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Optimal boundary control problems for a hierarchical age-structured two-species model

Abstract

This paper discusses the optimal control problems for a nonlinear age-structured two-species model, where elder individuals are more competitive than younger ones, and each species is described by a nonlinear integropartial differential equation with a global feedback boundary condition. First, we establish the existence of a unique nonnegative bounded solution by means of frozen coefficients and the fixed-point theorem. More importantly, we discuss the least deviation-cost problem and the most benefit-cost problem. For the least deviation-cost problem, the existence of an optimal strategy is established by means of Ekeland’s variational principle, and the minimum principle is obtained via an adjoint system. Meanwhile, the corresponding results for the most benefit-cost problem are given. In addition, some numerical experiment results are presented to examine the effects of parameters on the optimal policies and indexes.

Introduction

It is well known that populations consist of individuals with many structural differences, which include age, size, gender, and status. Structured population models distinguish individuals from one another according to these structural differences, to determine the birth, growth, and death rates, interaction with each other and with the environment, infectivity, etc. (see [1]). During the past one hundred years, age- and size-structured first-order partial differential equations have provided a main tool for modeling population systems and have been recently employed in economics. To name a few, see [212] on the well-posedness, asymptotic behaviors, and optimal control of population models with age structure, while [1, 1322] discussed the related problems of population systems with size structure.

Ecological studies show that there exist dominance ranks of individuals in many species, such as nonmammalian, primates, rodents, etc., (see [23]). Moreover, [24] pointed out that for many species the competition for resources that determine individual fertility and mortality is based on some hierarchy in the population that is related to individuals age or any other physiological structure. They also referred to this competitive phenomenon as “contest competition” (see [25]). Hierarchical structuring in a biotic population is the ranking of individuals one above the other based on their age, body size, height, or any other possible structuring variable that can affect their vital rates (see [26]). Further, Lomnicki [27] showed that the rank of an individual not only influences its access to some factors such as food resource, mates, shelter, and nesting, but also has some influence on the individual’s vital rates. Thus, the existence of hierarchy has an impact on the dynamic behaviors of the population. Just as Gurney and Nisbet [26] said, the hierarchy of ranks in a population is one of the significant forces to maintain its ecological stability and species persistence. During the past decade, both discrete-time and continuous hierarchical structured population models have been studied by several researchers (see [2837]). The hierarchical model is considered as a generalization of the age- and size-structure model. The key is to consider the so-called “internal environment” in the modeling. For example, Ackleh and Deng [29] discussed the following nonlinear hierarchical age-structured population model

$$ \textstyle\begin{cases} \frac{\partial p(a,t)}{\partial t}+\frac{\partial p(a,t)}{\partial a}=- \mu (a,t,Q(p)(a,t))p(a,t),& 0< a< \infty , t>0, \\ p(0,t)=\int _{0}^{\infty}\beta (a,t,Q(p)(a,t))p(a,t)\,{\mathrm{d}}a, & t>0, \\ p(a,0)=p^{0}(a),& 0\leq a< \infty , \end{cases} $$
(1.1)

where the environment \(Q(p)(a,t)\) is a function of the density p, which is given by

$$\begin{aligned} Q(p) (a,t)=\alpha \int _{0}^{a}p(r,t)\,{\mathrm{d}}r+ \int _{a}^{\infty}p(r,t) \,{\mathrm{d}}r, \quad 0\leq \alpha < 1. \end{aligned}$$
(1.2)

Parameter α is used to measure the degree of hierarchy in the population structure, and is called the hierarchical coefficient. More precisely, α is the weight of the lower ranks (i.e., age smaller than a) in the competition for resources. From [38], \(\alpha =0\) (i.e.,“contest competition”) implies an absolute hierarchical structure, whereas values of α tending to 1 mean that, when competing for resources, the effect of higher ranks is similar to the effect of lower ranks. Moreover, the limit case \(\alpha =1\) (i.e., “scramble competition”) corresponds to the environment given by the total size of population at time t, i.e., without any hierarchical structure.

Most studies on the hierarchical population model mainly discuss the existence and uniqueness of solutions (see [28, 30, 36, 37]), numerical approximation of solutions (see [28, 29, 32, 33, 37]), and the asymptotic behavior of solutions (such as stability and persistence, see [31, 34, 35, 38]). However, studies on optimal control problems of hierarchical species models are rather rare. As far as we know, so far only [39, 40] have investigated optimal control problems in hierarchical species. However, there is no investigation on the optimal boundary control of hierarchical multispecies model. The purpose of this paper is to make some contribution in this direction. In this paper, we will discuss the optimal control problems in a hierarchical system composed of two interacting species with age structure. The control variables describe the inflow rate of the baby individual from an external source (such as artificially stocking fry or artificially planting saplings), which appear in the boundary conditions.

Description of the problem

Our study is inspired by that of Ackleh and Deng [29]. To build our model, we assume that there are age structures in two species and they have the same life expectancy (i.e., maximal age) and is \(A\in (0,+\infty )\). Moreover, we assume that the mortality and fertility of species depend mainly on the number of elderly individuals in addition to the age of individuals. From [29], this phenomenon can be described as that the vital rates of an individual depend not only on its age, but also on an “internal environment”. Let \(p_{i}(a,t)\) be the density of species i with age a at time t (\(i=1,2\)). In a similar way as we developed (1.2), the “internal environment” \(E(p_{i})\) of species i can be defined as

$$ E(p_{i}) (a,t)=\alpha _{i} \int _{0}^{a}p_{i}(r,t)\,{ \mathrm{d}}r+ \int _{a}^{A}p_{i}(r,t) \,{ \mathrm{d}}r,\quad 0\leq \alpha _{i}< 1. $$
(2.1)

Here, constant \(\alpha _{i}\) is the degree of hierarchy. This means that younger individuals are less competitive than older ones. Moreover, when two species interact, we assume that the fertility of a species is only related to its own “internal environment”, and the mortality of one species is not only related to its own “internal environment”, but also dependent on the “internal environment” of the another species. Thus, we can assume that the fertility and mortality of species i are \(\beta _{i}(a,E(p_{i})(a,t))\) and \(\mu _{i}(a,E(p_{1})(a,t),E(p_{2})(a,t))\), respectively. Hence, we propose the following hierarchical age-structured two-species model with boundary control (\(i=1,2\))

$$ \textstyle\begin{cases} \frac{\partial p_{i}(a,t)}{\partial t}+ \frac{\partial p_{i}(a,t)}{\partial a}=-\mu _{i} (a,E(p_{1})(a,t),E(p_{2})(a,t) )p_{i}(a,t),& (a,t)\in Q, \\ p_{i}(0,t)=\int _{0}^{A}\beta _{i} (a,E(p_{i})(a,t) )p_{i}(a,t) \,{\mathrm{d}}a+u_{i}(t), &t\in (0,T), \\ p_{i}(a,0)=p_{i}^{0}(a),& a\in [0,A), \end{cases} $$
(2.2)

where \(Q=(0,A)\times (0,T)\) and \(T\in (0,+\infty )\) represents the control horizon. \(p_{i}^{0}(a)\) is the initial age distribution of species i. The control variable \(u_{i}(t)\) stands for the number of baby individuals influx into species i at time t. Moreover, we assume that the control variable \(u_{i}(t)\) belongs to

$$ u_{i}\in \mathcal{U}_{i}\doteq \bigl\{ v\in L^{\infty}(0,T) \mid 0 \leq v(t)\leq U_{i}, t\in (0,T) \bigr\} . $$

The positive constant \(U_{i}\) represents the maximum number of baby individuals influx into species i. Let \(\mathcal{U}\doteq{\mathcal{U}}_{1}\times{\mathcal{U}}_{2}= \{(v_{1},v_{2}) \in [L^{\infty}(0,T)]^{2} | 0\leq v_{i}(t)\leq U_{i}, t\in (0,T), i=1,2 \}\). For any \(u=(u_{1},u_{2})\in \mathcal{U}\), let \((p_{1}^{u}(a,t),p_{2}^{u}(a,t))\) be the solution of (2.1) and (2.2) with the control variable u.

Compared with known closely related ones, our model has the following features. It is clear that our model includes some existing two species models with age structure. If we take \(\mu _{i}(a,E(p_{1})(a,t),E(p_{2})(a,t))=\mu _{i}(a,t)-\lambda _{i}(a,t)P_{j}\) and \(\beta _{i}(a,E(p_{i})(a,t))=\beta _{i}(t)m_{i}(a,t)\) (\(i,j=1,2\); \(i\neq j\)), then our model can be reduced to the model discussed in [9]. Moreover, if there is only one species and \(u_{i}(t)\equiv 0\), our system can be reduced to the system (1.1) and (1.2). In [29], Ackleh and Deng assumed that the vital signs β and μ in (1.2) are, respectively, decreasing and increasing with respect to “internal environment” Q (i.e., \(-\infty <\beta _{Q}\leq 0\) and \(0\leq \mu _{Q}<\infty \)). Based on these, the authors proved that the system has a unique solution by establishing a comparison principle and constructing monotone sequences. Moreover, they gave conditions for extinction and persistence of the population.

