Optimal control for the degenerate elliptic logistic equation

1 OPTIMAL CONTROL FOR THE DEGENERATE ELLIPTIC LOGISTIC EQUATION M. Delgado1, J. A. Montero2and A. Su´arez1 1. Dpto. Ecuaciones Diferenciales y An´alisis Num´erico Fac. Matem´aticas, C/ Tarfia s/n C. P. 41012, Univ. Sevilla, Sevilla, Spain 2. Dpto. An´alisis Matem´atico C. P. 18071, Univ. Granada, Granada, Spain e-mails: [email protected], jmon[email protected] and [email protected] Abstract We consider the optimal control of the harvesting of the diffusive degenerate elliptic logistic equation. Under certain assumptions, we prove the existence and uniqueness of an optimal control. Moreover, the optimality system and a characterization of the optimal control are also derived. Sub-supersolution method, singular eigenvalue problem and differentiability with respect to the positive cone are the techniques used to get our results. Key Words. Degenerate logistic equation, Singular eigenvalue problems, Optimal control. AMS Classification. Primary 49J20, 49K20, 92D25, Secondary 35J65. Running head. Optimal control for degenerate logistic equation 2M. Delgado, J. A. Montero and A. Su´arez 1 Introduction This work considers the optimal harvesting control of a species whose state is governed by the degenerate (nonlinear slow diffusion) elliptic logistic equation, i.e.,          −∆wm= (a−f)w−ew2in Ω, w= 0 on ∂Ω, (1.1) where Ω is a bounded and regular domain of IRN,N≥1; m > 1; a, f and eare bounded functions with some restrictions that will be detailed below. Equ. (1.1) was introduced in populations dynamics by Gurtin and MacCamy in [5] describing the behaviour of a single species inhabiting in Ω and whose population density is w(x). Since the population is subject to homogeneous Dirichlet boundary conditions, we are assuming that Ω is fully surrounded by inhospitable areas. In such model, the positive function e(x) describes the limiting effects of crowding in the species and a(x) represents the growth rate of the species. The function f(x) denotes the distribution of control harvesting of the species. Since fwill be considered non-negative, observe that fleads by reducting the growth rate. Finally, the operator −∆ measures the diffusion, i.e., the moving rate of the species from high density regions to low density areas. In this case, m > 1 (nonlinear slow diffusion) means that the diffusion is slower than in the linear case m= 1, which gives rise to more realistic biological results, see [5]. To study (1.1), we make the change of variables wm=uand obtain          −∆u= (a−f)uα−euβin Ω, u= 0 on ∂Ω, (1.2) with α= 1/m and β= 2/m. Under hypothesis (H2) below, we prove that for each f, there exists a unique positive solution of (1.2), that it will be denoted by uf. The optimal control criteria is to maximize the payoff functional J(f) := ZΩ (λufh(f)−k(f)), where h∈C1(IR+; IR+), k∈C2(IR+; IR+) and λ > 0 will be considered as parameter. J represents the difference between economic revenue measured by RΩλufh(f) and the control cost measured by RΩk(f). Here, λdescribes the quotient between the price of the