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

Abstract

We study an optimal harvesting for a nonlinear age-spatial-structured population dynamic model, where the dynamic system contains an external mortality rate depending on the total population size. The total mortality consists of two types: the natural, and external mortality and the external mortality reflects the effects of external environmental causes. We prove the existence and uniqueness of solutions for the population dynamic model. We also derive a sufficient condition for optimal harvesting and some necessary conditions for optimality in an optimal control problem relating to the population dynamic model. The results may be applied to an optimal harvesting for some realistic biological models.

1. Introduction

Optimal control problems for the age-structured systems are of interest for many areas of application, as harvesting, cost control, birth control, and epidemic disease control [1–5]. Many authors studied some optimal harvesting problems for an age-dependent population dynamic system ([2, 6–8] and references therein). One of the aims of such optimal control problems is to find some conditions of optimality for some objective functionals.

Aniţa [2] considered the optimal harvesting problem for the following nonlinear age-dependent population dynamic model. Let 𝑢(𝑎,𝑡) be the population density of age 𝑎 at time 𝑡, and let 𝜇(𝑎) and 𝑣(𝑡) be the natural death rate of individuals of age 𝑎 and the harvesting rate, respectively. The evolution of an age-structured population subject to harvesting is described as a partial differential equation: 𝜕𝑢()+𝜕𝑡𝑎,𝑡𝜕𝑢()(𝑎)𝑢()(𝑃(𝑡𝑢()(𝑡)𝑢(),()×()𝑢()=𝜕𝑎𝑎,𝑡+𝜇𝑎,𝑡+Φ))𝑎,𝑡=−𝑣𝑎,𝑡in0,𝐴0,𝑇)(𝑇>00,𝑡^𝐴₀𝛽(𝑎)𝑢()()𝑃(𝑡)=𝑎,𝑡𝑑𝑎,in0,𝑇_𝐴0𝑢(𝑎,𝑡)𝑑𝑎,in(0,𝑇), (1.1) where 𝑃(𝑡) and Φ(𝑃(𝑡)) stand for the total population and the external mortality rate, respectively, 𝛽(𝑎) is the natural fertility-rate, and 𝐴 is the maximal age of the individual.

Now let us make the previous model a generalized model considering the location and the external environmental cause. Let Ω be a bounded domain in ℝ^𝑁 with a smooth boundary 𝜕Ω, 𝐴>0, 𝑇>0. We denote by 𝑢(𝑥,𝑡,𝑎) the distribution of individuals of age 𝑎≥0 at time 𝑡≥0 and location 𝑥 in Ω. Let 𝛽(𝑥,𝑡,𝑎)≥0 the natural fertility rate and 𝜇(𝑥,𝑡,𝑎,𝑢)≥0 be the natural death rate of individuals of age 𝑎 at time 𝑡 and location 𝑥 and density 𝑢, where we note that the death rate 𝜇 depends on the density 𝑢 but most of study for models of the age structured population has been done with the death rate dependnt on the time and age only [2, 4]. For a more realistic situation, it is natural to assume that the death rate depends on the density as well as the time and age. Moreover, we set that the total mortality consists of a natural mortality and an external mortality and that the total population has a special weight 𝑤(𝑥,𝑎) at age 𝑎 and location 𝑥: 𝑃_𝑤(𝑥,𝑡)=^𝐴₀𝑤(𝑥,𝛼)𝑢(𝑥,𝑡,𝛼)𝑑𝛼, (1.2) where 𝐴 is the maximal age of the individual. In this total population 𝑃_𝑤, the weight function 𝑤 gives the effects differently on each age under the external environmental causes: the virus, the climate change, the earthquake, and storm waves. So the external mortality rate Φ(𝑃_𝑤(𝑥,𝑡)) reflects the long-term or short-term effects of external environments such as the virus, the climate change, and the earthquake. We also assume that the flux of population, as emigration, takes the form 𝑘∇_𝑥𝑢(𝑥,𝑡,𝑎) with 𝑘>0, where ∇_𝑥 is the gradient vector with respect to the location variable 𝑥.

Now we consider the following nonlinear age-spatial-structured population dynamics model with external mortality: 𝜕𝑢()+𝜕𝑡𝑥,𝑡,𝑎𝜕𝑢()𝜕𝑎𝑥,𝑡,𝑎−𝑘Δ_𝑥𝑢(𝑥,𝑡,𝑎)+𝜇(𝑥,𝑡,𝑎,𝑢(𝑥,𝑡,𝑎))𝑢(𝑥,𝑡,𝑎)+Φ𝑃_𝑤()𝑢()()𝑢()()×(),𝑥,𝑡𝑥,𝑡,𝑎=−𝑣𝑥,𝑡,𝑎𝑥,𝑡,𝑎in𝑄=Ω×0,𝑇0,𝐴𝜕𝑢()()×(),𝑢()𝜕𝜂𝑥,𝑡,𝑎=0onΣ=𝜕Ω×0,𝑇0,𝐴𝑥,0,𝑎=𝑢₀()(),𝑢()=𝑥,𝑎inΩ×0,𝐴𝑥,𝑡,0^𝐴₀𝛽()𝑢()(),𝑃𝑥,𝑡,𝑎𝑥,𝑡,𝑎𝑑𝑎inΩ×0,𝑇_𝑤(𝑥,𝑡)=^𝐴₀𝑤(𝑥,𝛼)𝑢(𝑥,𝑡,𝛼)𝑑𝛼inΩ×(0,𝑇), (1.3) where Ω is a bounded domain in ℝ^𝑁 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: Find𝑣^∗∈𝒰suchthat𝐽𝑢^𝑣∗,𝑣^∗=sup_𝑣∈𝒰𝐽𝑢^𝑣,𝑣, (𝐏) where 𝐽(𝑢^𝑣,𝑣)≡∫_𝑄𝑣(𝑥,𝑡,𝑎)𝑔(𝑥,𝑡,𝑎)𝑢^𝑣(𝑥,𝑡,𝑎)𝑑𝑥𝑑𝑡𝑑𝑎, 𝑔 is a given bounded weight function 𝑢^𝑣 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,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.(𝐻3)Φ∶[0,∞)→[0,∞) is bounded and Lipschitz continuous, that is, there exists a constant 𝐿>0 such that Φ𝜓₁−Φ𝜓₂≤𝐿𝜓₁−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))∩𝐿²_loc(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^𝑣 which belongs to 𝐿^∞(𝑄).