Undoubtedly, the above works are important and helpful to understand the evolution of hierarchical species. However, it is also an important topic for human beings to use or transform nature (such as renewable-resource development, pest control, etc.) to serve human survival and development on the basis of understanding of nature. The research in this field corresponds to the control problem of the model mathematically. It is widely known that investigations on optimal control problems in hierarchical populations are rather rare. The purpose of this paper is to make some contribution in this direction.

On the one hand, if the system describes the interaction between pests or annoying animals (such as rodents), we always hope that each species can be reduced to an ideal distribution level that does not affect crop growth and will not become extinct. Thus, in this paper, we first consider the following optimization problem

$$ \min_{(u_{1},u_{2})\in \mathcal{U}}J(u_{1},u_{2}) \doteq \sum_{i=1}^{2} \biggl\{ \int _{0}^{A} \bigl[p_{i}^{u}(a,T)- \bar{p}_{i}(a) \bigr]^{2}\,{\mathrm{d}}a+\sigma _{i} \int _{0}^{T}u_{i}^{2}(t) \, { \mathrm{d}}t \biggr\} . $$
(2.3)

Here, the first integral represents the deviation between final state \(p_{i}^{u}(a,T)\) and ideal distribution \(\bar{p}_{i}(a)\), while the second one stands for the costs of control with factor \(\sigma _{i}\). Thus, an optimal policy for (2.3) is to manipulate the system with any given initial distribution as close as possible to an ideal distribution with least control cost. Hence, control problem (2.3) can be called the least deviation-cost problem.

On the other hand, if species in the system are of renewable resources (such as forest resources or fish resources), optimal harvesting is always one of the most fascinating topics. Thus, in this paper, we also consider the following optimization problem

$$ \max_{(u_{1},u_{2})\in \mathcal{U}}J(u_{1},u_{2}) \doteq \sum_{i=1}^{2} \biggl\{ \int _{0}^{A}g_{i}(a)p_{i}(a,T) \,{\mathrm{d}}a- \sigma _{i} \int _{0}^{T}u_{i}^{2}(t) \,{\mathrm{d}}t \biggr\} . $$
(2.4)

Here, function \(g_{i}(a)\) represents the price of individuals with age a in species i. The motivation for (2.4) is as follows: How to input baby individuals during the period \([0,T]\), such that one can obtain the most benefit by exploiting the final distributions of species with the least control cost? Hence, one can say optimization problem (2.4) is the most benefit-cost problem.

Similar to the hypothesis of model (2.1) in [2, p.30], here, we make the following assumptions (\(i=1,2\)). Denote \(R_{+}=[0,+\infty )\).

(A1):

\(0\leq \beta _{i}(a,x)\leq \beta _{i}^{*}\) for any \((a,x)\in (0,A)\times R_{+}\), where \(\beta _{i}^{*}\) is a positive constant. This is reasonable because the birth rate of any species is a bounded function. Moreover, for any \(M>0\), there is \(L_{1}(M)>0\) such that

$$ \bigl\vert \beta _{i}(a,x_{1})-\beta _{i}(a,x_{2}) \bigr\vert \leq L_{1}(M) \vert x_{1}-x_{2} \vert , \quad \text{for } a\in [0,A), x_{1},x_{2}\in [0,M]. $$

This means that the derivative of the birth rate of the species with respect to its “internal environment” is a bounded function.

(A2):

\(\mu _{i}(a,x,y)>0\) for any \((a,x,y)\in (0,A)\times R_{+}\times R_{+}\). For any \((x,y)\in R_{+}\times R_{+}\), \(\mu _{i}(\cdot ,x,y)\in L_{\mathrm{loc}}^{1}[0,A]\) and \(\int _{0}^{A}\mu _{i}(a,x,y)\,{\mathrm{d}}a=+\infty \). This condition will guarantee that there are no individuals with age beyond A for each species. Moreover, for any \(M>0\), there is \(L_{2}(M)>0\) such that for \(a\in [0,A)\) and \(x_{1},x_{2},y_{1},y_{2}\in [0,M]\)

$$ \bigl\vert \mu _{i}(a,x_{1},y_{1})-\mu _{i}(a,x_{2},y_{2}) \bigr\vert \leq L_{2}(M) \bigl( \vert x_{1}-x_{2} \vert + \vert y_{1}-y_{2} \vert \bigr). $$
(A3):

For any \(a\in (0,A)\), \(0\leq p_{i}^{0}(a)\leq p_{i}^{*}\) with positive constant \(p_{i}^{*}\). This ensures that the initial distribution of each species is bounded.

Well-posedness of the state system

The purpose of this section is to discuss the well-posedness of (2.1) and (2.2). Here, we assume \(T>A\) and omit the proof for the case \(T\leq A\) (which is analogous, but easier). Let \({\mathbf{X}}=[L^{\infty}(0,T;L^{1}(0,A))]^{2}\doteq L^{\infty}(0,T;L^{1}(0,A)) \times L^{\infty}(0,T;L^{1}(0,A))\). For any \(w=(w_{1},w_{2})\in{\mathbf{X}}\), define a new norm in X by

$$ \Vert w \Vert _{*} = \operatorname{Ess}\sup _{t\in (0,T)} \biggl\{ e^{-\lambda t} \biggl[ \int _{0}^{A} \bigl\vert w_{1}(a,t) \bigr\vert \,{ \mathrm{d}}a+ \int _{0}^{A} \bigl\vert w_{2}(a,t) \bigr\vert \,{ \mathrm{d}}a \biggr] \biggr\} , $$

for some \(\lambda >0\). It is clear that \(({\mathbf{X}},\|\cdot \|_{\ast})\) is a Banach space. Let \(M_{i}^{*}=\max \{(A\beta _{i}^{*}p_{i}^{*}+U_{i})\exp \{T\beta _{i}^{*} \}, p_{i}^{*} \}\) and \(M^{*}=\max \{M_{1}^{*}, M_{2}^{*} \}\). Further, define the solution space as follows

$$\begin{aligned} \mathcal{X}= \biggl\{ (w_{1},w_{2})\in{\mathbf{X}} \bigg\vert \textstyle\begin{array}{l} w_{i}(a,t)\geq 0 \text{ a.e. } (a,t)\in Q \text{ and } \int _{0}^{A}w_{i}(a,t) \, { \mathrm{d}}a\leq M^{*} \\ \text{a.e. } t\in (0,T), i=1,2 \end{array}\displaystyle \biggr\} . \end{aligned}$$

Note that \(\mathcal{X}\) is a nonempty closed subset in X. Hence, \((\mathcal{X},\|\cdot \|_{\ast})\) is also a Banach space. Moreover, for any \(w=(w_{1},w_{2})\in{\mathbf{X}}\), we denote

$$ \bigl\Vert w(\cdot ,t) \bigr\Vert _{[L^{1}(0,A)]^{2}} = \sum _{i=1}^{2} \bigl\Vert w_{i}(\cdot ,t) \bigr\Vert _{L^{1}(0,A)} = \sum_{i=1}^{2} \int _{0}^{A} \bigl\vert w_{i}(a,t) \bigr\vert \, { \mathrm{d}}a. $$

First, let \(q(a,t)=(q_{1}(a,t),q_{2}(a,t))\in \mathbf{{X}}\) be arbitrary but fixed, so is the function

$$ E(q_{i}) (a,t)=\alpha _{i} \int _{0}^{a}q_{i}(r,t)\,{ \mathrm{d}}r+ \int _{a}^{A}q_{i}(r,t) \,{ \mathrm{d}}r,\quad i=1,2. $$
(3.1)

Consider the following linear system

$$ \textstyle\begin{cases} \frac{\partial p_{i}(a,t)}{\partial t}+ \frac{\partial p_{i}(a,t)}{\partial a}=-\mu _{i} (a,E(q_{1})(a,t),E(q_{2})(a,t) )p_{i}(a,t),&(a,t)\in Q, \\ p_{i}(0,t)=\int _{0}^{A}\beta _{i} (a,E(q_{i})(a,t) )p_{i}(a,t) \,{\mathrm{d}}a+u_{i}(t),&t\in (0,T), \\ p_{i}(a,0)=p_{i}^{0}(a),&a\in [0,A). \end{cases} $$
(3.2)

Clearly, system (3.1) and (3.2) has a unique nonnegative solution \((p_{1}(a,t;q),p_{2}(a,t;q))\). Using the characteristic lines (see [2]), the solution has the form (\(i=1,2\)):

$$ p_{i}(a,t;q)= \textstyle\begin{cases} p_{i}(a-t,0)\Pi _{i}(a,t,t;q),& a\geq t; \\ b_{i}(t-a;q)\Pi _{i}(a,t,a;q),& a< t, \end{cases} $$
(3.3)

where \(b_{i}(t-a;q)\doteq p_{i}(0,t-a;q)\) and

$$ \Pi _{i}(a,t,s;q) = e^{-\int _{0}^{s}\mu _{i}(a-\tau ,E(q_{1})(a- \tau ,t-\tau ),E(q_{2})(a-\tau ,t-\tau ))\,{\mathrm{d}}\tau}. $$
(3.4)

