Optimal Harvesting for an Age-Spatial-Structured Population Dynamic Model with External Mortality

and Applied Analysis 3 ∂u ∂η x, t, a 0 on Σ ∂Ω × 0, T × 0, A , u x, 0, a u0 x, a in Ω × 0, A , u x, t, 0 ∫A 0 β x, t, a u x, t, a da in Ω × 0, T , Pw x, t ∫A 0 w x, α u x, t, α dα in Ω × 0, T , 1.3 whereΩ is a bounded domain inR with smooth boundary ∂Ω, v x, t, a is a harvesting rate, and w is a positive function in L∞ Ω × 0, A . We study an optimal control problem relating to the dynamic system 1.3 as follows: Find v∗ ∈ U such that J ( u ∗ , v∗ ) sup v∈U J u, v , P where J u, v ≡ Q v x, t, a g x, t, a u x, t, a dx dt da, g is a given boundedweight function u is the solution of the dynamic control system 1.3 , and U is the set of controllers given by U { v ∈ L2 Q : ν1 x, t, a ≤ v x, t, a ≤ ν2 x, t, a a.e., x, t, a ∈ Q } 1.4 for some ν1, ν2 ∈ L∞ Q , 0 ≤ ν1 x, t, a ≤ ν2 x, t, a , a.e., in Q. This problem P is called the primal problem. The objective functional J u, v in P represents the profit from harvesting, that is, the profit term is the proportion of the species harvested multiplied by the selling price dependent on age a at time t and location x. In a biological system, we may apply the dynamic system 1.3 to the fish, animal, and plant dynamic models. The purpose of this paper is to prove the existence and compactness of solutions for the dynamic system 1.3 and to investigate an optimal harvesting problem P for a nonlinear age-spatial-structured population dynamic model with external mortality. The optimal approach introduced in this work may be applicable in the realistic biological models with field data beyond the theoretical model. The paper is organized as follows. In Section 2, we obtain the existence, uniqueness, and compactness of solutions for the dynamic system 1.3 . In Section 3, we derive a sufficiently condition for the optimal control problem P . Finally, a necessary condition for the optimal control problem P is given in Section 4. 2. Existence, Uniqueness, and Compactness of Solutions In this work, we assume the following: H1 The fertility rate β ∈ L∞ Q , β x, t, a ≥ 0 a.e., x, t, a ∈ Q. H2 The mortality rate μ ∈ L∞ Ω × 0, T × 0, A × L∞ Q and μ is increasing and Lipschitz continuous with respect to the variable u. 4 Abstract and Applied Analysis H3 Φ : 0,∞ → 0,∞ is bounded and Lipschitz continuous, that is, there exists a constant L > 0 such that ∣∣Φ(ψ1) −Φ(ψ2)∣∣ ≤ L∣∣ψ1 − ψ2∣∣ 2.1 and Φ : 0,∞ → 0,∞ is continuously differentiable. H4 u0 ∈ L∞ Ω × 0, A , u0 x, a ≥ 0 a.e., x, a ∈ Ω × 0, A . H5 g ∈ L∞ Q , g x, t, a ≥ 0 a.e., x, t, a ∈ Q. H6 w is a nonnegative bounded and measurable function in L∞ Ω × 0, A with 0 ≤ w x, a ≤ 1 for all x, a ∈ Ω × 0, A . The existence of a