species and Optimal control for degenerate logistic equation 3 the cost of the control. The special case (quadratic functional) h(t) = tand k(t) = t2, was introduced in dynamics population by Leung and Stojanovic in [10] (see also [3], [9] and references therein). An optimal control is a function f∈ C, where Cis a suitable subset of L∞(Ω), such that J(f) = sup g∈C J(g). In the case m= 1, i.e., α= 1 and β= 2, and h(t) = tand k(t) = t2, this problem has been studied in detail in [3], [10] and [11]. In fact, some results of this work have been motivated by [3]. In these papers, under certain assumptions in the coefficients of the problem, the authors obtained the existence and uniqueness of the optimal control, as well as a characterization of the optimal control by means the solution of the optimality system. To obtain the results, the authors used mainly the sub-supersolution method, the derivability of the maps f7→ ufand f7→ J(f) and the expressions of their derivatives. When m > 1, i.e. α < 1, this derivability is rather difficult than in the case m= 1, because it involves linear elliptic and eigenvalue problems with unbounded potentials in a neighbourhood of ∂Ω. These difficulties have been solvented by using results of singular eigenvalue problems from [2] and [6], and some classical ones of Krasnoselskii, see [7]. They let us deduce the Fr´echet derivability from the Gˆateaux derivability with respect to the positive cone. Moreover, the introduction of the functions hand kin the payoff functional leads us to establish the hypotheses to assure the existence and uniqueness of the optimal control. An outline of this work is as follows: in Section 2 we introduce some notations and we give some results of the existence and uniqueness of the principal eigenvalue and of solution of a linear elliptic problems with unbounded potentials. In Section 3 we show the existence and uniqueness of positive solution of (1.2), collecting a result from [4]. Moreover, we study the derivability of the map f7→ ufgiving an explicit expression of that. In Section 4, we show that for λ sufficiently small there exists a unique optimal control. In the last Section we characterize the optimal control. This characterization provides us the optimality system and certain regularity of the optimal control. It is well known that this regularity can suggest numeric methods to approximate the optimal control, which are not considered in this work. 4M. Delgado, J. A. Montero and A. Su´arez 2 Preliminaries In this paper we use the following notation: Ω is a bounded domain in IRNwith a smooth boundary ∂Ω and γ∈(0,2) fixed. For any f∈L∞(Ω) we denote fM:= ess sup f fL:= ess inf f, L∞ +(Ω) := {f∈L∞(Ω) : fL≥0}L∞ −(Ω) := {f∈L∞(Ω) : fM≤0}. Moreover, we denote by Pthe non-negative cone of C1 0(Ω), whose interior is int(P) := {u∈C1 0(Ω) : u > 0 in Ω, ∂u/∂n < 0 on ∂Ω} where C1 0(Ω) = {u∈C1(Ω) : u= 0 on ∂Ω}and nis the outward unit normal at ∂Ω. Finally, for any Ω0⊂Ω, σΩ0 1and ϕΩ0 1stand for the principal eigenvalue and the corresponding positive eigenfunction of the operator −∆ and homogeneous