Proof. We will use the Banach fixed-point theorem for proof. Let 𝐿^𝑝₊(𝑄)={𝑢∈𝐿^𝑝(𝑄)∶𝑢≥0a.e.,in𝑄}. Denote by 𝜁 the mapping 𝜁∶𝑢↦𝑢^𝑢,𝑣, where 𝑢^𝑢,𝑣 is the solution of 𝜕𝑢()+𝜕𝑡𝑥,𝑡,𝑎𝜕𝑢()𝜕𝑎𝑥,𝑡,𝑎−𝑘Δ_𝑥𝑢()((𝑢()𝑃()𝑢()()𝑢()()×(),𝑥,𝑡,𝑎+𝜇𝑥,𝑡,𝑎,𝑢𝑥,𝑡,𝑎))𝑥,𝑡,𝑎+Φ𝑥,𝑡𝑥,𝑡,𝑎=−𝑣𝑥,𝑡,𝑎𝑥,𝑡,𝑎in𝑄=Ω×0,𝑇0,𝐴𝜕𝑢()()×(),𝑢()𝜕𝜂𝑥,𝑡,𝑎=0onΣ=𝜕Ω×0,𝑇0,𝐴𝑥,0,𝑎=𝑢₀()(),𝑢()=𝑥,𝑎,inΩ×0,𝐴𝑥,𝑡,0^𝐴₀𝛽()𝑢()(),𝑃𝑥,𝑡,𝑎𝑥,𝑡,𝑎𝑑𝑎inΩ×0,𝑇_𝑤(𝑥,𝑡)=^𝐴₀𝑤(𝑥,𝛼)𝑢(𝑥,𝑡,𝛼)𝑑𝛼Ω×(0,𝑇). (2.2) Then, the mapping 𝜁 is well defined form 𝐿²₊(𝑄) to 𝐿²₊(𝑄) (see Lemma 2 of [9]). For any 𝑢₁,𝑢₂∈𝐿²(𝑄), we denote 𝑃^𝑖_𝑤(𝑥,𝑡)=∫^𝐴₀𝑤(𝑥,𝛼)𝑢_𝑖(𝑥,𝑡,𝛼)𝑑𝛼, with (𝑥,𝑡)∈Ω×(0,𝑇) and 𝑖∈{1,2}. By definition of 𝜁, we get the following equation: _𝑄𝑡𝜕𝜁𝑢𝜕𝑡¹−𝜁𝑢₂+𝜕𝜁𝑢𝜕𝑎¹−𝜁𝑢₂−𝑘Δ_𝑥𝜁𝑢₁−𝜁𝑢₂+𝜇𝑥,𝑠,𝑎,𝜁𝑢₁𝜁𝑢₁−𝜇𝑥,𝑠,𝑎,𝜁𝑢₂𝜁𝑢₂+Φ𝑃¹_𝑤𝜁𝑢₁−Φ𝑃²_𝑤𝜁𝑢₂+𝑣𝜁𝑢₁−𝜁𝑢₂𝜁𝑢₁−𝜁𝑢₂=𝑑𝑥𝑑𝑠𝑑𝑎_𝑄𝑡𝜕𝜁𝑢𝜕𝑡¹−𝜁𝑢₂+𝜕𝜁𝑢𝜕𝑎¹−𝜁𝑢₂−𝑘Δ_𝑥𝜁𝑢₁−𝜁𝑢₂+𝜇𝑥,𝑠,𝑎,𝜁𝑢₁−𝜇𝑥,𝑠,𝑎,𝜁𝑢₂𝜁𝑢₁+𝜇𝑥,𝑠,𝑎,𝜁𝑢₂𝜁𝑢₁−𝜁𝑢₂+Φ𝑃¹_𝑤−Φ𝑃²_𝑤𝜁𝑢₁+Φ𝑃²_𝑤𝜁𝑢₁−𝜁𝑢₂+𝑣𝜁𝑢₁−𝜁𝑢₂𝜁𝑢₁−𝜁𝑢₂𝑑𝑥𝑑𝑠𝑑𝑎=0, (2.3) where 𝑄_𝑡=Ω×(0,𝑡)×(0,𝐴),𝑡∈(0,𝑇). Using the conditions (𝐻2) and (𝐻3), we get after some calculations that 𝜁𝑢₁−𝜁𝑢₂(𝑡)²_{𝐿2(Ω×(0,𝐴))}≤𝐶^𝑡₀(𝑢₁−𝑢₂)(𝑠)²_{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^𝑣 satisfies 0≤𝑢^𝑣(𝑥,𝑡,𝑎)≤𝑢(𝑥,𝑡,𝑎)a.e.,in𝑄 (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^𝑣_𝑤(𝑥,𝑡)=^𝐴₀𝑤(𝑥,𝛼)𝑢^𝑣(𝑥,𝑡,𝑎)𝑑𝛼inΩ×(0,𝑇). (2.6)

Lemma 2.2. The set {𝑃^𝑣_𝑤;𝑣∈𝒰} is relatively compact in 𝐿²(Ω×(0,𝑇)).