solution u to the dynamic system 1.3 is given by the following lemma see also 1 . Here we assume that a function u ∈ L2 Q belongs to C S;L2 Ω ∩ AC S;L2 Ω ∩ L2 S;H1 Ω ∩ Lloc S;L2 Ω , for almost any characteristic line S; a − t constant, t, a ∈ 0, T × 0, A . In addition, we assume that esssup |∂u/∂a| < ∞ or esssup |∂u/∂t| < ∞, which may be a natural biological condition for population dynamics. Lemma 2.1. Let the assumptions H1 – H6 hold. For any v ∈ U, the dynamic system 1.3 admits a unique and nonnegative solution u which belongs to L∞ Q . Proof. We will use the Banach fixed-point theorem for proof. Let L Q {u ∈ L Q : u ≥ 0 a.e., in Q}. Denote by ζ the mapping ζ : ũ → u, where u is the solution of ∂u ∂t x, t, a ∂u ∂a x, t, a − kΔxu x, t, a μ x, t, a, u x, t, a u x, t, a Φ ( P̃ x, t ) u x, t, a −v x, t, a u x, t, a in Q Ω × 0, T × 0, A , ∂u ∂η x, t, a 0 on Σ ∂Ω × 0, T × 0, A , u x, 0, a u0 x, a , in Ω × 0, A , u x, t, 0 ∫A 0 β x, t, a u x, t, a da in Ω × 0, T , P̃w x, t ∫A 0 w x, α ũ x, t, α dα Ω × 0, T . 2.2 Abstract and Applied Analysis 5 Then, the mapping ζ is well defined form L2 Q to L 2 Q see Lemma 2 of 9 . For any ũ1, ũ2 ∈ L2 Q , we denote P̃ i w x, t ∫A 0 w x, α ũi x, t, α dα, with x, t ∈ Ω × 0, T and i ∈ {1, 2}. By definition of ζ, we get the following equation: ∫and Applied Analysis 5 Then, the mapping ζ is well defined form L2 Q to L 2 Q see Lemma 2 of 9 . For any ũ1, ũ2 ∈ L2 Q , we denote P̃ i w x, t ∫A 0 w x, α ũi x, t, α dα, with x, t ∈ Ω × 0, T and i ∈ {1, 2}. By definition of ζ, we get the following equation: ∫ Qt [ ∂ ∂t ζũ1 − ζũ2 ∂ ∂a ζũ1 − ζũ2 − kΔx ζũ1 − ζũ2 μ x, s, a, ζũ1 ζũ1 − μ x, s, a, ζũ2 ζũ2 Φ ( P̃ 1 w ) ζũ1 −Φ ( P̃ 2 w ) ζũ2 v ζũ1 − ζũ2 ] ζũ1 − ζũ2 dx dsda ∫ Qt [ ∂ ∂t ζũ1 − ζũ2 ∂ ∂a ζũ1 − ζũ2 − kΔx ζũ1 − ζũ2 ( μ x, s, a, ζũ1 − μ x, s, a, ζũ2 ) ζũ1 μ x, s, a, ζũ2 ζũ1 − ζũ2 ( Φ ( P̃ 1 w ) −Φ ( P̃ 2 w )) ζũ1 Φ ( P̃ 2 w ) ζũ1 − ζũ2 v ζũ1 − ζũ2 ] ζũ1 − ζũ2 dx dsda 0, 2.3 where Qt Ω × 0, t × 0, A , t ∈ 0, T . Using the conditions H2 and H3 , we get after some calculations that ‖ ζũ1 − ζũ2 t ‖L2 Ω× 0,A ≤ C ∫ t 0 ‖ ũ1 − ũ2 s ‖L2 Ω× 0,A ds, 2.4 where C is a positive constant. For sufficiently small t, we get the existence of a unique fixed point for ζ. Since the solution u satisfies 0 ≤ u x, t, a ≤ u x, t, a a.e., in Q 2.5 and u ∈ L∞ Q is the solution of the dynamic system 1.3 corresponding to μ 0,Φ 0, we complete the proof. For v ∈ U, denote P w x, t ∫A 0 w x, α u x, t, a dα in Ω × 0, T . 