Thus, we can claim that \(b_{i}(t;q)\) (\(i=1,2\)) satisfies

$$ b_{i}(t;q)=F_{i}(t;q)+ \int _{0}^{t}K_{i}(t,a;q)b_{i}(t-a;q) \,{\mathrm{d}}a,\quad t\in (0,T), $$
(3.5)

where

$$\begin{aligned} &K_{i}(t,a;q)=\textstyle\begin{cases} \beta _{i} (a,E(q_{i})(a,t) )\Pi _{i}(a,t,a;q),& (a,t)\in Q; \\ 0, & \text{otherwise}, \end{cases}\displaystyle \end{aligned}$$
(3.6)
$$\begin{aligned} &F_{i}(t;q)=\textstyle\begin{cases} \int _{t}^{A}\beta _{i} (a,E(q_{i})(a,t) )p_{i}^{0}(a-t)\Pi _{i}(a,t,t;q) \,{\mathrm{d}}a+u_{i}(t), & t\in (0,A); \\ 0,& \text{otherwise}. \end{cases}\displaystyle \end{aligned}$$
(3.7)

Moreover, (A1)(A3) imply that

$$ 0\leq K_{i}(t,s;q)\leq \beta _{i}^{*},\qquad 0 \leq F_{i}(t;q)\leq A\beta _{i}^{*}p_{i}^{*}+U_{i}. $$
(3.8)

Lemma 3.1

For any \(q^{i}=(q_{1}^{i},q_{2}^{i})\in \mathcal{X}\) (\(i=1,2\)), there are positive constants \(M_{1}\) and \(M_{2}\) such that for \(0< t< T\)

$$\begin{aligned}& \bigl\vert F_{i} \bigl(t;q^{1} \bigr)-F_{i} \bigl(t;q^{2} \bigr) \bigr\vert \\& \quad \leq M_{1} \biggl[ \bigl\Vert q^{1}(\cdot ,t)-q^{2}( \cdot ,t) \bigr\Vert _{[L^{1}(0,A)]^{2}} + \int _{0}^{t} \bigl\Vert q^{1}( \cdot ,s)-q^{2}(\cdot ,s) \bigr\Vert _{[L^{1}(0,A)]^{2}}\, { \mathrm{{d}}}s \biggr], \end{aligned}$$
(3.9)
$$\begin{aligned}& \bigl\vert b_{i} \bigl(t;q^{1} \bigr)-b_{i} \bigl(t;q^{2} \bigr) \bigr\vert \\& \quad \leq M_{2} \biggl[ \bigl\Vert q^{1}(\cdot ,t)-q^{2}( \cdot ,t) \bigr\Vert _{[L^{1}(0,A)]^{2}} + \int _{0}^{t} \bigl\Vert q^{1}( \cdot ,s)-q^{2}(\cdot ,s) \bigr\Vert _{[L^{1}(0,A)]^{2}}\, { \mathrm{{d}}}s \biggr]. \end{aligned}$$
(3.10)

Proof

It is easy to show that (3.5) has a unique nonnegative solution. Moreover, using (3.5)–(3.8) and Gronwall’s inequality, we can derive that \(b_{i}(t;q)\leq B_{i}\) with positive constant \(B_{i}=(A\beta _{i}^{*}p_{i}^{*}+U_{i})\exp \{T\beta _{i}^{*}\}\) (\(i=1,2\)). For any \(q^{i}=(q_{1}^{i},q_{2}^{i})\in \mathcal{X}\), it follows from (3.1) that \((i,j=1,2)\)

$$ E \bigl(q_{j}^{i} \bigr) (a,t) = \alpha _{j} \int _{0}^{a}q_{j}^{i}(r,t) \,{\mathrm{d}}r+ \int _{a}^{A}q_{j}^{i}(r,t) \,{\mathrm{d}}r \leq \int _{0}^{A}q_{j}^{i}(r,t) \,{\mathrm{d}}r\leq M^{*}. $$
(3.11)

Moreover, we have

$$\begin{aligned}& \bigl\vert E \bigl(q_{j}^{1} \bigr) (a,t)-E \bigl(q_{j}^{2} \bigr) (a,t) \bigr\vert \\& \quad = \biggl\vert \alpha _{j} \int _{0}^{a} \bigl[q_{j}^{1}(r,t)-q_{j}^{2}(r,t) \bigr]\,{ \mathrm{d}}r+ \int _{a}^{A} \bigl[q_{j}^{1}(r,t)-q_{j}^{2}(r,t) \bigr]\,{\mathrm{d}}r \biggr\vert \\& \quad \leq \int _{0}^{A} \bigl\vert q_{j}^{1}(r,t)-q_{j}^{2}(r,t) \bigr\vert \,{\mathrm{d}}r= \bigl\Vert q_{j}^{1}( \cdot ,t)-q_{j}^{2}( \cdot ,t) \bigr\Vert _{L^{1}(0,A)}. \end{aligned}$$
(3.12)

Now, we prove inequality (3.9). From (A1)(A3), it follows that

$$\begin{aligned}& \bigl\vert \Pi _{i} \bigl(a,t ,t;q^{1} \bigr)-\Pi _{i} \bigl(a,t,t;q^{2} \bigr) \bigr\vert \\& \quad \leq L_{2} \bigl(M^{*} \bigr) \int _{0}^{t}\sum_{j=1}^{2} \bigl\vert E \bigl(q_{j}^{1} \bigr) (a- \tau ,t-\tau )-E \bigl(q_{j}^{2} \bigr) (a-\tau ,t-\tau ) \bigr\vert \,{\mathrm{d}} \tau \\& \quad = L_{2} \bigl(M^{*} \bigr) \int _{0}^{t}\sum_{j=1}^{2} \bigl\vert E \bigl(q_{j}^{1} \bigr) (a-t+s,s)-E \bigl(q_{j}^{2} \bigr) (a-t+s,s) \bigr\vert \,{ \mathrm{d}}s \\& \quad \leq L_{2} \bigl(M^{*} \bigr) \int _{0}^{t} \bigl\Vert q^{1}( \cdot ,s)-q^{2}(\cdot ,s) \bigr\Vert _{[L^{1}(0,A)]^{2}} \,{ \mathrm{d}}s\quad (i=1,2). \end{aligned}$$
(3.13)

Thus, from (3.7) and (3.11)–(3.13), we have

$$\begin{aligned} \bigl\vert F_{1} \bigl(t;q^{1} \bigr)-F_{1} \bigl(t;q^{2} \bigr) \bigr\vert \leq {}& L_{1} \bigl(M^{*} \bigr)p_{1}^{*} \int _{0}^{A} \bigl\vert E \bigl(q_{1}^{1} \bigr) (a,t)-E \bigl(q_{1}^{2} \bigr) (a,t) \bigr\vert \,{ \mathrm{d}}a \\ &{}+ p_{1}^{*}\beta _{1}^{*} \int _{0}^{A} \bigl\vert \Pi _{1} \bigl(a,t,t;q^{1} \bigr)- \Pi _{1} \bigl(a,t,t;q^{2} \bigr) \bigr\vert \,{\mathrm{d}}a \\ \leq {}& M_{1,1}D(t), \end{aligned}$$

where \(D(t)=\|q^{1}(\cdot ,t)-q^{2}(\cdot ,t)\|_{[L^{1}(0,A)]^{2}}+\int _{0}^{t} \|q^{1}(\cdot ,s)-q^{2}(\cdot ,s)\|_{[L^{1}(0,A)]^{2}}\,{\mathrm{d}}s\) and positive constant \(M_{1,1}=p_{1}^{*}A\max \{L_{1}(M^{*}),\beta _{1}^{*}L_{2}(M^{*}) \}\). Similarly, we also have

$$ \bigl\vert F_{2} \bigl(t;q^{1} \bigr)-F_{2} \bigl(t;q^{2} \bigr) \bigr\vert \leq M_{1,2}D(t), $$

where \(M_{1,2}=p_{2}^{*}A\max \{L_{1}(M^{*}),\beta _{2}^{*}L_{2}(M^{*}) \}\). Thus, the inequality (3.9) is true with \(M_{1}=\max \{M_{1,1}, M_{1,2} \}\).