Proof. For any 𝜀>0 small enough, we get that 𝑃_𝑤^𝑣,𝜀(𝑥,𝑡)=₀^𝐴−𝜀𝑤(𝑥,𝛼)𝑢^𝑣(𝑥,𝑡,𝛼)𝑑𝛼inΩ×(0,𝑇) (2.7) is a solution of 𝜕𝑃_𝑤^𝑣,𝜀𝜕𝑡−𝑘Δ_𝑥𝑃_𝑤^𝑣,𝜀=−₀^𝐴−𝜀𝑤(𝑥,𝛼)𝜕_𝛼𝑢^𝑣(𝑥,𝑡,𝛼)𝑑𝛼−₀^𝐴−𝜀𝜇𝑥,𝑡,𝛼,𝑢^𝑣(𝑥,𝑡,𝛼)𝑤(𝑥,𝛼)𝑢^𝑣()−𝑥,𝑡,𝛼𝑑𝛼_0𝐴−𝜀Φ𝑃^𝑣_𝑤(𝑥,𝑡,𝛼)𝑤(𝑥,𝛼)𝑢^𝑣(𝑥,𝑡,𝛼)𝑑𝛼−₀^𝐴−𝜀𝑣(𝑥,𝑡,𝛼)𝑤(𝑥,𝛼)𝑢^𝑣(𝑥,𝑡,𝛼)𝑑𝛼,𝜕𝑃_𝑤^𝑣,𝜀(),𝑃𝜕𝜂=0a.e.,𝜕Ω×0,𝑇_{𝑤𝑣,𝜀}(𝑥,0)=₀^𝐴−𝜀𝑤(𝑥,𝛼)𝑢₀(𝑥,𝛼)𝑑𝛼inΩ. (2.8) Using the condition (𝐻6), we obtain 𝜕𝑃_𝑤^𝑣,𝜀𝜕𝑡−𝑘Δ_𝑥𝑃_𝑤^𝑣,𝜀=−₀^𝐴−𝜀𝑤(𝑥,𝛼)𝜕𝑢^𝑣()𝜕𝛼𝑥,𝑡,𝛼𝑑𝛼−₀^𝐴−𝜀𝜇𝑥,𝑡,𝛼,𝑢^𝑣(𝑥,𝑡,𝛼)𝑤(𝑥,𝛼)𝑢^𝑣()−𝑥,𝑡,𝛼𝑑𝛼_0𝐴−𝜀Φ𝑃^𝑣_𝑤(𝑥,𝑡,𝛼)𝑤(𝑥,𝛼)𝑢^𝑣(𝑥,𝑡,𝛼)𝑑𝛼−₀^𝐴−𝜀𝑣(𝑥,𝑡,𝛼)𝑤(𝑥,𝛼)𝑢^𝑣()≤𝑥,𝑡,𝛼𝑑𝛼_0𝐴−𝜀𝑤(𝑥,𝛼)𝜕𝑢^𝑣()+𝜕𝛼𝑥,𝑡,𝛼𝑑𝛼₀^𝐴−𝜀𝜇𝑥,𝑡,𝛼,𝑢^𝑣(𝑥,𝑡,𝛼)𝑢^𝑣()+𝑥,𝑡,𝛼𝑑𝛼_0𝐴−𝜀Φ𝑃^𝑣_𝑤(𝑥,𝑡,𝛼)𝑤(𝑥,𝛼)𝑢^𝑣(𝑥,𝑡,𝛼)𝑑𝛼+₀^𝐴−𝜀𝑣(𝑥,𝑡,𝛼)𝑤(𝑥,𝛼)𝑢^𝑣(𝑥,𝑡,𝛼)𝑑𝛼≤sup_{[]𝛼∈0,𝐴−𝜀}𝜕𝑢^𝑣)+𝜕𝛼(𝐴−𝜀₀^𝐴−𝜀𝜇𝑥,𝑡,𝛼,𝑢^𝑣(𝑥,𝑡,𝛼)𝑢^𝑣()+𝑥,𝑡,𝛼𝑑𝛼_0𝐴−𝜀Φ𝑃^𝑣_𝑤(𝑥,𝑡,𝛼)𝑢^𝑣(𝑥,𝑡,𝛼)𝑑𝛼+₀^𝐴−𝜀𝑣(𝑥,𝑡,𝛼)𝑢^𝑣(𝑥,𝑡,𝛼)𝑑𝛼≤𝐶, (2.9) where we have used the fact that {𝑣𝑢^𝑣} and {𝜇(⋅,⋅,⋅,𝑢^𝑣)𝑢^𝑣} are bounded in 𝐿^∞(Ω×(0,𝑇)×(0,𝐴−𝜀)), {Φ(𝑃^𝑣_𝑤)𝑢^𝑣} is bounded in 𝐿^∞(Ω×(0,𝑇)×(0,𝐴−𝜀)) and {𝑢^𝑣(⋅,⋅,𝐴−𝜀)} is bounded in 𝐿^∞(Ω×(0,𝑇)).
Therefore, {(𝜕𝑃_𝑤^𝑣,𝜀/𝜕𝑡)−𝑘Δ_𝑥𝑃_𝑤^𝑣,𝜀} is bounded in 𝐿^∞(Ω×(0,𝑇)). By Aubin's compactness theorem that for any 𝜀>0, the set {𝑃_𝑤^𝑣,𝜀∶𝑣∈𝒰} is relatively compact in 𝐿²(Ω×(0,𝑇)). On the other hand, we get also 𝑃_𝑤^𝑣,𝜀(𝑥,𝑡)−𝑃^𝑣_𝑤(𝑥,𝑡)≤^𝐴_𝐴−𝜀𝑤(𝑥,𝛼)𝑢^𝑣(𝑥,𝑡,𝛼)𝑑𝛼≤𝜀‖𝑢‖_𝐿^∞_(𝑄). (2.10)
