure 1and a mixed finite element formulation with continuous piecewise P1-bubble and P1functions respectively for the velocity field and the pressure; for details, see [12,13]. The data uid are the following: u1d=∇ × ψ1d,where ψ1dis the solution to the problem −∆ψ1d= 1 in O1, ψ1d= 0 on ∂O1(4.25) and u2d≡0. That means that the “desired” (ideal) configurations correspond to a uniformly rotating flow in O1and a fluid at rest in O2(see Fig. 2). For ALG 8 and ALG 9, the computation of the steps ρn ihas been performed by applying a standard dichotomy process. This provides acceptable values after a reduced amount of iterates. 24 E. FERN´ ANDEZ-CARA AND I. MAR´ IN-GAYTE Figure 3. Test 1 – The computed velocity fields computed with ALG 8 for various Re. Here, a= 1.2 and µ= 2 −a= 0.8. In the case of ALG 7,ALG 8 and ALG 9, the stopping test has been kun+1 −unkL∞+kpn+1 −pnkL∞+kqn+1 −qnkL∞≤ε, with ε= 10−6Re. This choice establishes a criterion with uniform severity. This has also been the stopping criterion for the external iterates in ALG 10. For the internal loops (indexed by k), the stopping test has been kun,k+1 −un,kkL∞+kϕn,k+1 −ϕn,kkL∞≤ε. BI-OBJECTIVE OPTIMAL CONTROL OF SOME PDES 25 Figure 4. Test 1 – The computed velocity field with ALG 10 for Re = 141. Now, a= 1.8 and µ= 2 −a= 0.2. The mesh is composed of 6003 vertices and 11684 triangles; maxΩ|u|= 3.17. Figure 5. Test 1 – The numbers of iterates for various values of Re,aand µ= 2 −a. In all cases, the starting control pair has been (f0 1, f0 2) = (0,0). On the other hand, for all choices of aand Re, all the algorithms provide the same solution. Some results are depicted in Figure 3. There, we can see the computed velocity fields corresponding to some Nash quasi-equilibria, for a= 1.2 and various Re. As expected, the appearance of the velocity field in O1 (resp. O2) reminds us of u1d(resp. u2d). We also present in Figure 4the velocity field computed with ALG 10, a finer mesh and a higher Re. In order to compare the behavior of the Gradient-like, Conjugate Gradient-like and Newton methods, we present some related numerical values in the Tables in Figure 5. Specifically, we have indicated there the numbers of iterates needed in each case to fulfill the stopping test and produce an approximation. In the case of ALG 10, we have included the total numbers of iterates (that is, the iterates needed for initialization have been taken into account). Also, we have included in Figure 6a Table with a comparison of the computation times and required numbers of iterates for each method. The convergence of the numerical aproximations toward a quasi-equilibrium has been checked in the following way. We have considered several meshes corresponding to triangles of typical sizes 0.2,0.13,0.1, .. . , as indicated in Table 1. For h= 0.05, with 23327 vertices and 46012 triangles, we have solved (4.1)–(4.2) for a= 1.8