Next, we prove inequality (3.10). From (A1)(A3) and (3.5), it follows that

$$\begin{aligned}& \bigl\vert b_{1} \bigl(t; q^{1} \bigr)-b_{1} \bigl(t;q^{2} \bigr) \bigr\vert \\& \quad \leq \bigl\vert F_{1} \bigl(t;q^{1} \bigr)-F_{1} \bigl(t;q^{2} \bigr) \bigr\vert + \int _{0}^{t}K_{1} \bigl(t,t-s;q^{2} \bigr) \bigl\vert b_{1} \bigl(s;q^{1} \bigr)-b_{1} \bigl(s;q^{2} \bigr) \bigr\vert \,{\mathrm{d}}s \\& \qquad {} + \int _{0}^{t} \bigl\vert K_{1} \bigl(t,t-s;q^{1} \bigr)-K_{1} \bigl(t,t-s;q^{2} \bigr) \bigr\vert b_{1} \bigl(s;q^{1} \bigr) \,{ \mathrm{d}}s \\& \quad \leq M_{1}D(t)+L_{1} \bigl(M^{*} \bigr)B_{1}T \bigl\Vert q_{1}^{1}(\cdot ,t)-q_{1}^{2}( \cdot ,t) \bigr\Vert _{L^{1}(0,A)} \\& \qquad {} +\beta _{1}^{*} \int _{0}^{t} \bigl\vert b_{1} \bigl(s;q^{1} \bigr)-b_{1} \bigl(s;q^{2} \bigr) \bigr\vert \,{\mathrm{d}}s \\& \qquad {} +L_{2} \bigl(M^{*} \bigr)B_{1}T\beta _{1}^{*} \int _{0}^{t}\sum_{j=1}^{2} \bigl\Vert q_{j}^{1}( \cdot ,s)-q_{j}^{2}( \cdot ,s) \bigr\Vert _{L^{1}(0,A)}\,{\mathrm{d}}s \\& \quad \leq M_{3}D(t)+\beta _{1}^{*} \int _{0}^{t} \bigl\vert b_{1} \bigl(s;q^{1} \bigr)-b_{1} \bigl(s;q^{2} \bigr) \bigr\vert \,{\mathrm{d}}s, \end{aligned}$$

where \(M_{3}=M_{1}+B_{1}T\max \{L_{1}(M),\beta _{1}^{*}L_{2}(M) \}\). Then, using Gronwall’s inequality, we can obtain

$$ \bigl\vert b_{1} \bigl(t;q^{1} \bigr)-b_{1} \bigl(t;q^{2} \bigr) \bigr\vert \leq M_{2,1}D(t), $$

where \(M_{2,1}=M_{3} [1+\beta _{1}^{*}\exp \{\beta _{1}^{*}T \}(1+T) ]\). Similarly, we also have

$$ \bigl\vert b_{2} \bigl(t; q^{1} \bigr)-b_{2} \bigl(t;q^{2} \bigr) \bigr\vert \leq M_{2,2}D(t), $$

where \(M_{2,2}= [M_{1}+B_{2}T\max \{L_{1}(M),\beta _{2}^{*}L_{2}(M) \} ] [1+\beta _{2}^{*}\exp \{\beta _{2}^{*}T \}(1+T) ]\). Hence, (3.10) is true with \(M_{2}=\max \{M_{2,1}, M_{2,2} \}\). The proof is complete. □

Now, for any \(q=(q_{1},q_{2})\in \mathcal{X}\), we define the mapping \(\mathcal{A}: \mathcal{X}\rightarrow{\mathbf{X}}\) by

$$ \mathcal{A}q=\mathcal{A}(q_{1},q_{2})= \bigl( \mathcal{A}_{1}(q_{1},q_{2}), \mathcal{A}_{2}(q_{1},q_{2}) \bigr), $$

where \(\mathcal{A}_{i}(q_{1},q_{2})\) (\(i=1,2\)) is defined by the right-hand sides of (3.3). Moreover, for any \(q\in \mathcal{X}\), it is easy to show that \(\mathcal{A}q\in \mathcal{X}\). Thus, if \(\mathcal{A}\) is a contractive mapping on Banach space \((\mathcal{X}, \|\cdot \|_{\ast})\), then system (2.1) and (2.2) has a unique solution.

Theorem 3.2

Let (A1)(A3) hold. Then, system (2.1) and (2.2) has a unique solution, which is nonnegative and bounded.

Proof

For any \(q^{i}=(q_{1}^{i},q_{2}^{i})\in \mathcal{X}\) (\(i=1,2\)), it follows from Lemma 3.1 that

$$\begin{aligned}& \int _{0}^{A} \bigl\vert \mathcal{A}_{1} \bigl(q_{1}^{1},q_{2}^{1} \bigr)-\mathcal{A}_{1} \bigl(q_{1}^{2},q_{2}^{2} \bigr) \bigr\vert (a,t) \,{\mathrm{d}}a \\& \quad \leq \int _{0}^{t} \bigl\vert b_{1} \bigl(s;q^{1} \bigr)-b_{1} \bigl(s;q^{2} \bigr) \bigr\vert \,{\mathrm{d}}s + B_{1} \int _{0}^{t} \bigl\vert \Pi _{1} \bigl(a,t,a;q^{1} \bigr)-\Pi _{1} \bigl(a,t,a;q^{2} \bigr) \bigr\vert \,{\mathrm{d}}a \\& \qquad {} + p_{1}^{*} \int _{t}^{A} \bigl\vert \Pi _{1} \bigl(a,t,t;q^{1} \bigr)-\Pi _{1} \bigl(a,t,t;q^{2} \bigr) \bigr\vert \,{\mathrm{d}}a \\& \quad \leq M_{2} \int _{0}^{t} \biggl[ \bigl\Vert q^{1}(\cdot ,s)-q^{2}(\cdot ,s) \bigr\Vert _{[L^{1}(0,A)]^{2}} + \int _{0}^{s} \bigl\Vert q^{1}( \cdot ,\tau )-q^{2}(\cdot ,\tau ) \bigr\Vert _{[L^{1}(0,A)]^{2}} \,{ \mathrm{d}}\tau \biggr] \,{\mathrm{d}}s \\& \qquad {} + B_{1}L_{2} \bigl(M^{*} \bigr) \int _{0}^{t} \int _{0}^{a}\sum_{j=1}^{2} \bigl\vert E \bigl(q_{j}^{1} \bigr) (a- \tau ,t-\tau )-E \bigl(q_{j}^{2} \bigr) (a-\tau ,t-\tau ) \bigr\vert \,{\mathrm{d}} \tau \,{\mathrm{d}}a \\& \qquad {} + p_{1}^{*}L_{2} \bigl(M^{*} \bigr) \int _{t}^{A} \int _{0}^{t}\sum_{j=1}^{2} \bigl\vert E \bigl(q_{j}^{1} \bigr) (a- \tau ,t-\tau )-E \bigl(q_{j}^{2} \bigr) (a-\tau ,t-\tau ) \bigr\vert \,{\mathrm{d}} \tau \,{\mathrm{d}}a \\& \quad \leq H_{1} \int _{0}^{t} \bigl\Vert q^{1}( \cdot ,s)-q^{2}(\cdot ,s) \bigr\Vert _{[L^{1}(0,A)]^{2}} \,{ \mathrm{d}}s, \end{aligned}$$

where \(H_{1}=M_{2}(1+T)+B_{1}L_{2}(M^{*})T+p_{1}^{*}L_{2}(M^{*})A\). Similarly, there is a constant \(H_{2}=M_{2}(1+T)+B_{2}L_{2}(M^{*})T+p_{2}^{*}L_{2}(M^{*})A\) such that

$$ \int _{0}^{A} \bigl\vert \mathcal{A}_{2} \bigl(q_{1}^{1},q_{2}^{1} \bigr)-\mathcal{A}_{2} \bigl(q_{1}^{2},q_{2}^{2} \bigr) \bigr\vert (a,t) \,{\mathrm{d}}a \leq H_{2} \int _{0}^{t} \bigl\Vert q^{1}( \cdot ,s)-q^{2}(\cdot ,s) \bigr\Vert _{[L^{1}(0,A)]^{2}} \,{ \mathrm{d}}s. $$

Thus,

$$ \bigl\Vert \bigl(\mathcal{A}q^{1} \bigr) (\cdot ,t)- \bigl( \mathcal{A}q^{2} \bigr) (\cdot ,t) \bigr\Vert _{[L^{1}(0,A)]^{2}} \leq M_{4} \int _{0}^{t} \bigl\Vert q^{1}( \cdot ,s)-q^{2}(\cdot ,s) \bigr\Vert _{[L^{1}(0,A)]^{2}} \,{ \mathrm{d}}s, $$
(3.14)

where \(M_{4}=2M_{2}(1+T)+L_{2}(M)T(B_{1}+B_{2})+L_{2}(M)A(p_{1}^{*}+p_{2}^{*})\). Hence, for any \(q^{i}=(q_{1}^{i},q_{2}^{i})\in \mathcal{X}\) (\(i=1,2\)), it follows from (3.14) that

$$\begin{aligned} \bigl\Vert \mathcal{A}q^{1}-\mathcal{A}q^{2} \bigr\Vert _{*} &= \operatorname{Ess}\sup_{t\in (0,T)} \bigl\{ e^{-\lambda t} \bigl\Vert \bigl(\mathcal{A}q^{1} \bigr) ( \cdot ,t)- \bigl(\mathcal{A}q^{2} \bigr) ( \cdot ,t) \bigr\Vert _{[L^{1}(0,A)]^{2}} \bigr\} \\ &\leq M_{4}\operatorname{Ess}\sup_{t\in (0,T)} \biggl\{ e^{-\lambda t} \int _{0}^{t} \bigl\Vert q^{1}( \cdot ,s)-q^{2}(\cdot ,s) \bigr\Vert _{[L^{1}(0,A)]^{2}}\,{ \mathrm{d}}s \biggr\} \\ &\leq \frac{M_{4}}{\lambda} \bigl\Vert q^{1}-q^{2} \bigr\Vert _{*}. \end{aligned}$$

