In this paper our aim is to present a survey of known results of an optimal control problem with concave non-quadratic cost functional and a state equation arising from population dynamics. First, we study in detail the state equation, and then we show existence and uniqueness of optimal control and also a numerical approximation of the optimal control.Ministerio de Ciencia y Tecnologí

Suárez Fernández, Antonio

idUS. Depósito de Investigación Universidad de Sevilla

Matema´tica Contemporaˆnea, Vol 32, 219-233c©2007, Sociedade Brasileira de Matema´ticaSTUDY OF THE OPTIMAL HARVESTING CONTROLFOR AN ELLIPTIC PROBLEM WITH SLOWDIFFUSIONA. Sua´rezTo Professor Luis Adauto Medeiros on the occasion of his eightieth birthdayAbstractIn this paper our aim is to present a survey of known results of anoptimal control problem with concave non-quadratic cost functional anda state equation arising from population dynamics. First, we study indetail the state equation, and then we show existence and uniquenessof optimal control and also a numerical approximation of the optimalcontrol.1 IntroductionConsider a regular domain Ω of IRN , N ≥ 1, and the following payoff functionalJ : L∞+ (Ω) 7→ IR, J(f) :=M∫Ωfuf −K∫Ωf2,where L∞+ (Ω) := {f ∈ L∞(Ω) : f(x) ≥ 0 a. a. x ∈ Ω}, M and K are positiveconstants and uf is the maximal solution (see Section 2) of the state equation,the degenerate or slow diffusion logistic equation−∆u = (a− f)uα − buβ in Ω,u = 0 on ∂Ω.(1.1)The author thanks to the Spanish Ministry of Science and Technology under grantBFM2003-06446.Key Words. Degenerate logistic equation, optimal control, optimality system.AMS Classification. 49J20, 49K20, 35J65, 92D25.220 A. SUA´REZThroughout this paper a, f and b are bounded functions with a strictly positive,b non-negative and nontrivial, a− f can change sign and α and β satisfy0 < α < 1, α < β. (1.2)The solutions of (1.1) can be regarded as the steady states solutions of thecorresponding time dependent model. In such case, u(x) stands for the popu-lation density and Ω for the inhabiting area. Since the population is subject tohomogeneous Dirichlet boundary conditions, we are assuming that the environ-ment surrounded Ω is lethal. In such model, a(x) represents the growth rate ofthe species and the positive function b(x) describes the intro-specific pressureof the species: in the region where b ≡ 0 the species is free of this pressureand so u grows according to the Malthus law, while in its complement u growsaccording to a logistic or Verhulst law. The function f(x), will be considerednon-negative, denotes the distribution of control harvesting of the species byreducing the growth rate.With respect to the functional, J represents the difference between economicrevenue measured by∫Ωfuf and the control cost measured by∫Ωf2. Theconstants M and K denote the price of the species and the cost of the control,respectively.Our objective is to know if it is possible to maximize the benefit, that is,mathematically we want to show:a) The existence of an optimal control, i. e., to find a function f0 ∈ L∞+ (Ω)such thatJ(f0) = supf∈L∞+(Ω)J(f).b) Under what conditions is this optimal control unique?STUDY OF THE OPTIMAL HARVESTING CONTROL 221c) Approximate numerically the optimal control.In population dynamics, it seems that the first work in optimal controlis [16], where the case α = 1, b strictly positive and homogeneous Neumannboundary conditions is considered. In such work, a bounded set of controls isconsidered, the existence of optimal control is shown, as well as the associatedoptimality system is obtained. Finally, some computational experiments aremade. After that, and always in the case α = 1 and b strictly positive, in [9],[2] and [18], and under homogeneous Dirichelt boundary conditions and controlspace L∞+ (Ω), the