Combining these two results, we conclude the relative compactness of {𝑃^𝑣_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=sup_𝑣∈𝒰𝐽(𝑢^𝑣,𝑣). Then, we have 0≤𝐽𝑢^𝑣,𝑣≤_𝑄𝜈₂(𝑥,𝑡,𝑎)𝑔(𝑥,𝑡,𝑎)𝑢(𝑥,𝑡,𝑎)𝑑𝑥𝑑𝑡𝑑𝑎, (3.1) where 𝑢∈𝐿^∞(𝑄) is the solution of the dynamic system (1.3) corresponding to 𝜇=0 and Φ=0. Now let {𝑣_𝑛}_𝑛∈𝑁^∗⊂𝒰 be a sequence such that 1𝑑−𝑛<𝐽𝑢^𝑣𝑛,𝑣_𝑛≤𝑑. (3.2) Since 0≤𝑢^𝑣𝑛(𝑥,𝑡,𝑎)≤𝑢(𝑥,𝑡,𝑎) a.e., in 𝑄, we conclude that there exists a subsequence such that 𝑢^𝑣𝑛→𝑢^∗weaklyin𝐿²(𝑄). (3.3) For a strong convergence to 𝑢^∗, we consider the sequence {𝑢_𝑛}_𝑛∈𝑁^∗ such that 𝑢_𝑛(𝑥,𝑡,𝑎)=^𝑘𝑛_𝑖=𝑛+1𝜆^𝑛_𝑖𝑢^𝑣𝑖(𝑥,𝑡,𝑎),𝜆^𝑛_𝑖≥0,^𝑘𝑛_𝑖=𝑛+1𝜆^𝑛_𝑖=1, (3.4) where 𝑘_𝑛>0 is an increasing sequence of integer numbers.
Let the totality 𝑇₁={𝑢∣𝑢=∑^𝑘𝑛_𝑖=𝑛+1𝜆^𝑛_𝑖𝑢^𝑣𝑖,𝜆^𝑛_𝑖≥0,∑^𝑘𝑛_𝑖=𝑛+1𝜆^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/2forsome𝜉∈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 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∈𝑇1𝑓(𝜉)≤sup_𝜉∈𝑇𝑓(𝜉)≤sup_𝜉∈𝑇𝑝(𝜉)=1<𝛿⁻¹=𝑓𝛿⁻¹𝑢₀=𝑓𝑢^∗. (3.5) This is contradiction to 𝑢^∗=𝑤−lim_𝑛→∞𝑢^𝑣𝑛. Therefore, 𝑢_𝑛 converges strongly to 𝑢^∗ in 𝐿²(𝑄). Consider now the sequence of controls: 𝑣_𝑛()=∑𝑥,𝑡,𝑎^{𝑘𝑛𝑖=𝑛+1}𝜆^𝑛_𝑖𝑣_𝑖(𝑥,𝑡,𝑎)𝑢^𝑣𝑖(𝑥,𝑡,𝑎)∑^𝑘𝑛_𝑖=𝑛+1𝜆^𝑛_𝑖𝑢^𝑣𝑖(𝑥,𝑡,𝑎)if∑^𝑘𝑛_𝑖=𝑛+1𝜆^𝑛_𝑖𝑢^𝑣𝑖()𝜈𝑥,𝑡,𝑎≠0,₁(𝑥,𝑡,𝑎)if∑^𝑘𝑛_𝑖=𝑛+1𝜆^𝑛_𝑖𝑢^𝑣𝑖(𝑥,𝑡,𝑎)=0. (3.6) This control 𝑣_𝑛 is an element of the set 𝒰. So we can take a