boundary Dirichlet condition with kϕΩ0 1k∞= 1. In particular, we write σ1:= σΩ 1and ϕ1:= ϕΩ 1. Assume (H1) M∈L∞ loc(Ω) verifying M(x)dΩ(x)γ∈L∞(Ω), where dΩ(x) := dist(x, ∂Ω). Given σ∈IR and f∈L∞(Ω), we consider the following problems          −∆u+M(x)u=σu in Ω, u= 0 on ∂Ω, (2.1)          −∆u+M(x)u=fin Ω, u= 0 on ∂Ω. (2.2) Remark 2.1 Observe that we are not assuming that M∈L∞(Ω) and that a weak solution of (2.2) or an associated eigenfunction to the eigenvalue σof (2.1) are well defined by the Hardy inequality, see for instance [8]. The next result follows from [2] and [6]. We include it for the reader’s convenience. Theorem 2.2 Assume that Msatisfies (H1). Then: Optimal control for degenerate logistic equation 5 a) There exists a unique principal eigenvalue (i.e., a real eigenvalue with an associated eigenfunction in int(P)), which is simple and we denote it by σ1(−∆ + M). Moreover, it satisfies σ1(−∆ + M) = inf u∈H1 0(Ω)\{0}        ZΩ |∇u|2+ZΩ M(x)u2 ZΩ u2        . b) (Strong Maximum Principle) σ1(−∆ + M)>0if and only if v∈W2,p(Ω) ∩C1(Ω), with p > N such that v6= 0,−∆v+M(x)v≥0in Ω,v≥0on ∂Ω, then v∈int(P). By the variational characterization of σ1(−∆ + M), it follows: Proposition 2.3 a) (Monotonicity respect to the potential) Assume that Mi,i= 1,2satisfy (H1) and M1≤M2. Then σ1(−∆ + M1)≤σ1(−∆ + M2). b) (Continuity respect to the potential) Assume that Mn, M,n∈IN satisfy (H1) with ZΩ Mnϕ2→ZΩ Mϕ2,as n→ ∞ and for all ϕ∈H1 0(Ω).(2.3) Then, σ1(−∆ + Mn)→σ1(−∆ + M)as n→ ∞. The following estimate will play an important role in the next sections. Lemma 2.4 Assume that Mn, M,n∈IN satisfy (H1),σ1(−∆+M)>0and (2.3). Then, there exist a positive constant C0<1(independient of n) and n0(C0)∈IN such that C0ZΩ |∇u|2≤ZΩ |∇u|2+ZΩ Mnu2∀u∈H1 0(Ω),∀n≥n0.(2.4) Proof: Since σ1(−∆ + KM)→σ1(−∆ + M)>0 as K↓1, there exists K0>1 such that σ1(−∆ + K0M)>0. Let C0be such that K0= 1/(1 −C0). To prove (2.4) it is sufficient to show that σ1(−∆ + K0Mn)≥0 for n≥n0. But σ1(−∆ + K0Mn)→σ1(−∆ + K0M)>0. 2 The following result shows that (2.2) possesses a unique solution. 6M. Delgado, J. A. Montero and A. Su´arez Theorem 2.5 Assume that Msatisfies (H1) and σ1(−∆+M)>0. Then, there exists a unique solution u∈C1,κ(Ω), for some κ∈(0,1), of (2.2). Moreover, there exists a constant K > 0 (independient of f) such that kukC1,κ(Ω) ≤Kkfk∞.(2.5) Proof: For v∈C1 0(Ω) we consider the problem          −∆u=−M(x)vin Ω, u= 0 on ∂Ω. (2.6) By Proposition 2.3 in [6], there exists a unique solution u∈C2(Ω)∩C1,κ(Ω), for some κ∈(0,1), of (2.6) with kukC1,κ(Ω) ≤K1kvkC1(Ω). Define G1:C1 0(Ω) 7→ C1,κ 0(Ω), v7→ G1(v) the unique solution of (2.6). We have shown that G1 is bounded. For h∈L∞(Ω) we consider the problem          −∆u=h(x) in Ω, u= 0 on ∂Ω. (2.7) It is well known that fixed h∈L∞(Ω), there exists a unique solution u∈W2,p(Ω) of (2.7) for all p > 1, and kukC1,κ(Ω) ≤K1kukW2,p(Ω) ≤K2khk∞. We can define the map G2:L∞(Ω) 7→ C1,κ 0(Ω), h7→ G2(h) the unique solution of (2.7). We have got that G2is bounded. Now, if we define H:C1 0(Ω) 7→ C1 0(Ω), H(u) := u−G1(u), denote by i:C1,κ 0(Ω) 7→ C1 0(Ω) the compact imbedding and we pose G:= H◦i:C1,κ 0(Ω) 7→ C1,κ 0(Ω), then we can rewrite (2.2) as G(u) = G2(f) Optimal control for degenerate logistic equation 7 being Ga compact pertubation of the identity. Since σ1(−∆ + M)>0, Gis inyective. The Fredholm’s Theorem provides us the existence and uniqueness of solution u∈C1,κ 0(Ω) of (2.2) satisfying (2.5). 