existence and uniqueness of optimal control are proved. Also,via optimality system, the optimal control is numerically approximated.However, the case α < 1 has not been considered previously in our view (see[20] for a related problem). Being α < 1, some important technical difficultiesappear, and over all, a drastic change in the results on the state equation.Indeed, for the case α = 1 any non-negative and nontrivial solution is in factpositive; however in the case α < 1 solutions u ≥ 0, u 6= 0 and no positive inthe whole domain exist.First, we enunciate the main results concerning to the state equation. Then,our aim will be to summary the papers [4] and [5] where the optimal controlproblem in the case α < 1 was studied. We show the existence of optimalcontrol, and for λ small the uniqueness of optimal control and a characterizationof the optimal control via the optimality system is also derived, which furtherleads to approximating the optimal control.222 A. SUA´REZ2 State equationEquation (1.1) includes the following one (under the change of variable wm = u)−∆wm = (a− f)w − bw2 in Ω,w = 0 on ∂Ω,(2.1)which was introduced in population dynamics by Gurtin and MacCamy in [10]for describing the dynamics of biological populations whose mobility dependsupon their density. In this context, m > 1 (nonlinear slow diffusion) meansthat the diffusion is slower than in the linear case m = 1, which gives rise tomore realistic biological results, see [10].By biological meaning, we are only looking for non-negative and nontriv-ial solutions. Moreover, by elliptic regularity, any bounded solution of (1.1)belongs to C1,α(Ω).We first recall the case m = 1 in (2.1) (or equivalently α = 1 in (1.1)). Weneed previously some notation: given a regular domain D ⊂ IRN and a functiong ∈ L∞(D), we denote by λD1 (−∆+ g) the first eigenvalue of the problem−∆u+ g(x)u = λu in D, u = 0 on D.It holds (see Theorem 3.5 in [8]):Proposition 2.1. Assume α = 1 and that B0 := int{x ∈ Ω : b(x) = 0} isregular.a) Any solution u ≥ 0, u 6= 0 of (1.1) is in fact u > 0 in the whole domainΩ.b) There exists a positive solution of (1.1) if, and only if,λΩ1 (−∆+ f − a) < 0 < λB01 (−∆+ f − a).STUDY OF THE OPTIMAL HARVESTING CONTROL 223When there exists, the solution is unique and globally stable.Observe that either the solution is trivial u ≡ 0 or it is positive in all the domain.On the other hand, the existence of solution depends on a global condition,specifically of the behavior of a − f in a determined domain. Moreover, incase of existence, there is uniqueness. Finally, we would like to recall that ifλB01 (−∆+ f − a) ≤ 0 the solutions corresponding to the associated parabolicproblem are not bounded, see [8].A drastic change occurs whenm > 1, due to that strong maximum principledoes not hold in general, appearing solutions u ≥ 0, u 6= 0 and not strictlypositive, that is the setΩ0 := {x ∈ Ω : u(x) = 0} 6= ∅,this set is called dead core. Moreover, as we can see in the following result, allthe solutions are bounded and the multiplicity of solutions appears; as well asthe condition assuring the existence of solution is local. Given g ∈ L∞(Ω) wedenote bygM := ess supΩg gL := ess infΩg.We can summarize the main result as follows:Theorem 2.2. a) There exists a maximal solution of (1.1), denoted byuf ≥ 0. Moreover, the map f 7→ uf is continuous and decreasing, and so‖uf‖∞ ≤ ‖u0‖∞ ≤ K, for some K independent of f . (2.2)b) The maximal solution uf ≥ 0, it is not trivial if, and only if, (a−f)M > 0.c) If (a−f)L > 0 then uf > 0 and it is the unique positive solution of (1.1).Moreover, if there exists a positive solution, this is the unique positivesolution.224 A. SUA´REZd) We can choose functions f such that uf has dead core.e) If the dead core of uf has k connected components (k ≥ 1), then thereexist