2.6 Lemma 2.2. The set {Pv w;v ∈ U} is relatively compact in L2 Ω × 0, T . Proof. For any ε > 0 small enough, we get that P w x, t ∫A−ε 0 w x, α u x, t, α dα in Ω × 0, T 2.7 6 Abstract and Applied Analysis is a solution of ∂P w ∂t − kΔxP w − ∫A−ε 0 w x, α ∂αu x, t, α dα − ∫A−ε 0 μ x, t, α, u x, t, α w x, α u x, t, α dα − ∫A−ε 0 Φ P w x, t, α w x, α u v x, t, α dα − ∫A−ε 0 v x, t, α w x, α u x, t, α dα, ∂P w ∂η 0 a.e., ∂Ω × 0, T , P w x, 0 ∫A−ε 0 w x, α u0 x, α dα in Ω. 2.8 Using the condition H6 , we obtain ∂P w ∂t − kΔxP w − ∫A−ε 0 w x, α ∂u ∂α x, t, α dα − ∫A−ε 0 μ x, t, α, u x, t, α w x, α u x, t, α dα − ∫A−ε 0 Φ P w x, t, α w x, α u v x, t, α dα − ∫A−ε 0 v x, t, α w x, α u x, t, α dα ≤ ∣∣∣∣ ∫A−ε 0 w x, α ∂u ∂α x, t, α dα ∣∣∣∣ ∣∣∣∣ ∫A−ε 0 μ x, t, α, u x, t, α u x, t, α dα ∣∣∣∣ ∣∣∣∣ ∫A−ε 0 Φ P w x, t, α w x, α u v x, t, α dα ∣∣∣∣ ∣∣∣∣ ∫A−ε 0 v x, t, α w x, α u x, t, α dα ∣∣∣∣ ≤ sup α∈ 0,A−ε ∣∣∣∂u v ∂α ∣∣∣ A − ε ∣∣∣∣ ∫A−ε 0 μ x, t, α, u x, t, α u x, t, α dα ∣∣∣∣ ∣∣∣∣ ∫A−ε 0 Φ P w x, t, α u v x, t, α dα ∣∣∣∣ ∣∣∣∣ ∫A−ε 0 v x, t, α u x, t, α dα ∣∣∣∣ ≤ C, 2.9 where we have used the fact that {vuv} and {μ ·, ·, ·, u uv} are bounded in L∞ Ω × 0, T × 0, A− ε , {Φ P w uv} is bounded in L∞ Ω× 0, T × 0, A− ε and {uv ·, ·, A− ε } is bounded in L∞ Ω × 0, T . Therefore, { ∂P w /∂t −kΔxP w } is bounded in L∞ Ω× 0, T . By Aubin’s compactness theorem that for any ε > 0, the set {P w : v ∈ U} is relatively compact in L2 Ω × 0, T . On the other hand, we get also ∣∣Pv,ε w x, t − P w x, t ∣∣ ≤ ∫A A−ε w x, α u x, t, α dα ≤ ε‖u‖L∞ Q . 2.10 Abstract and Applied Analysis 7 Combining these two results, we conclude the relative compactness of {Pv w : v ∈ U} in L2 Q .and Applied Analysis 7 Combining these two results, we conclude the relative compactness of {Pv w : v ∈ U} in L2 Q . 3. Existence of the Optimal Solution Now, we show the existence of the optimal solution for the primal problem P . Theorem 3.1. Let the assumptions H1 – H6 hold. Then, the primal problem P has at least one optimal pair. Proof. Let d supv∈UJ u , v . Then, we have 0 ≤ J u, v ≤ ∫ Q ν2 x, t, a g x, t, a û x, t, a dx dt da, 3.1 where û ∈ L∞ Q is the solution of the dynamic system 1.3 corresponding to μ 0 and Φ 0. Now let {vn}n∈N∗ ⊂ U be a sequence such that d − 1 n < J un , vn ≤ d. 3.2 Since 0 ≤ un x, t, a ≤ û x, t, a a.e., in Q, we conclude that there exists a subsequence such that un −→ u∗ weakly in L2 Q . 3.3 For a strong convergence to u∗, we consider the sequence {un}n∈N∗ such that un x, t, a kn ∑ i n 1 λni u vi x, t, a , λni ≥ 0, kn ∑ i n 1 λni 1, 3.4 where kn > 0 is an increasing sequence of integer numbers. Let the totality T1 {u | u ∑kn i n 1 λ n i u vi , λni ≥ 0, ∑kn i n 1 λ n i 1}, and we assume that 0 ∈ T1. For any ε > 0, suppose that ‖u∗ − ξ‖L2 Q > ε > 0 for every ξ ∈ T1. Then, the set T {y ∈ L2 Q ; ‖y − ξ‖L2 Q ≤ ε/2 for some ξ ∈ T1} is a convex neighborhood of 0 of L2 Q and ‖u∗ − y‖L2 Q > ε/2 for all y ∈ T . Let p y be the Minkowski functional of T . Note here that if we choose u∗ δ−1u0 with p u0 1 and 0 < δ < 1, then we get p u∗ p δ−1u0 δ−1p u0 δ−1 > 1. Consider a real linear subspace X1 {ξ ∈ L2 Q ; ξ γu0,−∞ < γ < ∞} and put f1 ξ γ for ξ γu0 ∈ X1. This real linear functional f1 on X1 satisfies f1 ξ ≤ p ξ on X1. Thus, by the Hahn-Banach extension theorem, there exists a real linear extension f of f1 defined on the real linear space L2 Q such that f ξ ≤ p ξ on L2 Q . T is a neighborhood 8 Abstract and Applied Analysis of 0, the Minkowski functional p ξ is continuous in ξ. Hence, f is a continuous real linear functional defined on the real linear normed space L2 Q . Moreover, we have sup ξ∈T1 f ξ ≤ sup ξ∈T f ξ ≤ sup ξ∈T p ξ 1 < δ−1 f ( δ−1u0 ) f u∗ . 3.5 This is contradiction to u∗ w − limn→∞ un . Therefore, un converges strongly to u∗ in L2 Q . Consider now the sequence of controls: vn x, t, a ⎪⎪⎨ ⎪⎩ ∑kn i n 1 λ n i vi x, t, a u vi x, t, a ∑kn i n 1 λ n i u vi x, t, a if kn ∑ i n 1 λni u vi x, t, a / 0, ν1 x, t, a if kn ∑ i n 1 λni u vi x, t, a 0. 3.6 This control vn is an element of the set U. So we can take a subsequence, also denoted by {vn}n∈N∗ such that vn −→ v∗ weakly in L2 Q . 3.7 By Lemma 2.2, we obtain Pn w −→ P ∗ w in L2 Ω × 0, T 3.8 and since un → u∗ weakly in L2 Q , we get P ∗ w x, t ∫A 0 w x, α u∗ x, t, α dα. 3.9 Obviously, un is a solution of ∂u ∂t x, t, a ∂u ∂a x, t, a − kΔxu x, t, a μ x, t, a, u x, t, a u x, t, a kn ∑ i n 1 λni Φ ( Pi w x, t ) ui x, t, a −vn x, t, a u x, t, a , in Q, 3.10 ∂u ∂η x, t, a 0 on Σ, u x, t, 0 ∫A 0 β x, t, a u x, t, a da in Ω × 0, T , u x, 0, a u0 x, a in Ω × 0, A , Pi w x, t ∫A 0 w x, α ui x, t, α dα in Ω × 0, T . 3.11 Abstract and Applied Analysis 9 By conditions of Φ and λ, we get ∥∥∥∥ kn ∑ i n 1 λni Φ ( Pi w ) ui −Φ P ∗ w u∗ ∥∥∥∥ L2 Q ∥∥∥∥ kn ∑ i n 1 λni Φ P vi w ui − kn ∑ n 1 λni Φ P ∗ w u ∗ ∥∥∥∥ L2 Qand Applied Analysis 9 By conditions of Φ and λ, we get ∥∥∥∥ kn ∑ i n 1 λni Φ ( Pi w ) ui −Φ P ∗ w u∗ ∥∥∥∥ L2 Q ∥∥∥∥ kn ∑ i n 1 λni Φ P vi w ui − kn ∑ n 1 λni Φ P ∗ w u ∗ ∥∥∥∥ L2 Q ≤ ∥∥∥∥ kn ∑ i n 1 λni Φ P vi w ui − u∗ ∥∥∥∥ L2 Q ∥∥∥∥ kn ∑ i n 1 λni u ∗(Φ Pi w −Φ P ∗ w ) ∥∥∥∥ L2 Q ≤ M ∥∥∥∥∥ kn ∑ i n 1 λni u vi − u∗ ∥∥∥∥∥ L2 Q ‖u‖L2 Q kn ∑ i n 1 λni ∥∥Φ(Pvi w ) −Φ P ∗ w ∥L2 Q ≤ M ∥∥∥∥ kn ∑ i n 1 λni u vi − u∗ ∥∥∥∥ L2 Q ‖u‖L2 Q kn ∑ i n 1 λni L ∥∥Pvi w − P ∗ w∥L2 Q −→ 0 as n −→ ∞, 3.12 where M sup ‖Φ · ‖L2 Q and L is the Lipschitz constant. Therefore, we have kn ∑ i n 1 λni Φ ( Pi w ) ui −→ Φ P ∗ w u∗ in L2 Q . 3.13 By H3 and β ∈ L∞ Q , we obtain ∫A 0 β x, t, a u x, t, a da −→ ∫A 0 β x, t, a u∗ x, t, a da. 3.14 Since un → u∗ in L2 Q , we have kn ∑ i n 1 λni Φ ( Pi w x, t ) ui x, t, a −→ Φ P ∗ w x, t u∗ x, t, a . 3.15 Passing to the limit in 3.10 , we obtain that u∗ is the solution of the dynamic system 1.3 corresponding to v∗. Therefore, we have