2 The next result is an easy consequence of Theorem 2.2 b). Lemma 2.6 a) Assume that Msatisfies (H1) and σ1(−∆ + M)>0. Consider fi∈L∞(Ω), i= 1,2with f1≤f2and let ui,i= 1,2be the respective solutions of (2.2). Then, u1≤u2. b) Assume that Mi,i= 1,2satisfy (H1) and M1≤M2with σ1(−∆ + M1)>0. Let ui, i= 1,2be the respective solutions of (2.2). Then, u2≤u1. 3 The degenerate logistic equation Consider          −∆u=buα−euβin Ω, u= 0 on ∂Ω, (3.1) and assume that (H2) 0 < α < 1≤β, b ∈L∞ +(Ω)\{0}, e ∈ A, where A:= {f∈L∞(Ω) : fL>0}. The next result has been proved in [4] when b, e ∈Cν(Ω), ν∈(0,1). The proof is also valid in this case. Theorem 3.1 Assume (H2). The following assertions are true: a) There exists a unique strictly positive solution ubof (3.1). Moreover, by elliptic regularity ub∈W2,p(Ω), p > 1, and so ub∈C1,κ(Ω) ∩int(P), with 0< κ ≤1−N/p. b) We have the following a priori bound, kubk∞≤µbM eL¶1/(β−α) .(3.2) 8M. Delgado, J. A. Montero and A. Su´arez c) If bL>0, then there exists ε0>0such that for all ε≤ε0, it holds εϕ1(x)≤ub(x)c.p.d. x∈Ω where ε0>0satisfies bL−σ1ε1−α−eMεβ−α= 0. d) If bL= 0, since bM>0there exists a ball B:= B(x0, r)such that bL,B >0in B, where bL,B is the essential infimum of bin B. Hence, εϕB 1≤ubc.p.d. in Bfor all ε≤ε1and where ε1>0satisfies bL,B −σB 1ε1−α−eM,Bεβ−α= 0. Remark 3.2 By (H2),(3.1) satisfies the strong maximum principle and then there exist two positive constants k1, k2such that k1dΩ(x)≤ub(x)≤k2dΩ(x)∀x∈Ω.(3.3) The following result plays an important role along the work. Theorem 3.3 Assume (H2). Then, the map b∈A⊂L∞(Ω) 7→ ub∈int(P)⊂C1 0(Ω) is increasing, continuous and C1. For the proof of this result we use the following elementary lemma. Lemma 3.4 a) Let α∈(0,1] and 0< t1< t2be. Then αtα−1 2(t2−t1)≤tα 2−tα 1≤αtα−1 1(t2−t1). b) Let β∈[1,+∞)and 0≤t1< t2be. Then βtβ−1 1(t2−t1)≤tβ 2−tβ 1≤βtβ−1 2(t2−t1). Proof of Theorem 3.3: It follows easily that the map is increasing. For the continuity, let bn, b ∈ A be such that bn→bin L∞, then (bn)M→bM. Hence, fixed δ > 0 there exists n0∈IN such that for n≥n0 kubnk∞≤µ(bn)M eL¶1/(β−α) ≤µbM+δ eL¶1/(β−α) =C(independient of n), and so, the sequence {ubn}is bounded in W2,p(Ω), p > 1. There exists a subsequence, relabeled by n, such that ubn→uin C1,κ(Ω), κ < 1−N/p. Moreover, uis a weak solution of (3.1). It Optimal control for degenerate logistic equation 9 remains to prove that u=ub. By the uniqueness of positive solution of (3.1), it suffices to prove that u > 0. Since bM>0, there exist x0∈Ω, r0>0, such that (bn)≥(bn)L,B >0 c.p.d. in B=B(x0, r0), for n≥n0. By Theorem 3.1 d), we have that there exist εn>0 such that εnϕB 1≤ubnc.p.d. in Bwhere εnis such that (bn)L,B −σB 1ε1−α n−eM,Bεβ−α n= 0. Since (bn)L,B →bL,B, it follows that εn→ε > 0 where εis such that bL,B −σB 