exactly 2k − 1 solutions of (1.1).Equation (1.1) was previously studied in [11] and [19], where basically para-graph e) and fist part of a) were proved. The rest of results have been provedin [6] and [7].3 Optimal controlRescaling the functional J , we can considerJ : L∞+ (Ω) 7→ IR, J(f) :=∫Ωλfuf − f2,with λ > 0. First we can prove the following result, which provides us abound for any optimal controls and it shows that at the optimal control thecorresponding solution uf has not dead core:Lemma 3.1. Let f be an optimal control. Then,f ≤ λK. (3.1)Moreover, uf > 0 in Ω.With respect to the existence of optimal control, thank to (3.1) we can takea maximizing sequence and pass to the limit, obtainingTheorem 3.2. There exists an optimal control, i. e., f ∈ L∞+ (Ω) such thatJ(f) = supg∈L∞+(Ω)J(g).Moreover, the benefit is positive, i.e., supg∈L∞+(Ω)J(g) > 0.STUDY OF THE OPTIMAL HARVESTING CONTROL 225In order to obtain the uniqueness result, we will use two different ways.First, again thank to the bound (3.1), we have that the set of optimal controlsbelongs to a convex, specifically to the setI := [0, λK].Hence, if we show that the functional J is concave, we will deduce the unique-ness of optimal control. For that, we calculate the derivative of J in the fol-lowing set:C := {f ∈ L∞(Ω) : (a− f)L > 0}.Proposition 3.3. J is Fre´chet differentiable andJ ′(f)(g) =∫Ω(λuf − λuαfPf − 2f)g, ∀f ∈ C, ∀g ∈ L∞(Ω), (3.2)where for any f ∈ C, Pf ∈ C10 (Ω) is the unique solution of−∆Pf +Mf (x)Pf = f in Ω,Pf = 0 on ∂Ω,(3.3)beingMf = −α(a− f)uα−1f + βbuβ−1f .Some remarks are in order: observe that (3.3) is a linear elliptic equationbut with a potential Mf blowing up at the boundary of Ω. In spite of that,we have solved this difficulty because Mf blows up in a controlled way, due tothat f ∈ C. Indeed, since f ∈ C it follows by Theorem 2.2 that uf > 0 and infact uf belongs to the interior of the positive cone of C10 (Ω). This allows us tosolve (3.3), see for instance Theorem 2.5 in [5].On the other hand, having the derivative of J only in C is not a restriction,because thank to (3.1) there exists λ0 > 0 such that for λ < λ0 it holds thatf ∈ C. (3.4)226 A. SUA´REZThe next result is essential in our work, it is a complex technical result(Theorem 4.6 in [5]):Theorem 3.4. There exists Λ > 0 such that for 0 < λ < Λ the mapsf ∈ [0, λK] 7→ uf , Pf , uαfPfare Lipschitz continuous, with the Lipschitz constants independent of λ, in thefollowing sensemax{‖uf − ug‖∞, ‖Pf − Pg‖∞, ‖uαfPf − uαgPg‖∞} ≤ L‖f − g‖2.We are ready to show:Theorem 3.5. There exists Λ0 > 0 such that if λ < Λ0, there exists an uniqueoptimal control.Idea of the proof: Take λ < λ0 of (3.4), and so f ∈ C. Take also λ < Λ ofTheorem 3.4, then using Proposition 3.3 we get(J ′(f)− J ′(g))(f−g)=∫Ω[λ(uf−ug)+λ(uαgPg−uαfPf )−2(f−g)](f−g).Observe that, by the Ho¨lder inequality∫Ω(uf − ug)(f − g)≤(∫Ω(uf − ug)2)1/2(∫Ω(f − g)2)1/2≤ CL∫Ω(f − g)2.where L is the Lipschitz constant of Theorem 3.4 and C a positive constant.Thus, we deduce(J ′(f)− J ′(g))(f − g) ≤∫Ω(λLC − 2)(f − g)2 < 0.This conclude that J is concave, and the uniqueness follows.As a consequence of (3.2), we get easilySTUDY OF THE OPTIMAL HARVESTING CONTROL 227Corollary 3.6. Let f ∈ L∞+ (Ω) an optimal control. Then for λ < λ0,f =12λ(uf − uαfPf )+.We can derive now the optimality system, for that we use the above expres-sion and substitute it in (2.1), and the equation (3.3).Proposition 3.7. There exists Λ1 > 0 such that if λ ≤ Λ1, thenPf ≤ u1−αf .So, if f is an optimal control, we havef =λ2uf (1− uα−1f Pf ). (3.5)As consequence, any optimal control f can be expressed as in (3.5), where thepair (uf , Pf ) := (u, P ) satisfies−∆u = uα(a− λ2u+ λ2uαP − buβ−α) in Ω,−∆P+(−αauα−1+βbuβ−1)P = λ2(u− uαP (1+α)+αu2α−1P 2) in Ω,u=P =0 on ∂Ω,(3.6)and u > 0.The former result