subsequence, also denoted by {𝑣_𝑛}_𝑛∈𝑁^∗ such that 𝑣_𝑛→𝑣^∗weaklyin𝐿²(𝑄). (3.7) By Lemma 2.2, we obtain 𝑃^𝑣𝑛_𝑤→𝑃^∗_𝑤in𝐿²(Ω×(0,𝑇)) (3.8) and since 𝑢^𝑣𝑛→𝑢^∗ weakly in 𝐿²(𝑄), we get 𝑃^∗_𝑤(𝑥,𝑡)=^𝐴₀𝑤(𝑥,𝛼)𝑢^∗(𝑥,𝑡,𝛼)𝑑𝛼. (3.9) Obviously, 𝑢_𝑛 is a solution of 𝜕𝑢()+𝜕𝑡𝑥,𝑡,𝑎𝜕𝑢()𝜕𝑎𝑥,𝑡,𝑎−𝑘Δ_𝑥𝑢()((𝑢()+𝑥,𝑡,𝑎+𝜇𝑥,𝑡,𝑎,𝑢𝑥,𝑡,𝑎))𝑥,𝑡,𝑎_{𝑘𝑛𝑖=𝑛+1}𝜆^𝑛_𝑖Φ𝑃^𝑣𝑖_𝑤(𝑥,𝑡)𝑢^𝑣𝑖(𝑥,𝑡,𝑎)=−𝑣_𝑛(𝑥,𝑡,𝑎)𝑢(𝑥,𝑡,𝑎),in𝑄,𝜕𝑢()𝑢()𝜕𝜂𝑥,𝑡,𝑎=0onΣ,𝑥,𝑡,0=^𝐴₀𝛽()𝑢()(),𝑢()𝑥,𝑡,𝑎𝑥,𝑡,𝑎𝑑𝑎inΩ×0,𝑇𝑥,0,𝑎=𝑢₀()(),𝑃𝑥,𝑎inΩ×0,𝐴_𝑣𝑖𝑤(𝑥,𝑡)=^𝐴₀𝑤(𝑥,𝛼)𝑢^𝑣𝑖(𝑥,𝑡,𝛼)𝑑𝛼inΩ×(0,𝑇). (3.10)(3.11) By conditions of Φ and 𝜆, we get ^𝑘𝑛_𝑖=𝑛+1𝜆^𝑛_𝑖Φ𝑃^𝑣𝑖_𝑤𝑢^𝑣𝑖−Φ𝑃^∗_𝑤𝑢^∗_𝐿2(𝑄)=^𝑘𝑛_𝑖=𝑛+1𝜆^𝑛_𝑖Φ(𝑃^𝑣𝑖_𝑤)𝑢^𝑣𝑖−^𝑘𝑛_𝑛+1𝜆^𝑛_𝑖Φ(𝑃^∗_𝑤)𝑢^∗_𝐿2(𝑄)≤^𝑘𝑛_𝑖=𝑛+1𝜆^𝑛_𝑖Φ(𝑃^𝑣𝑖_𝑤)(𝑢^𝑣𝑖−𝑢^∗)_𝐿2(𝑄)+^𝑘𝑛_𝑖=𝑛+1𝜆^𝑛_𝑖𝑢^∗Φ(𝑃^𝑣𝑖_𝑤)−Φ(𝑃^∗_𝑤)_𝐿2(𝑄)≤𝑀^𝑘𝑛_𝑖=𝑛+1𝜆^𝑛_𝑖𝑢^𝑣𝑖−𝑢^∗_𝐿2(𝑄)+𝑢^∗_𝐿2^𝑘_(𝑄)^𝑛_𝑖=𝑛+1𝜆^𝑛_𝑖Φ𝑃^𝑣𝑖_𝑤−Φ𝑃^∗_{𝑤𝐿2(𝑄)}≤𝑀^𝑘𝑛_𝑖=𝑛+1𝜆^𝑛_𝑖𝑢^𝑣𝑖−𝑢^∗_𝐿2(𝑄)+𝑢^∗_𝐿2^𝑘_(𝑄)^𝑛_𝑖=𝑛+1𝜆^𝑛_𝑖𝐿𝑃^𝑣𝑖_𝑤−𝑃^∗_{𝑤𝐿2(𝑄)}→0as𝑛→∞, (3.12) where 𝑀=sup‖Φ(⋅)‖_𝐿²_(𝑄) and 𝐿 is the Lipschitz constant. Therefore, we have ^𝑘𝑛_𝑖=𝑛+1𝜆^𝑛_𝑖Φ𝑃^𝑣𝑖_𝑤𝑢^𝑣𝑖→Φ𝑃^∗_𝑤𝑢^∗in𝐿²(𝑄). (3.13) By (𝐻₃) and 𝛽∈𝐿^∞(𝑄), we obtain ^𝐴₀𝛽(𝑥,𝑡,𝑎)𝑢(𝑥,𝑡,𝑎)𝑑𝑎→^𝐴₀𝛽(𝑥,𝑡,𝑎)𝑢^∗(𝑥,𝑡,𝑎)𝑑𝑎. (3.14) Since 𝑢_𝑛→𝑢^∗ in 𝐿²(𝑄), we have ^𝑘𝑛_𝑖=𝑛+1𝜆^𝑛_𝑖Φ𝑃^𝑣𝑖_𝑤(𝑥,𝑡)𝑢^𝑣𝑖(𝑥,𝑡,𝑎)→Φ𝑃^∗_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 ^𝑘𝑛_𝑖=𝑛+1𝜆^𝑛_𝑖𝑄𝑣_𝑖(𝑥,𝑡,𝑎)𝑔(𝑥,𝑡,𝑎)𝑢^𝑣𝑖(𝑥,𝑡,𝑎)𝑑𝑥𝑑𝑡𝑑𝑎→𝐽𝑢^∗,𝑣^∗=𝑑as𝑛→∞. (3.16)