Choose λ such that \(\lambda >M_{4}\). Then, \(\mathcal{A}\) is a contraction on Banach space \((\mathcal{X},\|\cdot \|_{\ast})\). By the Banach fixed-point theorem, \(\mathcal{A}\) has a unique fixed point, which is the solution of system (2.1) and (2.2). Moreover, from the definition of solution space \(\mathcal{X}\), it follows that the solution is nonnegative and bounded. □

Theorem 3.3

Let (A1)(A3) hold. Then, the solution \(p^{u}=(p_{1}^{u},p_{2}^{u})\) of system (2.1) and (2.2) is continuous with respect to the control variable \(u=(u_{1},u_{2})\in \mathcal{U}\), that is, for any \(u^{i}=(u_{1}^{i},u_{2}^{i})\in \mathcal{U}\) (\(i=1,2\)), there is a constant \(\bar{M}>0\) such that

$$ \sum_{j=1}^{2} \bigl\Vert p_{j}^{1}( \cdot ,t)-p_{j}^{2}( \cdot ,t) \bigr\Vert _{L^{1}(0,A)} \leq \bar{M} \int _{0}^{t}\sum_{j=1}^{2} \bigl\vert u_{j}^{1}(s)-u_{j}^{2}(s) \bigr\vert \,{ \mathrm{d}}s, $$

where \(p^{i}=(p_{1}^{i},p_{2}^{i})\) is the solution of system (2.1) and (2.2) corresponding to \(u^{i}\).

Proof

For any \(u=(u_{1},u_{2})\in \mathcal{U}\), the solution of system (2.1) and (2.2) has the form

$$ p_{i}(a,t)= \textstyle\begin{cases} p_{i}^{0}(a-t)\Pi _{i}(a,t,t),& a\geq t; \\ b_{i}(t-a)\Pi _{i}(a,t,a),& a< t, \end{cases} $$
(3.15)

where \(b_{i}(t-a)\doteq p_{i}(0,t-a)\) and

$$ \Pi _{i}(a,t,s) = e^{-\int _{0}^{s}\mu _{i}(a-\tau ,E(p_{1})(a-\tau ,t- \tau ),E(p_{2})(a-\tau ,t-\tau ))\,{\mathrm{d}}\tau}. $$
(3.16)

Similarly, \(b_{i}(t)\) (\(i=1,2\)) satisfies

$$ b_{i}(t)=F_{i}(t)+ \int _{0}^{t}K_{i}(t,a)b_{i}(t-a) \,{\mathrm{d}}a,\quad t \in (0,T). $$
(3.17)

Here,

$$\begin{aligned} &K_{i}(t,a)= \textstyle\begin{cases} \beta _{i} (a,E(p_{i})(a,t) )\Pi _{i}(a,t,a),& (a,t)\in Q; \\ 0, & \text{otherwise}, \end{cases}\displaystyle \end{aligned}$$
(3.18)
$$\begin{aligned} &F_{i}(t)= \textstyle\begin{cases} \int _{t}^{A}\beta _{i} (a,E(p_{i})(a,t) )p_{i}^{0}(a-t)\Pi _{i}(a,t,t) \,{\mathrm{d}}a+u_{i}(t), & t\in (0,A); \\ 0,& \text{otherwise}. \end{cases}\displaystyle \end{aligned}$$
(3.19)

Note that \(p^{i}=(p_{1}^{i},p_{2}^{i})\) is a solution of system (2.1) and (2.2) corresponding to \(u^{i}\in \mathcal{U}\) (\(i=1,2\)). It follows from (3.17) that

$$\begin{aligned}& \bigl\vert b_{1}^{1} (t)-b_{1}^{2}(t) \bigr\vert \\& \quad \leq p_{1}^{*}L_{1} \bigl(M^{*} \bigr) \int _{t}^{A} \bigl\vert E \bigl(p_{1}^{1} \bigr) (a,t)-E \bigl(p_{1}^{2} \bigr) (a,t) \bigr\vert \,{ \mathrm{d}}a+ \bigl\vert u_{1}^{1}(t)-u_{1}^{2}(t) \bigr\vert \\& \qquad {} + p_{1}^{*}\beta _{1}^{*} \int _{t}^{A} \bigl\vert \Pi _{1}^{1}(a,t,t)- \Pi _{1}^{2}(a,t,t) \bigr\vert \,{\mathrm{d}}a + \beta _{1}^{*} \int _{0}^{t} \bigl\vert b_{1}^{1}(s)-b_{1}^{2}(s) \bigr\vert \,{\mathrm{d}}s \\& \qquad {} + B_{1} \int _{0}^{t} \bigl\vert \beta _{1} \bigl(a,E \bigl(p_{1}^{1} \bigr) (a,s) \bigr) \Pi _{1}^{1}(a,s,a)-\beta _{1} \bigl(a,E \bigl(p_{1}^{2} \bigr) (a,s) \bigr)\Pi _{1}^{2}(a,s,a) \bigr\vert \,{\mathrm{d}}s \\& \quad \leq \bigl(p_{1}^{*}A+B_{1}T \bigr)L_{1} \bigl(M^{*} \bigr) \bigl\Vert p_{1}^{1}( \cdot ,t)-p_{1}^{2}( \cdot ,t) \bigr\Vert _{L^{1}(0,A)} + \bigl\vert u_{1}^{1}(t)-u_{1}^{2}(t) \bigr\vert \\& \qquad {} + \beta _{1}^{*} \int _{0}^{t} \bigl\vert b_{1}^{1}(s)-b_{1}^{2}(s) \bigr\vert \,{ \mathrm{d}}s \\& \qquad {} + p_{1}^{*}\beta _{1}^{*}L_{2} \bigl(M^{*} \bigr) \int _{t}^{A} \int _{0}^{t}\sum_{j=1}^{2} \bigl\Vert p_{j}^{1}(\cdot ,t-\tau )-p_{j}^{2}( \cdot ,t-\tau ) \bigr\Vert _{L^{1}(0,A)} \,{\mathrm{d}}\tau \,{\mathrm{d}}a \\& \qquad {} + B_{1}\beta _{1}^{*}L_{2} \bigl(M^{*} \bigr) \int _{0}^{t} \int _{0}^{a}\sum_{j=1}^{2} \bigl\Vert p_{j}^{1}(\cdot ,s-\tau )-p_{j}^{2}( \cdot ,s-\tau ) \bigr\Vert _{L^{1}(0,A)} \,{\mathrm{d}}\tau \,{\mathrm{d}}s \\& \quad \leq M_{5} \bigl\Vert p^{1}(\cdot ,t)-p^{2}(\cdot ,t) \bigr\Vert _{[L^{1}(0,A)]^{2}} + \bigl\vert u_{1}^{1}(t)-u_{1}^{2}(t) \bigr\vert +\beta _{1}^{*} \int _{0}^{t} \bigl\vert b_{1}^{1}(s)-b_{1}^{2}(s) \bigr\vert \,{ \mathrm{d}}s, \end{aligned}$$
(3.20)

where \(M_{5}=(p_{1}^{*}A+B_{1}T)L_{1}(M^{*})+(p_{1}^{*}+B_{1})AT\beta _{1}^{*}L_{2}(M^{*})\). Using Gronwall’s inequality, we can derive that

$$\begin{aligned} \bigl\vert b_{1}^{1}(t)-b_{1}^{2}(t) \bigr\vert \leq {}& \bigl\vert u_{1}^{1}(t)-u_{1}^{2}(t) \bigr\vert +M_{5} \bigl\Vert p^{1}(\cdot ,t)-p^{2}(\cdot ,t) \bigr\Vert _{[L^{1}(0,A)]^{2}} \\ &{}+\exp \bigl\{ \beta _{1}^{*}T \bigr\} \int _{0}^{t} \bigl[ \bigl\vert u_{1}^{1}(s)-u_{1}^{2}(s) \bigr\vert \\ &{}+M_{5} \bigl\Vert p^{1}(\cdot ,s)-p^{2}( \cdot ,s) \bigr\Vert _{[L^{1}(0,A)]^{2}} \bigr]\,{ \mathrm{d}}s. \end{aligned}$$
(3.21)