says that, when λ is small enough, if f is an optimalcontrol, then (uf , Pf ) is a solution of (3.6). We are going to prove now that, fora range of λ, there exists a unique positive solution of (3.6) verifying u1−α ≥ Pand so the unique optimal control will bef =λ2(u− uαP ).Observe that the system (3.6) is nonlinear in the variables u and P with somenonlinearities singular. So, the study of this system is difficult, an even morethe uniqueness, that as it is well know is one of the hardest problem in ellipticproblems. We can show the following result:228 A. SUA´REZTheorem 3.8 (Uniqueness of optimal control). There exists Λ2 > 0 suchthat for λ ≤ Λ2, (3.6) possesses a unique positive solution (u, P ) satisfyingu1−α ≥ P .Proof: Define the map:T : I := [0, λK] ⊂ L∞+ (Ω) 7→ L∞+ (Ω), f 7→ T (f) =λ2(uf − uαfPf ).By Theorem 3.4, for λ < Λ, T is a Lipschitz continuous function with a Lip-schitz constant of the form λL/2, where L is the corresponding one for thefunction f 7→ uf − uαfPf . So, we can choose Λ2 := min{Λ,2L} such that forλ ≤ Λ2, T is a contractive function.Assume that there exist two positive solutions (ui, Pi), i = 1, 2 of (3.6) withu1−αi ≥ Pi. We definefi =λ2(ui − uαi Pi) ∈ I, i = 1, 2.Hence, by (3.6) and the uniqueness of uf (see Theorem 2.2 e)), we have thatui = ufi , Pi = Pfi , ⇒ T (fi) = fi i = 1, 2.Since T is contractive, it follows that f1 = f2, and again by the uniqueness ofuf we have that uf1 = uf2 , hence u1 = u2, and so P1 = P2. This completesthe proof.Now, we want to approximate the optimal control, for which we approximatethe solutions of (3.6). We are going to use the sub-supersolution method,which provides not only the existence of solution, but sequences approachingthe solutions.STUDY OF THE OPTIMAL HARVESTING CONTROL 229We can define (see [5]) a function B(x, u, p) increasing in p, and, dependingon the values of α and β, functions C(x, u, p) decreasing in u and D(x, u, p)increasing in u such that the optimality system (3.6) can be written as−∆u = B(x, u, p) in Ω,−∆p = C(x, u, p) +D(x, u, p) in Ω,u = p = 0 on ∂Ω.Now, we are going to apply the sub-supersolution method to this system andobtainProposition 3.9. There exists a pair (u, u) − (p, p) of sub-supersolution of the above system, and so there exist sequencesu = u1 ≤ u2 ≤ ... ≤ un ≤ un ≤ un−1 ≤ ... ≤ u1 = u,p = p1 ≤ p2 ≤ ... ≤ pn ≤ pn ≤ pn−1 ≤ ... ≤ p1 = p,andun ր u∗, un ց u∗, pn ր p∗, pn ց p∗,and for any solution (u, p) of (3.6), we getun ≤ u∗ ≤ u ≤ u∗ ≤ un, pn ≤ p∗ ≤ p ≤ p∗ ≤ pnand (u∗, u∗, p∗, p∗) verifies the following system−∆u∗ = B(x, u∗, p∗) in Ω,−∆u∗ = B(x, u∗, p∗) in Ω,−∆p∗ = C(x, u∗, p∗) +D(x, u∗, p∗) in Ω,−∆p∗ = C(x, u∗, p∗) +D(x, u∗, p∗) in Ω,u∗ = u∗ = p∗ = p∗ = 0, on ∂Ω.(3.7)230 A. SUA´REZClearly, if (u, p) is the solution of the optimality system (3.6), then (u, u, p, p)is a solution of (3.7). So, to complete the iterative approximation and the con-vergence of the sequences {un}, {un}, {pn}, {pn} to the unique solution, (u, p),of the optimality system, it is sufficient to prove the uniqueness of the solutionfor the system (3.7).With a similar argument to the used in Theorem 3.8, we haveTheorem 3.10. There exists a positive constant Λ4 such that, if λ ≤ Λ4, thenthe system (3.7) possesses a unique solution.Remark 3.11. a) The existence and uniqueness results are still true formore general functionals asJ(g) :=∫Ω(λh(g)ug − k(g)),where h ∈ C1(IR+; IR+), k ∈ C2(IR+; IR+); h(s) = 0 if, and only if, s = 0,and k(s) = 0 if, and only if, s = 0; h is concave and k strictly convexsatisfying k′′(s) ≥ k0 > 0 for some k0. Moreover:(H) limt→0k(t)h(t)= 0, limt→+∞k(t)h(t)= +∞.b) Different optimal control problems have been analyzed when the stateequation is a system, see for instance the works [17], [13], [15], [14],[1], [3] and [12]. In all of them, the control appears in both equations,being an interesting open problem the case with the control acting in onlyone equation.References[1] Arino, O.; Montero, J. A., Optimal control of a nonlinear elliptic popula-tion system, Proc. Edinburgh Math. Soc., 43 (2000), 225-241.STUDY OF THE OPTIMAL HARVESTING CONTROL 231[2] Can˜ada, A.; Ga´mez, J. L.; Montero, J. A., Study of an optimal controlproblem for diffusive nonlinear elliptic equations of logistic type, SIAM J.Control Optim., 36 (1998), 1171-1189.