4. Necessary Conditions for Optimality

In this section, we study a necessary condition of optimality for the primal problem (𝐏).

Theorem 4.1. Let the assumptions (𝐻1)–(𝐻6) hold. Suppose that (𝑢^∗,𝑣^∗) is an optimal pair for the primal problem (𝐏). If 𝑝 is the solution of −𝜕𝑝()−𝜕𝑡𝑥,𝑡,𝑎𝜕𝑝()𝜕𝑎𝑥,𝑡,𝑎−𝑘Δ_𝑥𝑝(𝑥,𝑡,𝑎)+𝜇_𝑢𝑥,𝑡,𝑎,𝑢^∗(𝑥,𝑡,𝑎)𝑢^∗(𝑥,𝑡,𝑎)𝑝(𝑥,𝑡,𝑎)+𝜇𝑥,𝑡,𝑎,𝑢^∗(𝑥,𝑡,𝑎)𝑝(𝑥,𝑡,𝑎)+Φ𝑃^∗_𝑤()𝑝()()𝑥,𝑡𝑥,𝑡,𝑎+𝑤𝑥,𝑎^𝐴₀Φ_𝑃𝑃^∗_𝑤(𝑥,𝑡)𝑢^∗(𝑥,𝑡,𝛼)𝑝(𝑥,𝑡,𝛼)𝑑𝛼−𝛽(𝑥,𝑡,𝑎)𝑝(𝑥,𝑡,0)=−𝑣^∗(𝑥,𝑡,𝑎)𝑔(𝑥,𝑡,𝑎)−𝑣^∗(𝑥,𝑡,𝑎)𝑝(𝑥,𝑡,𝑎)in𝑄,𝜕𝑝()𝜕𝜂𝑥,𝑡,𝑎=0onΣ,𝑝(𝑥,𝑇,𝑎)=𝑝(𝑇)=0inΩ×(0,𝐴),𝑝(𝑥,𝑡,𝐴)=𝑝(𝐴)=0inΩ×(0,𝑇),𝑃^∗_𝑤(𝑥,𝑡)=^𝐴₀𝑤(𝑥,𝛼)𝑢^∗(𝑥,𝑡,𝛼)𝑑𝛼inΩ×(0,𝑇), (4.1)(4.2)(4.3)(4.4)(4.5) then we have 𝑣^∗()=𝜈𝑥,𝑡,𝑎¹(𝑥,𝑡,𝑎)if𝑔+𝑝(𝑥,𝑡,𝑎)<0and𝑢^∗()∈𝜈≠0,∀𝜈𝑥,𝑡,𝑎₁(𝑥,𝑡,𝑎),𝜈₂(𝑥,𝑡,𝑎)if𝑔+𝑝(𝑥,𝑡,𝑎)=0or𝑢^∗𝜈=0,₂(𝑥,𝑡,𝑎)if𝑔+𝑝(𝑥,𝑡,𝑎)>0and𝑢^∗≠0. (4.6) Here, 𝜈₁,𝜈₂ are the given functions in the control set 𝒰 which is introduced in the introduction, 𝜇_𝑢 is the derivative of 𝜇 with respect to 𝑢, and Φ_𝑃 is the derivative of Φ with respect to 𝑃=𝑃^∗_𝑤.