1ε1−α−eM,Bεβ−α= 0, and so εϕB 1≤uc.p.d. in Band then u > 0. For the derivability we use the Implicit Function Theorem. Fixed p > N, we define the map F:A × U ⊂ L∞(Ω) ×C1 0(Ω) 7→ Lp(Ω) where U:= W2,p(Ω) ∩int(P), as F(b, u) := −∆u−buα+euβ. Ais an open set in L∞(Ω) and it is well known, see [1], that Uis also open in C1 0(Ω). It is clear that F(b0, ub0) = 0. We show that Fis C1, for which it is sufficient to show it for the second component. We calculate the Gˆateaux derivative respect to this, which will be denoted by DGF. Let (b, u)∈ A × U and ξ∈C1 0(Ω) be, then DGF(b, u)ξ:= lim ε→0 F(b, u +εξ)− F(b, u) ε=−∆ξ−blim ε→0 (u+εξ)α−uα ε+elim ε→0 (u+εξ)β−uβ ε. We claim that: (u+εξ)β−uβ ε→βuβ−1ξand (u+εξ)α−uα ε→αuα−1ξin Lp(Ω) as ε→0.(3.4) Assume ε↓0. Using Lemma 3.4, to prove (3.4) it is sufficient to show that (u+εξ)β−1ξ→uβ−1ξand (u+εξ)α−1ξ→uα−1ξin Lp(Ω) as ε↓0. The first one is true because β≥1. For the second one, we have k[(u+εξ)α−1−uα−1]ξkp=k[(u+εξ)α−(u+εξ)uα−1]µξ u+εξ ¶kp.(3.5) Since u∈int(P), there exist ε0>0 and k(ε) such that u+εξ ∈int(P) for ε≤ε0and k(ε) := inf x∈Ω u(x) + εξ(x) dΩ(x)>0. 16 M. Delgado, J. A. Montero and A. Su´arez Proof: Let f∈L∞ +(Ω) be. By (H5), there exists t0>0 such that k(t0)/h(t0) = λK. We consider g:= min{f, t0}, and we will prove that J(g)> J(f), whence the result follows. By definition, g≤fand then ug≥uf. If x0∈Ω is such that f(x0) = g(x0) then λug(x0)h(g(x0)) −k(g(x0)) ≥λuf(x0)h(f(x0)) −k(f(x0)). On the other hand, if f(x0)> g(x0) = t0>0, then by (3.2) λug(x0)h(g(x0)) −k(g(x0)) ≤λKh(t0)−k(t0) = 0, and so by (H5), we get 0≥λug(x0)h(g(x0)) g(x0)−k(g(x0)) g(x0)> λuf(x0)h(f(x0)) f(x0)−k(f(x0)) f(x0). Then, J(g) = Z{f=g} λh(g)ug−k(g) + Z{f>g} λh(g)ug−k(g)≥Z{f=g} λh(f)uf−k(f)+ +Z{f>g} (λh(g) gug−k(g) g)g > Z{f=g} λh(f)uf−k(f) + Z{f>g} λh(f)uf−k(f) = J(f). 2 For the uniqueness, we use the argument described in Section 6 in [3]. Firstly, we prove the next result. Proposition 4.4 Let J:D:= {f∈L∞(Ω) : (a−f)∈ A} ⊂ L∞(Ω) 7→ IR be. Then Jis Fr´echet continuously differentiable and J0(f)(g) = ZΩ (λh0(f)uf−λuα fPf−k0(f))g, ∀f∈ D,∀g∈L∞(Ω),(4.1) where for any f∈ D,Pf∈C1 0(Ω) is the unique solution of          −∆Pf+Mf(x)Pf=h(f)in Ω, Pf= 0 on ∂Ω, (4.2) being Mf:= −α(a−f)uα−1 f+βeuβ−1 f. Optimal control for degenerate logistic equation 17 To prove this result, we need some previous ones. For f∈ D and g∈L∞(Ω), let ξf,g be the unique solution of          −∆ξ+Mf(x)ξ=−guα fin Ω, ξ= 0 on ∂Ω. (4.3) Observe that (4.2) and (4.3) have a unique solution because σ1(−∆ + Mf)>0 (see (3.10) and (3.11)) and Theorem 2.5. Lemma 4.5 The map f∈ D 7→ Pf∈C1 0(Ω) is continuous. Proof: Fixed p > N, we consider the map G:D × (C1 0(Ω) ∩W2,p(Ω)) 7→ Lp(Ω) defined by G(f, P) = −∆P+MfP−h(f). Observe that Gis continuous. Indeed, the continuity of the map f7→ MfPfollows with a similar argument to the one used in the proof of Theorem 3.3 to show that the map DGFis continuous. On the other hand, it is clear that G(f, Pf) = 0. Given ξ∈C1 0(Ω) ∩W2,p(Ω) is easy to prove that D2G(f, Pf)ξ=−∆ξ+Mfξ. Moreover, as in (3.10), σ1(−∆ + Mf)>0 and so D2G(f, Pf) is non singular. The Implicit Function Theorem completes the proof. 