[3] Can˜ada, A.; Magal, P.; Montero, J. A., Optimal control of harvesting ina nonlinear elliptic systems arising fron population dynamics, J. Math.Anal. Appl., 254 (2001), 571-586.[4] Delgado, M., Montero, J. A. Sua´rez, A., Optimal control for the degenerateelliptic logistic equation, Appl. Math. Optim., 45 (2002), 325-345.[5] Delgado, M.; Montero, J. A.; Sua´rez, A., Study of the optimal harvestingcontrol and the optimality system for an elliptic problem, SIAM J. ControlOptim., 42 (2003), 1559-1577.[6] Delgado, M.; Sua´rez, A., On the existence of dead cores for degenerateLotka-Volterra models, Proc. Royal Society of Edin. A, 130 (2000), 743-766.[7] Delgado, M.; Sua´rez, A., On the structure of the positive solutions of thelogistic equation with nonlinear diffusion, J. Math. Anal. Appl., 268(2002), 200-216.[8] Fraile, J. A.; Koch-Medina, P.; Lo´pez-Go´mez, J.; Merino, S., Ellipticeigenvalue problems and unbounded continua of positive solutions of asemilinear elliptic equation, J. Diff. Eqns., 127 (1996), 295-319.[9] Ga´mez, J. L.; Montero, J. A., Uniqueness of the optimal control for aLotka-Volterra control problem with a large crowding effect, ESAIM Con-trol Optim. Calc. Var. , 2 (1997), 1-12.232 A. SUA´REZ[10] Gurtin, M. E.; MacCamy, R. C., On the diffusion of biological populations,Math. Biosci., 33 (1977), 35-49.[11] Herna´ndez, J., Positive solutions for the logistic equation with unboundedlimitations, in “Reaction diffusion systems”, Lecture Notes in Pure andApplied Mathematics, 194 (1997), 183-197.[12] Lenhart, S.; Montero, J. A., Optimal control of harvesting in a parabolicsystem modelling two subpopulations, Math. Models Methods Appl. Sci.,11 (2001), 1129-1141.[13] Leung, A. W., Optimal harvesting-coefficient control of steady-state prey-predator diffusive Volterra-Lotka systems, Appl. Math. Optim., 31 (1995),219-241.[14] Leung, A. W., Positive solutions for systems of PDE and optimal control,Non. Anal., 47 (2001) 1345-1356.[15] Leung, A. W.; Chen, G. S., Optimal control of multigroup neutron fissionsystems, Appl. Math. Optim., 40 (1999), 39-60.[16] Leung, G. S.; Stojanovic, S., Optimal control for elliptic Volterra-Lotkatype equations, J. Math. Anal. Appl., 173 (1993), 603-619.[17] He, F.; Leung, G. S.; Stojanovic, S., Periodic optimal control for competingparabolic Volterra-Lotka-type systems, J. Comp. App. Math., 52 (1994),199-217.[18] Montero, J. A., A uniqueness result for an optimal control problem on adiffusive elliptic Volterra-Lotka type equation, J. Math. Anal. Appl., 243(2000), 13-31.STUDY OF THE OPTIMAL HARVESTING CONTROL 233[19] Pozio, M. A.; Tesei, A., Support properties of solutions for a class of de-generate parabolic problems, Comm. Partial Diff. Eqns., 12 (1987) 47-75.[20] Stojanovic, S., Modeling and minimization of extinction in Volterra-Lotkatype equations with free boundaries, J. Diff. Eqns., 134 (1997), 320-342.Dpto. de Ecuaciones Diferenciales y Ana´lisis Nume´ricoFac. de Matema´ticas, Univ. de SevillaCalle Tarfia s/n, 41012-Sevilla, SpainE-mail: suarez@us.es

Study of the optimal harvesting control for an elliptic problem with slow diffusion

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Study of the optimal harvesting control for an elliptic problem with slow diffusion

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