Proof. Since (𝑢^∗,𝑣^∗) is an optimal pair for the primal problem (𝐏) we get _𝑄𝑣^∗(𝑥,𝑡,𝑎)𝑔(𝑥,𝑡,𝑎)𝑢^∗()≥𝑥,𝑡,𝑎𝑑𝑥𝑑𝑡𝑑𝑎_𝑄𝑣^∗(𝑥,𝑡,𝑎)+𝜆𝑣(𝑥,𝑡,𝑎)𝑔(𝑥,𝑡,𝑎)𝑢^{𝑣∗+𝜆𝑣}(𝑥,𝑡,𝑎)𝑑𝑥𝑑𝑡𝑑𝑎 (4.7) for all 𝜆>0 and for all 𝑣∈𝐿^∞(𝑄) such that 𝑣(𝑥,𝑡,𝑎)≤0if𝑣^∗(𝑥,𝑡,𝑎)=𝜈₂(),𝑣()𝑥,𝑡,𝑎𝑥,𝑡,𝑎≥0if𝑣^∗(𝑥,𝑡,𝑎)=𝜈₁(𝑥,𝑡,𝑎). (4.8) Let 𝑣^𝜆=𝑣^∗+𝜆𝑣 and 𝑢^𝜆 be the solution of the dynamic system (1.3) corresponding to 𝑣^𝜆. Then the above equality implies _𝑄𝑣^∗()𝑔()𝑢𝑥,𝑡,𝑎𝑥,𝑡,𝑎^𝜆(𝑥,𝑡,𝑎)−𝑢^∗(𝑥,𝑡,𝑎)𝜆+𝑑𝑥𝑑𝑡𝑑𝑎_𝑄𝑣(𝑥,𝑡,𝑎)𝑔(𝑥,𝑡,𝑎)𝑢^𝜆(𝑥,𝑡,𝑎)𝑑𝑥𝑑𝑡𝑑𝑎≤0. (4.9) Let 𝑧(𝑥,𝑡,𝑎)=lim_𝜆→0((𝑢^𝜆(𝑥,𝑡,𝑎)−𝑢^∗(𝑥,𝑡,𝑎))/𝜆) be the solution to 𝜕𝑧()+𝜕𝑡𝑥,𝑡,𝑎𝜕𝑧()𝜕𝑎𝑥,𝑡,𝑎−𝑘Δ_𝑥𝑧(𝑥,𝑡,𝑎)+𝜇_𝑢𝑥,𝑡,𝑎,𝑢^∗(𝑥,𝑡,𝑎)𝑢^∗(𝑥,𝑡,𝑎)𝑧(𝑥,𝑡,𝑎)+𝜇𝑥,𝑡,𝑎,𝑢^∗(𝑥,𝑡,𝑎)𝑧(𝑥,𝑡,𝑎)+Φ_𝑃𝑃^∗_𝑤(𝑥,𝑡)𝑢^∗(𝑥,𝑡,𝑎)^𝐴₀𝑤()𝑧()𝑃𝑥,𝛼𝑥,𝑡,𝛼𝑑𝛼+Φ^∗_𝑤(𝑥,𝑡)𝑧(𝑥,𝑡,𝑎)=−𝑣^∗(𝑥,𝑡,𝑎)𝑧(𝑥,𝑡,𝑎)−𝑣(𝑥,𝑡,𝑎)𝑢^∗(𝑥,𝑡,𝑎)in𝑄,𝜕𝑧𝑧()=𝜕𝜂=0inΣ,𝑥,𝑡,0^𝐴₀𝛽()𝑧()(),𝑧()(),𝑃𝑥,𝑡,𝑎𝑥,𝑡,𝑎𝑑𝑎,inΩ×0,𝑇𝑥,0,𝑎=0inΩ×0,𝐴_∗𝑤(𝑥,𝑡)=^𝐴₀𝑤(𝑥,𝛼)𝑢^∗(𝑥,𝑡,𝛼)𝑑𝛼inΩ×(0,𝑇). (4.10) Since 𝑢^𝜆(𝑥,𝑡,𝑎)→𝑢^∗(𝑥,𝑡,𝑎) in 𝐿^∞(0,𝑇;𝐿²(Ω)×(0,𝐴)) as 𝜆→0, after some simple calculations and passing to the limit 𝜆→0 in (4.9) then we obtain _𝑄𝑣^∗()𝑔()𝑧()+𝑥,𝑡,𝑎𝑥,𝑡,𝑎𝑥,𝑡,𝑎𝑑𝑥𝑑𝑡𝑑𝑎_𝑄𝑣(𝑥,𝑡,𝑎)𝑔(𝑥,𝑡,𝑎)𝑢^∗(𝑥,𝑡,𝑎)𝑑𝑥𝑑𝑡𝑑𝑎≤0 (4.11) for all 𝜆>0 and for all 𝑣∈𝐿^∞(𝑄) such that 𝑣(𝑥,𝑡,𝑎)≤0if𝑣^∗(𝑥,𝑡,𝑎)=𝜈₂()𝑣()𝑥,𝑡,𝑎𝑥,𝑡,𝑎≥0if𝑣^∗(𝑥,𝑡,𝑎)=𝜈₁(𝑥,𝑡,𝑎). (4.12) Multiplying (4.1) by 𝑧 and integrating over 𝑄, we get _𝑄𝑣(𝑥,𝑡,𝑎)𝑢^∗(𝑥,𝑡,𝑎)𝑔+𝑝(𝑥,𝑡,𝑎)𝑑𝑥𝑑𝑡𝑑𝑎≤0 (4.13) for all 𝜆>0 and for all 𝑣∈𝐿^∞(𝑄) such that 𝑣(𝑥,𝑡,𝑎)≤0if𝑣^∗(𝑥,𝑡,𝑎)=𝜈₂()𝑣()𝑥,𝑡,𝑎𝑥,𝑡,𝑎≥0if𝑣^∗(𝑥,𝑡,𝑎)=𝜈₁(𝑥,𝑡,𝑎). (4.14)
By inequality (4.13), we get 𝑢^∗(𝑔+𝑝)∈𝑁_𝒰(𝑣^∗), where 𝑁_𝒰(𝑣^∗) is the normal cone at 𝒰 in 𝑣^∗. Therefore, if 𝑢^∗(𝑥,𝑡,𝑎)≠0, then 𝑣^∗()=𝜈𝑥,𝑡,𝑎¹()()()∈𝜈𝑥,𝑡,𝑎if𝑔+𝑝𝑥,𝑡,𝑎<0,∀𝜈𝑥,𝑡,𝑎₁(𝑥,𝑡,𝑎),𝜈₂(()𝜈𝑥,𝑡,𝑎)if𝑔+𝑝𝑥,𝑡,𝑎=0,₂(𝑥,𝑡,𝑎)if𝑔+𝑝(𝑥,𝑡,𝑎)>0, (4.15) and if 𝑢^∗(𝑥,𝑡,𝑎)=0, then 𝑣^∗ is any arbitrary value belonging to the interval [𝜈₁(𝑥,𝑡,𝑎),𝜈₂(𝑥,𝑡,𝑎)]. This completes the proof of Theorem 4.1.

From now on, let 𝑇>0,𝐴>0 and Ω be a bounded domain in ℝ^𝑁 with a smooth boundary 𝜕Ω. We consider an optimal control problem: find seek𝑣^∗∈𝒰suchthat𝐽𝑢^∗,𝑣^∗=max_𝑣∈𝒰𝐽(𝑢,𝑣) (𝐏𝟏) subject to 𝜕𝑢()+𝜕𝑡𝑥,𝑡,𝑎𝜕𝑢()𝜕𝑎𝑥,𝑡,𝑎−𝑘Δ_𝑥𝑢()((𝑢()+𝑥,𝑡,𝑎+𝜇𝑥,𝑡,𝑎,𝑢𝑥,𝑡,𝑎))𝑥,𝑡,𝑎_𝐴0𝑤()𝑢()𝑢()()𝑢()()×(),𝑥,𝛼𝑥,𝑡,𝛼𝑑𝛼𝑥,𝑡,𝑎=−𝑣𝑥,𝑡,𝑎𝑥,𝑡,𝑎in𝑄=Ω×0,𝑇0,𝐴𝜕𝑢()()×(),𝑢()𝜕𝜂𝑥,𝑡,𝑎=0onΣ=𝜕Ω×0,𝑇0,𝐴𝑥,0,𝑎=𝑢₀()(),𝑢()=𝑥,𝑎inΩ×0,𝐴𝑥,𝑡,0^𝐴₀𝛽(𝑥,𝑡,𝑎)𝑢(𝑥,𝑡,𝑎)𝑑𝑎inΩ×(0,𝑇), (4.16) where 𝐽(𝑢,𝑣)=∫_𝑄𝑣(𝑥,𝑡,𝑎)𝑔(𝑥,𝑡,𝑎)𝑢(𝑥,𝑡,𝑎)𝑑𝑥𝑑𝑡𝑑𝑎.