2 The next result is due by Krasnoselskii, see [7], where we send for the definitions of the following concepts. Lemma 4.6 Let Ebe a Banach space ordered by a generating positive cone P,Fa Banach space and T:E7→ F. Assume that the Gˆateaux derivative of Twith respect to P, denoted by DG,P T, exists and it is continuous in a neighbourhood of x0∈E. Then, the Fr´echet derivative coincides with the Gˆateaux derivative and Tis C1near x0. Recall that Pis generating if E=P−P. It is well known, see Proposition 1.7 in [1], that if int(P)6=∅, then Pis generating. Proof of Proposition 4.4: Firstly, we compute the Gˆateaux derivative respect to the cone, denoted by DG,P J. Let g∈L∞ +(Ω), f∈ D and ε > 0 be such that f+εg ∈ D. Using Lemma 3.5 and (4.3) DG,P J(f)g:= lim ε↓0 J(f+εg)−J(f) ε=ZΩ λξf,gh(f) + λh0(f)ufg−k0(f)g. 18 M. Delgado, J. A. Montero and A. Su´arez By the equation that satisfy ξf,g yPf(see (4.3) and (4.2)), it follows that ZΩ h(f)ξf,g +ZΩ guα fPf= 0, and so, DG,P J(f)(g) = ZΩ (λh0(f)uf−λuα fPf−k0(f))g, ∀g∈L∞ +(Ω). Let fn→f∈ D be in L∞and g∈L∞(Ω). Then, by Theorem 3.3 and Lemma 4.5 it follows sup kgk∞≤1 |DG,P J(fn)(g)−DG,P J(f)(g)| ≤ ≤sup kgk∞≤1ZΩ |λ(h0(fn)ufn−h0(f)uf)−λ(uα fnPfn−uα fPf)−(k0(fn)−k0(f))g| → 0. and so, DG,P Jis continuous. Applying Lemma 4.6, the Gˆateaux derivative coincides with the Fr´echet derivative and that the map is C1.2 The next result shows that some maps involved in (4.1) are Lipschitz continuous. Lemma 4.7 Assume (H3) −(H5). There exists Λ>0such that for 0<λ<Λthe maps f∈[0, Tλ]7→ uf, Pf, uα fPf∈L∞(Ω) are Lipschitz continuous. Proof: Let f, g ∈[0, Tλ] be, by the monotony of the map f7→ uf, it follows that 0< uTλ≤uf, ug≤u0 for λsuch that a−Tλ>0, that is λ < λ0for some λ0(see Remark 4.3 a)). To the end of the proof we take λ < λ0. By the Mean Value Theorem, uα f−uα g=αξα−1(f, g)(uf−ug), uβ f−uβ g=βηβ−1(f, g)(uf−ug) with 0< uTλ≤min{uf, ug} ≤ ξ(f, g), η(f, g)≤max{uf, ug} ≤ u0. (4.4) Let w:= uf−ugbe. Then, wsatisfies (−∆ + N(f, g))w= (g−f)uα g,in Ω, w= 0 on ∂Ω, where N(f, g) := −α(a−f)ξα−1(f, g) + βeηβ−1(f, g). Using f≥0 and (4.4), it follows that N(f, g)≥ −αaξα−1(f, g) + βeηβ−1(f, g)≥ −αauα−1 Tλ+eβuβ−1 Tλ. It is not hard to show that as λ↓0 ZΩ (−αauα−1 Tλ+eβuβ−1 Tλ)ϕ2→ZΩ (−αauα−1 0+eβuβ−1 0)ϕ2∀ϕ∈H1 0(Ω), Optimal control for degenerate logistic equation 19 and so, by Proposition 2.3 we obtain that σ1(−∆ + N(f, g)) ≥σ1(−∆−αauα−1 Tλ+eβuβ−1 Tλ)→σ1(−∆−αauα−1 0+eβuβ−1 0)>0 as λ↓0. Hence, there exists λ1>0 such that N(f, g)≥ −αauα−1 Tλ1+eβuβ−1 Tλ1(4.5) and σ1(−∆ + N(f, g)) ≥σ1(−∆−αauα−1 Tλ1+eβuβ−1 Tλ1)>0.(4.6) Then, by (4.5), (4.6) and Lemma 2.6, we have that w≤ψ1where ψ1is the unique solution of          −∆u+ (−αauα−1 Tλ1+eβuβ−1 Tλ1)u= (g−f)uα gin Ω, u= 0 on ∂Ω. (4.7) Interchanging fand g, we get that −w≤ψ2where ψ2is the unique solution of (4.7) with second member (f−g)uα f. Then, taking into account that ufpossesses a priori bound independient of f(see (3.2)) and Theorem 2.5, it follows that kuf−ugk∞=kwk∞≤max{kψ1k∞,kψ2k∞} ≤ max{kψ1kC1(Ω),kψ2kC1(Ω)} ≤ Ckf−gk∞.