Further, for \(a\geq t\), from (3.12), (3.15), and (3.16), it follows that

$$ \bigl\vert p_{1}^{1}(a,t)-p_{1}^{2}(a,t) \bigr\vert \leq p_{1}^{*}L_{2} \bigl(M^{*} \bigr) \int _{0}^{t} \bigl\Vert p^{1}( \cdot ,s)-p^{2}(\cdot ,s) \bigr\Vert _{[L^{1}(0,A)]^{2}}\,{ \mathrm{d}}s. $$
(3.22)

For \(a< t\), from (3.12), (3.15), and (3.16), it follows that

$$\begin{aligned} \bigl\vert p_{1}^{1}(a,t)-p_{1}^{2}(a,t) \bigr\vert \leq{} & B_{1}L_{2} \bigl(M^{*} \bigr) \int _{0}^{t} \bigl\Vert p^{1}( \cdot ,s)-p^{2}(\cdot ,s) \bigr\Vert _{[L^{1}(0,A)]^{2}}\,{ \mathrm{d}}s \\ &{}+ \bigl\vert b_{1}^{1}(t-a)-b_{1}^{2}(t-a) \bigr\vert . \end{aligned}$$
(3.23)

Thus, from (3.21)–(3.23), it follows that

$$\begin{aligned}& \bigl\Vert p_{1}^{1} (\cdot ,t)-p_{1}^{2}( \cdot ,t) \bigr\Vert _{L^{1}(0,A)} \\& \quad = \int _{0}^{t} \bigl\vert p^{1}(a,t)-p^{2}(a,t) \bigr\vert \,{\mathrm{d}}a+ \int _{t}^{A} \bigl\vert p^{1}(a,t)-p^{2}(a,t) \bigr\vert \,{\mathrm{d}}a \\& \quad \leq \int _{0}^{t} \bigl\vert b_{1}^{1}(s)-b_{1}^{2}(s) \bigr\vert \,{\mathrm{d}}s +H_{3}AL_{2} \bigl(M^{*} \bigr) \int _{0}^{t} \bigl\Vert p^{1}( \cdot ,s)-p^{2}(\cdot ,s) \bigr\Vert _{[L^{1}(0,A)]^{2}}\,{ \mathrm{d}}s \\& \quad \leq \bigl(1+\exp \bigl\{ \beta _{1}^{*}T \bigr\} T \bigr) \int _{0}^{t} \bigl\vert u_{1}^{1}(s)-u_{1}^{2}(s) \bigr\vert \,{\mathrm{d}}s \\& \qquad {} +M_{6} \int _{0}^{t} \bigl\Vert p^{1}( \cdot ,s)-p^{2}(\cdot ,s) \bigr\Vert _{[L^{1}(0,A)]^{2}} \,{ \mathrm{d}}s, \end{aligned}$$

where \(H_{3}=\max \{B_{1},p_{1}^{*}\}\) and \(M_{6}=M_{5}(1+\exp \{\beta _{1}^{*}T\})+H_{3}AL_{2}(M^{*})\). Similarly, there exists a positive constant \(M_{7}\) such that

$$\begin{aligned} \bigl\Vert p_{2}^{1}(\cdot ,t)-p_{2}^{2}( \cdot ,t) \bigr\Vert _{L^{1}(0,A)} \leq {}& \bigl(1+ \exp \bigl\{ \beta _{2}^{*}T \bigr\} T \bigr) \int _{0}^{t} \bigl\vert u_{2}^{1}(s)-u_{2}^{2}(s) \bigr\vert \,{ \mathrm{d}}s \\ &{}+ M_{7} \int _{0}^{t} \bigl\Vert p^{1}( \cdot ,s)-p^{2}(\cdot ,s) \bigr\Vert _{[L^{1}(0,A)]^{2}} \,{ \mathrm{d}}s. \end{aligned}$$

Thus, we have

$$\begin{aligned} \sum_{j=1}^{2} \bigl\Vert p_{j}^{1}( \cdot ,t)-p_{j}^{2}( \cdot ,t) \bigr\Vert _{L^{1}(0,A)} \leq {}& \bigl(1+\exp \bigl\{ \beta ^{*}T \bigr\} T \bigr) \int _{0}^{t}\sum_{j=1}^{2} \bigl\vert u_{j}^{1}(s)-u_{j}^{2}(s) \bigr\vert \,{\mathrm{d}}s \\ &{}+M_{8} \int _{0}^{t} \bigl\Vert p^{1}( \cdot ,s)-p^{2}(\cdot ,s) \bigr\Vert _{[L^{1}(0,A)]^{2}} \,{ \mathrm{d}}s, \end{aligned}$$

where \(\beta ^{*}=\max \{\beta _{1}^{*},\beta _{2}^{*}\}\) and \(M_{8}=\max \{M_{6},M_{7}\}\). Then, Gronwall’s inequality implies that

$$ \sum_{j=1}^{2} \bigl\Vert p_{j}^{1}( \cdot ,t)-p_{j}^{2}( \cdot ,t) \bigr\Vert _{L^{1}(0,A)} \leq H_{4} \int _{0}^{t}\sum_{j=1}^{2} \bigl\vert u_{j}^{1}(s)-u_{j}^{2}(s) \bigr\vert \,{ \mathrm{d}}s, $$

where \(H_{4}=(1+\exp \{\beta ^{*}T\}T)(1+\exp \{M_{8}T\}T)\). The proof is complete. □

Remark 3.4

In this section, we first prove that the model has a unique nonnegative bounded solution by constructing a suitable solution space and equivalent norm, and using the freezing-coefficient method and fixed-point theory. The boundedness of the solution ensures that the species described in the system will not grow indefinitely. Then, using the basic inequality theory, we obtain the continuous dependence of solutions on the control variable, which will play a key role in the following discussions.

Least deviation-cost problem

In this section, we will discuss the optimization problem (2.3). Here and after, we denote by \(\mathcal{T}_{\mathcal{U}_{i}}(u_{i})\) and \(\mathcal{N}_{\mathcal{U}_{i}}(u_{i})\) the tangent cone and the normal cone of \(\mathcal{U}_{i}\) at \(u_{i}\), respectively. Moreover, from [41], we have the following result.

Lemma 4.1

For any \(v_{i}\in \mathcal{T}_{\mathcal{U}_{i}}(u_{i})\), suppose that \(\vartheta _{i}(t)\in L^{1}(0,T)\) satisfies

$$\begin{aligned} \sum_{i=1}^{2} \int _{0}^{T} \bigl[\vartheta _{i}(t)v_{i}(t)+ \rho _{i} \bigl\vert v_{i}(t) \bigr\vert \bigr]\,{\mathrm{d}}t\geq 0. \end{aligned}$$

Then, there exists \(\theta _{i}\in L^{\infty}(0,T)\) such that \(\|\theta _{i}\|_{L^{\infty}(0,T)}\leq 1\) and \(\rho _{i}\theta _{i}-\vartheta _{i}\in \mathcal{N}_{\mathcal{U}_{i}}(u_{i})\).

Minimum principle

In this subsection, we will establish first-order necessary conditions of optimality in the form of an Euler–Lagrange system.

Theorem 4.2

Let \(u^{*}=(u_{1}^{*},u_{2}^{*})\) and \(p^{*}=(p_{1}^{*},p_{2}^{*})\). If \((u^{*},p^{*})\) is an optimal pair for the optimization problem (2.3), then optimal strategies \(u_{i}^{*}\) must be structured as

$$ u_{i}^{*}(t) = \mathcal{F}_{i} \biggl( \frac{q_{i}(0,\cdot )}{\sigma _{i}} \biggr) (t) \doteq \textstyle\begin{cases} 0,& \textit{if } q_{i}(0,t)< 0, \\ \frac{q_{i}(0,t)}{\sigma _{i}},& \textit{if } 0\leq \frac{q_{i}(0,t)}{\sigma _{i}}< U_{i}, \\ U_{i},& \textit{if } \frac{q_{i}(0,t)}{\sigma _{i}} \geq U_{i}, \end{cases} $$
(4.1)