We note that this is a special case of the optimal control problem by the dynamic system (1.3) (Φ=𝐼). Then, we obtain the necessary condition for optimality.

Theorem 4.2. Let the assumptions (𝐻1),(𝐻2),(𝐻4),(𝐻5), and (𝐻6) hold. Suppose that (𝑢^∗,𝑣^∗) is an optimal pair for the problem (𝐏𝟏). If 𝑝 is a solution of the adjoint system (𝐴𝐸): −𝜕𝑝()−𝜕𝑡𝑥,𝑡,𝑎𝜕𝑝()𝜕𝑎𝑥,𝑡,𝑎−𝑘Δ_𝑥𝑝(𝑥,𝑡,𝑎)+𝜇_𝑢𝑥,𝑡,𝑎,𝑢^∗(𝑥,𝑡,𝑎)𝑢^∗(𝑥,𝑡,𝑎)𝑝(𝑥,𝑡,𝑎)+𝜇𝑥,𝑡,𝑎,𝑢^∗(𝑥,𝑡,𝑎)𝑝(𝑥,𝑡,𝑎)+^𝐴₀𝑤()𝑝()()𝑥,𝛼𝑑𝛼𝑥,𝑡,𝑎+𝑤𝑥,𝑎^𝐴₀𝑢^∗(𝑥,𝑡,𝛼)𝑝(𝑥,𝑡,𝛼)𝑑𝛼−𝛽(𝑥,𝑡,𝑎)𝑝(𝑥,𝑡,0)=−𝑣^∗(𝑥,𝑡,𝑎)𝑔(𝑥,𝑡,𝑎)−𝑣^∗(𝑥,𝑡,𝑎)𝑝(𝑥,𝑡,𝑎)in𝑄,𝜕𝑝()𝑝()(𝑇)(),𝑝()(𝐴)(),𝜕𝜂𝑥,𝑡,𝑎=0onΣ,𝑥,𝑇,𝑎=𝑝=0inΩ×0,𝐴𝑥,𝑡,𝐴=𝑝=0inΩ×0,𝑇 (4.17) then we have 𝑣^∗()=𝜈𝑥,𝑡,𝑎¹(𝑥,𝑡,𝑎)if𝑔+𝑝(𝑥,𝑡,𝑎)<0and𝑢^∗()∈𝜈≠0,∀𝜈𝑥,𝑡,𝑎₁(𝑥,𝑡,𝑎),𝜈₂(𝑥,𝑡,𝑎)if𝑔+𝑝(𝑥,𝑡,𝑎)=0or𝑢^∗𝜈=0,₂(𝑥,𝑡,𝑎)if𝑔+𝑝(𝑥,𝑡,𝑎)>0and𝑢^∗≠0. (4.18)

Remark 4.3. We may consider natural death rates 𝜇(𝑢)=𝜆𝑒^𝜆𝑢,𝑢≥0,𝜆>0;𝜇(𝑢)=𝑐₀+𝑐₁√𝑢,𝑢≥0,𝑐_𝑖≥0(𝑖=1,2) (4.19) as examples which satisfy our hypotheses. Here 𝜇 is the increasing function of 𝑢. It is natural to assume that the mortality rate depends on density of individuals 𝑢 as well as the total population. Also, we can consider the weight function which gives an effect on age as follows: 𝑤(𝛼)=1√𝑒2𝜋^−𝛼2/2,0≤𝛼≤𝐴. (4.20)

Remark 4.4. Let a functional 𝐾 and the set of controllers 𝒱 be defined by 𝐾𝑝(𝑥,𝑡,𝑎),𝑢(𝑥,𝑡,𝑎)=sup_𝑣∈𝒰𝑣(𝑥,𝑡,𝑎)𝑔(𝑥,𝑡,𝑎)𝑢(𝑥,𝑡,𝑎)+𝑝(𝑥,𝑡,𝑎)𝑣(𝑥,𝑡,𝑎)𝑢(𝑥,𝑡,𝑎),𝒱=𝑝∈𝑊^1,∞(𝑄)∣𝑝isapositivesolutionof(AE)inTheorem4.2, (4.21) respectively.
We introduce another control problem (𝐃) corresponding to the primal problem (𝐏), which is called the dual problem: inf_{𝑝∈𝒱𝑄}𝐾𝑝()()()×()𝑥,𝑡,𝑎,𝑢𝑥,𝑡,𝑎+𝑝𝑥,𝑡,𝑎𝜕𝑢𝑥,𝑡,𝑎+()𝜕𝑡𝜕𝑢𝑥,𝑡,𝑎𝜕𝑎−𝑘Δ_𝑥𝑢()((()𝑃𝑥,𝑡,𝑎+𝜇𝑥,𝑡,𝑎,𝑢𝑥,𝑡,𝑎))×𝑢𝑥,𝑡,𝑎+Φ_𝑤(𝑥,𝑡)𝑢(𝑥,𝑡,𝑎)𝑑𝑥𝑑𝑡𝑑𝑎 (𝐃) subject to the adjoint system (4.17).
Then, we can establish a duality theorem saying that the primal problem (𝐏) is equal to the dual problem (𝐃), which is the result in [10].

Acknowledgement

This work was supported by a 2-Year Research Grant of Pusan National University.