(4.8) This shows that the map f7→ ufis Lipschitz. Before proving the Lipschitz character of the map f∈[0, Tλ]7→ Pf, we see that Pf≤ P in Ω, (4.9) where P ∈ C1 0(Ω), independient of f. Indeed, let f∈[0, Tλ] be, then Mf≥ −αauα−1 Tλ+βeuβ−1 Tλ, and so, using again Lemma 2.6 b), Pf≤ P where Pis the unique solution of          −∆u+ (−αauα−1 Tλ1+eβuβ−1 Tλ1)u=Tin Ω, u= 0 on ∂Ω, where T:= max f∈[0,Tλ]max x∈Ω h(f(x)).This implies (4.9). We will prove now that the map is Lipschitz. Let f, g ∈[0, Tλ] and z:= Pf−Pgbe. Then z satisfies −∆z+Mfz=T(f, g),in Ω, z= 0 on ∂Ω, 20 M. Delgado, J. A. Montero and A. Su´arez where T(f, g) = h(f)−h(g) + Pg[α(a−f)(uα−1 f−uα−1 g)−βe(uβ−1 f−uβ−1 g)] + α(g−f)Pguα−1 g. Applying again the Mean Value Theorem, we get uα−1 f−uα−1 g= (α−1)ξα−2(f, g)(uf−ug), uβ−1 f−uβ−1 g= (β−1)ηβ−2(f, g)(uf−ug) 0< uTλ≤min{uf, ug} ≤ ξ(f, g), η(f, g)≤max{uf, ug} ≤ u0. (4.10) Hence, T(f, g) = h(f)−h(g) + Pg[α(α−1)(a−f)ξα−2−β(β−1)eηβ−2](uf−ug) + α(g−f)Pguα−1 g. By a similar argument to the used in the proof of (4.8), we obtain kPf−Pgk∞=kzk∞≤CkT(f, g)k∞.(4.11) Since P ∈ C1 0(Ω), and using (3.3), (3.7), (4.9) and (4.10), we obtain kα(f−g)Pguα−1 gk∞≤Ckf−gk∞kPguα−1 Tλ1k∞ ≤Ckf−gk∞kα−1 1kPdα−1 Ωk∞ ≤Ckf−gk∞kdα Ωk∞kPkC1(Ω) ≤Ckf−gk∞with Cindependient of fand g. On the other hand, using (4.8), (4.9) and (4.10) kα(α−1)(a−f)Pgξα−2(uf−ug)k∞≤CkPξα−2(uf−ug)k∞ ≤CkPξα−2max{|ψ1|,|ψ2|}k∞ ≤CkPdα−2 Ωmax{|ψ1|,|ψ2|}k∞ ≤CkPkC1(Ω)kdα Ωk∞max{kψ1kC1(Ω),kψ2kC1(Ω)} ≤Ckf−gk∞ with Cindependient of fand g. Analogously it can be treated the term −eβ(β−1)Pgηβ−2(uf− ug). Then, since his Lipschitz in [0, Tλ] and by (4.11), it follows that the map f7→ Pfis Lipschitz. Optimal control for degenerate logistic equation 21 Let f, g ∈[0, Tλ] be, we have kuα fPf−uα gPgk∞≤ k(uα f−uα g)Pfk∞+kuα g(Pf−Pg)k∞. By the Mean Value Theorem, k(uα f−uα g)Pfk∞=kαξα−1Pf(uf−ug)k∞≤CkϕkC1(Ω)kf−gk∞≤Ckf−gk∞. It is sufficient to take Λ := min{λ0, λ1}. This completes the proof. 2 Theorem 4.8 Assume (H3) −(H5). Then, there exists Λ0>0such that if λ < Λ0there exists a unique optimal control. Proof: Let f∈ C be an optimal control, then by Lemma 4.2 f∈I:= [0, Tλ]∞. We take λ < Λ (the constant obtained in Lemma 4.7) and sufficiently small λsuch that I⊂ C. In I, convex, the strictly concave character of Jis equivalent to the monotony of J0. Hence, by (4.1), for f, g ∈I, we have that (J0(f)−J0(g))(f−g) = ZΩ [λ(h0(f)uf−h0(g)ug) + λ(uα gPg−uα fPf)−(k0(f)−k0(g))](f−g)≤ ≤ZΩ (λL −k0)(f−g)2<0, taking λ < k0/L := Λ1, where Lthe Lipschitz constant of the maps h0,f7→ uf,f7→ Pfand f7→ uα fPf(see Lemma 4.7). 2 5 Regularity of the optimal control and optimality system In this section we consider the special case h(t) = tand k(t) = t2, which satisfy clearly (H4) and (H5). Moreover, in this case Tλ=λK. The following result provides us of a caracterization of an optimal control. It follows as Theorem 3.1 in [10], using now our Lemma 3.5. Lemma 5.1 Assume f∈ C and (H3). If fis an optimal control, then f=λ 2uf(1 −uα−1 fPf)+. 22 M. Delgado, J. A. Montero and A. Su´arez The next result says us that the optimal control is a H¨older continuous function when λis small and it lets us write the optimality system. Proposition 5.2 Assume (H3). There exists Λ1such that if λ≤Λ1, then Pf≤u1−α f. So, if f is an optimal control, we have that f=λ 2uf(1 −uα−1 fPf).