where \(q=(q_{1},q_{2})\) is the solution of the adjoint system

$$ \textstyle\begin{cases} \frac{\partial q_{i}}{\partial t}+\frac{\partial q_{i}}{\partial a} = \alpha _{i}\int _{a}^{A} [\sum_{j=2}^{3}\mu _{ij} (r,E(p_{1}^{*})(r,t),E(p_{2}^{*})(r,t) )q_{i}(r,t)p_{i}^{*}(r,t) \\ \hphantom{\frac{\partial q_{i}}{\partial t}+\frac{\partial q_{i}}{\partial a} ={}}{}- \beta _{i2} (r,E(p_{i}^{*})(r,t) )q_{i}(0,t)p_{i}^{*}(r,t) ]\, {\mathrm{{d}}}r \\ \hphantom{\frac{\partial q_{i}}{\partial t}+\frac{\partial q_{i}}{\partial a} ={}}{}+\int _{0}^{a} [\sum_{j=2}^{3}\mu _{ij} (r,E(p_{1}^{*})(r,t),E(p_{2}^{*})(r,t) )q_{i}(r,t)p_{i}^{*}(r,t) \\ \hphantom{\frac{\partial q_{i}}{\partial t}+\frac{\partial q_{i}}{\partial a} ={}}{}- \beta _{i2} (r,E(p_{i}^{*})(r,t) )q_{i}(0,t)p_{i}^{*}(r,t) ] \, {\mathrm{d}}r \\ \hphantom{\frac{\partial q_{i}}{\partial t}+\frac{\partial q_{i}}{\partial a} ={}}{}+ [\mu _{i} (a,E(p_{1}^{*})(a,t),E(p_{2}^{*})(a,t) )q_{i}(a,t)- \beta _{i} (a,E(p_{i}^{*})(a,t) )q_{i}(0,t) ], \\ q_{i}(a,T)=\bar{p}_{i}(a)-p_{i}^{*}(a,T), \qquad q_{i}(A,t)=0, \\ E(p_{i}^{*})({a,t})=\alpha _{i}\int _{0}^{a}p_{i}^{*}(r,t)\,{\mathrm{{d}}}r+ \int _{a}^{A}p_{i}^{*}(r,t)\,{\mathrm{{d}}}r, \quad (a,t)\in Q, i=1,2, \end{cases} $$
(4.2)

in which \(\mu _{i2}\) and \(\beta _{i2}\) are partial derivatives of \(\mu _{i}\) and \(\beta _{i}\) with respect to the second variable, respectively; \(\mu _{i3}\) is a partial derivative of \(\mu _{i}\) with respect to the third variable.

Proof

The existence of a unique bounded solution to (4.2) can be treated in the same manner as that for (2.1) and (2.2). For any \(v_{i}\in \mathcal{T}_{\mathcal{U}_{i}}(u_{i}^{*})\) (see [41]), we have \(u_{i}^{\varepsilon}\doteq u_{i}^{*}+\varepsilon v_{i}\in \mathcal{U}_{i}\) (\(i=1,2\)) for sufficiently small \(\varepsilon >0\). Let \(p^{\varepsilon}=(p_{1}^{\varepsilon},p_{2}^{\varepsilon})\) be a solution of (2.1) and (2.2) corresponding to \(u=u^{\varepsilon}\). Then, the optimality of \(u^{*}\) implies \(J(u^{*})\leq J(u^{\varepsilon})\). Hence,

$$\begin{aligned} &\int _{0}^{A} \frac{[p_{i}^{\varepsilon}(a,T)-p_{i}^{*}(a,T)] [p_{i}^{\varepsilon}(a,T)+p_{i}^{*}(a,T)-2\bar{p}_{i}(a)]}{\varepsilon} \,{ \mathrm{d}}a \\ &\quad {}+ \int _{0}^{T} \frac{[2u_{i}^{*}(t)+\varepsilon v_{i}(t)]\varepsilon v_{i}}{\varepsilon} \,{ \mathrm{d}}t \geq 0. \end{aligned}$$
(4.3)

Using the Theorem 3.3 and passing to the limit \(\varepsilon \rightarrow 0^{+}\), we can derive that

$$ \sum_{i=1}^{2} \biggl\{ \int _{0}^{A}z_{i}(a,T) \bigl[p_{i}^{*}(a,T)-\bar{p}_{i}(a) \bigr] \,{\mathrm{d}}a + \sigma _{i} \int _{0}^{T}u_{i}^{*}(t)v_{i}(t) \,{\mathrm{d}}t \biggr\} \geq 0, $$
(4.4)

in which \(z_{i}(a,t)=\lim_{\varepsilon \rightarrow 0^{+}} \frac{p_{i}^{\varepsilon}(a,t)-p_{i}^{*}(a,t)}{\varepsilon}\) (\(i=1,2\)). Via a similar discussion to that in [2], \(z_{1}(a,t)\) and \(z_{2}(a,t)\) do make sense. Further, it follows from (2.1) and (2.2) that \((z_{1},z_{2})\) satisfies

$$ \textstyle\begin{cases} \frac{\partial z_{i}}{\partial t}+\frac{\partial z_{i}}{\partial a} = - \sum_{j=2}^{3}\mu _{ij} (a,E(p_{1}^{*})(a,t),E(p_{2}^{*})(a,t) )E(z_{j})(a,t)p_{i}^{*}(a,t) \\ \hphantom{\frac{\partial z_{i}}{\partial t}+\frac{\partial z_{i}}{\partial a} ={}}{}-\mu _{i} (a,E(p_{1}^{*})(a,t),E(p_{2}^{*})(a,t) )z_{i}(a,t), \\ z_{i}(0,t)=v_{i}(t)+\int _{0}^{A} [\beta _{i2} (a,E(p_{i}^{*})(a,t) )p_{i}^{*}(a,t)E(z_{i})(a,t) \\ \hphantom{z_{i}(0,t)={}}{}+\beta _{i} (a,E(p_{i}^{*})(a,t) )z_{i}(a,t) ]\,{\mathrm{d}}a, \\ z_{i}(a,0)=0, \\ E(z_{i})(a,t)=\alpha _{i}\int _{0}^{a}z_{i}(r,t)\,{\mathrm{d}}r+\int _{a}^{A}z_{i}(r,t) \,{\mathrm{d}}r, \quad (a,t)\in Q, i=1,2. \end{cases} $$
(4.5)

Multiplying the first equation in (4.5) by \(q_{i}(a,t)\) and integrating on Q, we have

$$\begin{aligned}& \int _{0}^{A} z_{i}(a,T) \bigl[p_{i}^{*}(a,T)-\bar{p}_{i}(a) \bigr] \,{ \mathrm{d}}a \\& \quad = - \int _{Q} \bigl[\beta _{i2} \bigl(a,E \bigl(p_{i}^{*} \bigr) (a,t) \bigr)E(z_{i}) (a,t)p_{i}^{*} + \beta _{i} \bigl(a,E \bigl(p_{i}^{*} \bigr) (a,t) \bigr)z_{i} \bigr]q_{i}(0,t)\,{\mathrm{d}}a \,{\mathrm{d}}t \\& \qquad {} + \int _{Q} \Biggl\{ \sum_{j=2}^{3} \mu _{ij} \bigl(a,E \bigl(p_{1}^{*} \bigr) (a,t),E \bigl(p_{2}^{*} \bigr) (a,t) \bigr)E(z_{j}) (a,t)p_{i}^{*}(a,t) \\& \qquad {} + \mu _{i} \bigl(a,E \bigl(p_{1}^{*} \bigr) (a,t),E \bigl(p_{2}^{*} \bigr) (a,t) \bigr)z_{i} \Biggr\} q_{i}(a,t)\,{\mathrm{d}}a\,{ \mathrm{d}}t - \int _{0}^{T}v_{i}(t)q_{i}(0,t) \,{\mathrm{d}}t \\& \qquad {} - \int _{Q}z_{i} \biggl(\frac{\partial q_{i}}{\partial t}+ \frac{\partial q_{i}}{\partial a} \biggr)\,{\mathrm{d}}a\,{\mathrm{d}}t. \end{aligned}$$

Hence, we can obtain

$$\begin{aligned}& \sum_{i=1}^{2} \int _{0}^{A}z_{i}(a,T) \bigl[p_{i}^{*}(a,T)-\bar{p}_{i}(a) \bigr] \,{\mathrm{d}}a \\& \quad = \sum_{i=1}^{2} \int _{Q}z_{i}(a,t)\alpha _{i} \int _{a}^{A} \Biggl[ \sum _{j=2}^{3}\mu _{ij} \bigl(r,E \bigl(p_{1}^{*} \bigr) (r,t),E \bigl(p_{2}^{*} \bigr) (r,t) \bigr)q_{i}(r,t)p_{i}^{*}(r,t) \\& \qquad {} -\beta _{i2} \bigl(r,E \bigl(p_{i}^{*} \bigr) (r,t) \bigr)q_{i}(0,t)p_{i}^{*}(r,t) \Biggr]\,{\mathrm{d}}r\,{\mathrm{d}}a\,{\mathrm{d}}t \\& \qquad {} + \sum_{i=1}^{2} \int _{Q}z_{i}(a,t) \int _{0}^{a} \Biggl[\sum _{j=2}^{3} \mu _{ij} \bigl(r,E \bigl(p_{1}^{*} \bigr) (r,t),E \bigl(p_{2}^{*} \bigr) (r,t) \bigr)q_{i}(r,t)p_{i}^{*}(r,t) \\& \qquad {} -\beta _{i2} \bigl(r,E \bigl(p_{i}^{*} \bigr) (r,t) \bigr)q_{i}(0,t)p_{i}^{*}(r,t) \Biggr]\,{\mathrm{d}}r\,{\mathrm{d}}a\,{\mathrm{d}}t \\& \qquad {} + \sum_{i=1}^{2} \int _{Q}z_{i} \bigl[\mu _{i} \bigl(a,E \bigl(p_{1}^{*} \bigr),E \bigl(p_{2}^{*} \bigr) (a,t) \bigr)q_{i} - \beta _{i} \bigl(a,E \bigl(p_{i}^{*} \bigr) (a,t) \bigr)q_{i}(0,t) \bigr]\,{ \mathrm{d}}a\,{\mathrm{d}}t \\& \qquad {} - \sum_{i=1}^{2} \int _{Q}z_{i} \biggl( \frac{\partial q_{i}}{\partial t}+ \frac{\partial q_{i}}{\partial a} \biggr)\,{\mathrm{d}}a\,{\mathrm{d}}t - \sum _{i=1}^{2} \int _{0}^{T}v_{i}(t)q_{i}(0,t) \,{\mathrm{d}}t. \end{aligned}$$