(5.1) Proof: Let fbe an optimal control. For λ < λ0:= aL/K, we have that uf≥uλK>0. As in Lemma 4.7, it follows the existence of λ1such that there exists a unique positive solution ψof          −∆ψ+ (−aαuα−1 λ1K+βeuβ−1 λ1K)ψ=Kin Ω, ψ= 0 on ∂Ω. By Lemma 2.6 and (3.2), it follows that Pf≤λψ for λ≤λ1. (5.2) We define now λ2:= inf x∈Ω u1−α λ1K ψ≤inf x∈Ω u1−α f ψ. Observe that λ2>0. Indeed, since ψand uλ1Kare positive functions, it follows the existence of a constant k > 0 such that u1−α λ1K ψ> kd−α Ω>0. Taking Λ1:= min{λ0, λ1, λ2}and taking into account (5.2) and the definition of λ2, it follows Pf≤u1−α f, and as a consequence of Lemma 5.1, we obtain (5.1). 2 The following result is an easy consequence of the previous result and it provides us with the optimality system. Corollary 5.3 Assume (H3) and λ≤Λ1. Then any optimal control fmay be expressed as in (5.1), where the pair (uf, Pf) := (u, P)satisfies                      −∆u=uα(a−λ 2u+λ 2uαP−euβ−α)in Ω, −∆P+ (−αauα−1+βeuβ−1)P=λ 2(u−uαP(1 + α) + αu2α−1P2)in Ω, u=P= 0 on ∂Ω, and u > 0. Optimal control for degenerate logistic equation 23 Acknowledgments. M. Delgado and A. Su´arez thank to CICYT of Spain (MAR98-0486) and J. A. Montero thanks to ”Junta de Andaluc´ıa” (FQM116) and DGESIC (PB98-1343) by the partial financial support to the elaboration of this work. References [1] Amann H (1976) Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Review 18:620-709. [2] Bertsch M, Rostamian R (1985) The principle of linearized stability for a class of degenerate diffusion equations. J. Diff. Eqns. 57:373-405. [3] Ca˜nada A, G´amez JL, Montero JA (1998) Study of an optimal control problem for diffusive nonlinear elliptic equations of logistic type. SIAM J. Control Optim. 36:1171-1189. [4] Delgado M, Su´arez A (2000) On the existence of dead cores for degenerate Lotka-Volterra models. Proc. Royal Society of Edin. 130 A:743-766. [5] Gurtin ME, MacCamy RC (1977) On the diffusion of biological populations. Math. Biosci. 33:35-49. [6] Hern´andez J, Mancebo F, Vega de Prada JM On the linearization of some singular nonlinear elliptic problem and applications, to appear in Ann. Inst. H. Poincare Anal. Non-Linearie. [7] Krasnoselskii MA (1964) Positive solutions of operator equations. Noordhoff, Groningen. [8] Kufner A (1980) Weighted Sobolev Spaces. Text zur Mathematik, 31, Teubner, Leipzig. [9] Leung AW (1995) Optimal harvesting-coefficient control of steady-state prey-predator diffusive Volterra-Lotka systems. Appl. Math. Optim. 31:219-241. [10] Leung AW, Stojanovic S (1993) Optimal control for elliptic Volterra-Lotka type equations. J. Math. Anal. Appl. 173:603-619. [11] Montero JA (2000) A uniqueness result for an optimal control problem on a diffusive elliptic Volterra-Lotka type Equation. J. Math. Anal. Appl. 243:13-31. [12] Kavian O (1993) Introduction `a la th´eorie des points critiques et applications aux probl`emes elliptiques. Springer-Verlag, Paris.