This, together with (4.2), yields

$$ \sum_{i=1}^{2} \int _{0}^{A}z_{i}(a,T) \bigl[p_{i}^{*}(a,T)-\bar{p}_{i}(a) \bigr] \,{\mathrm{d}}a = -\sum_{i=1}^{2} \int _{0}^{T}v_{i}(t)q_{i}(0,t) \,{ \mathrm{d}}t. $$
(4.6)

It follows from (4.4) and (4.6) that \(\sum_{i=1}^{2}\int _{0}^{T} [q_{i}(0,t)-\sigma _{i}u_{i}^{*}(t) ]v_{i}(t)\,{\mathrm{d}}t\leq 0\) holds for all \(v\in \mathcal{T}_{\mathcal{U}}(u^{*})\). Thus, \(q_{i}(0,\cdot )-\sigma _{i}u_{i}^{*}\in \mathcal{N}_{\mathcal{U}_{i}}(u_{i}^{*})\) (\(i=1,2\)), which implies the conclusion of this theorem. The proof is complete. □

Existence of a unique optimal policy

In this subsection, we study the existence and uniqueness of the optimal control. Define the embedding mapping \(\varphi : [L^{1}(0,T)]^{2}\rightarrow (-\infty ,+\infty ]\) by

$$\begin{aligned} &\tilde{J}(u_{1},u_{2}) \\ &\quad =\textstyle\begin{cases} \sum_{i=1}^{2} \{\int _{0}^{A} [p_{i}^{u}(a,T)- \bar{p}_{i}(a) ]^{2}\,{\mathrm{d}}a + \sigma _{i}\int _{0}^{T}u_{i}^{2}(t) \,{\mathrm{d}}t \},& (u_{1},u_{2})\in \mathcal{U}, \\ +\infty ,& (u_{1},u_{2})\notin \mathcal{U}. \end{cases}\displaystyle \end{aligned}$$

Lemma 4.3

The mapping is lower semicontinuous.

Proof

Let \(\{(u_{1}^{n},u_{2}^{n})\}\) be a sequence in \([L^{1}(0,T)]^{2}\) such that \((u_{1}^{n},u_{2}^{n})\rightarrow (u_{1},u_{2})\) as \(n\rightarrow \infty \). Without loss of generality, we assume that \((u_{1}^{n},u_{2}^{n})\in \mathcal{U}\doteq \mathcal{U}_{1}\times \mathcal{U}_{2}\) for any \(n\geq 1\). Moreover, let \((p^{n}_{1},p^{n}_{2})\) and \((p_{1},p_{2})\) be solutions of (2.1) and (2.2) corresponding to \((u_{1}^{n},u_{2}^{n})\) and \((u_{1},u_{2})\), respectively. From Theorem 3.3, it follows that

$$ p_{1}^{n}(\cdot ,t)\rightarrow p_{1}( \cdot ,t) \quad \text{and}\quad p_{2}^{n}( \cdot ,t) \rightarrow p_{2}(\cdot ,t)\quad \text{for any }t\in (0,T) $$

as \(n\rightarrow \infty \). Riesz’s theorem implies that there is a subsequence, denoted still by \(\{(u_{1}^{n},u_{2}^{n})\}\), such that, for any \((a,t)\in Q\) and \(t\in (0,T)\),

$$\begin{aligned} & \bigl(u_{1}^{n}(t) \bigr)^{2}\rightarrow \bigl(u_{1}(t) \bigr)^{2} \quad \text{and}\quad \bigl(u_{2}^{n}(t) \bigr)^{2} \rightarrow \bigl(u_{2}(t) \bigr)^{2} , \end{aligned}$$
(4.7)
$$\begin{aligned} &p_{1}^{n}(a,t)\rightarrow p_{1}(a,t) \quad \text{and}\quad p_{2}^{n}(a,t) \rightarrow p_{2}(a,t), \end{aligned}$$
(4.8)

as \(n\rightarrow \infty \). From (4.7), using Lebesgue’s dominated convergence theorem, we obtain

$$ \lim_{n\rightarrow \infty} \int _{0}^{T} \bigl(u_{i}^{n}(t) \bigr)^{2} \, {\mathrm{d}}t= \int _{0}^{T} \bigl(u_{i}(t) \bigr)^{2}\,{\mathrm{d}}t \quad (i=1,2). $$

On the other hand, it follows from (4.8) and Theorem 3.3 that

$$\begin{aligned}& \biggl\vert \int _{0}^{A} \bigl[p_{i}^{n}(a,T)- \bar{p}_{i}(a) \bigr]^{2}\,{\mathrm{d}}a- \int _{0}^{A} \bigl[p_{i}(a,T)- \bar{p}_{i}(a) \bigr]^{2}\,{\mathrm{d}}a \biggr\vert \\& \quad \leq \int _{0}^{A} \bigl\vert p_{i}^{n}(a,T)-p_{i}(a,T) \bigr\vert \cdot \bigl\vert p_{i}^{n}(a,T)+p_{i}(a,T)-2 \bar{p}_{i}(a) \bigr\vert \,{\mathrm{d}}a \\& \quad \rightarrow 0 \quad \text{as } n\rightarrow +\infty . \end{aligned}$$

Thus, \(\lim_{n\rightarrow \infty}\int _{0}^{A}[p_{i}^{n}(a,T)-\bar{p}_{i}(a)]^{2} \,{\mathrm{d}}a = \int _{0}^{A}[p_{i}(a,T)-\bar{p}_{i}(a)]^{2}\,{\mathrm{d}}a\) (\(i=1,2\)). It follows from Fatou’s lemma that

$$ \lim_{n\rightarrow +\infty}\inf_{(u_{1}^{n},u_{2}^{n})\in \mathcal{U}}\tilde{J} \bigl(u_{1}^{n},u_{2}^{n} \bigr) \geq \tilde{J}(u_{1},u_{2}), $$

which shows that \(\tilde{J}(u_{1},u_{2})\) is lower semicontinuous. □

In a similar manner to that in Theorem 3.3, we can obtain the following result.

Lemma 4.4

For any \(u=(u_{1},u_{2})\), \(v=(v_{1},v_{2})\in \mathcal{U}\), let \((p_{1}^{u},p_{2}^{u})\) and \((p_{1}^{v},p_{2}^{v})\) be solutions of (2.1) and (2.2) corresponding to u and v, respectively. For (4.2), there is a constant \(C\doteq C(U,T)>0\), which depends on \(U=\max \{U_{1},U_{2}\}\) and T, such that

$$ \sum_{i=1}^{2} \bigl[ \bigl\Vert q_{i}^{u}(\cdot ,t)-q_{i}^{v}( \cdot ,t) \bigr\Vert _{L^{ \infty}(0,A)}+ \bigl\vert q_{i}^{u}(0,t)-q_{i}^{v}(0,t) \bigr\vert \bigr] \leq C \Vert u-v \Vert _{[L^{ \infty}(0,T)]^{2}}, $$

holds for all \(t\in (0,T)\). Here, \((q_{1}^{u},q_{2}^{u})\) and \((q_{1}^{v},q_{2}^{v})\) are solutions to (4.2) with \((p_{1}^{*},p_{2}^{*})\) replaced by \((p_{1}^{u},p_{2}^{u})\) and \((p_{1}^{v},p_{2}^{v})\), respectively.

Theorem 4.5

If \(\max \{\sigma _{1}^{-1},\sigma _{2}^{-1} \}C(U,T)<1\), then optimization problem (2.3) has a unique solution.

Proof

From Lemma 4.3 and Ekeland’s variational principle, for each \(\varepsilon >0\), there exists \((u_{1}^{\varepsilon},u_{2}^{\varepsilon})\in \mathcal{U}\) such that

$$\begin{aligned} &\tilde{J} \bigl(u_{1}^{\varepsilon},u_{2}^{\varepsilon} \bigr)\leq \inf_{(u_{1},u_{2}) \in \mathcal{U}}\tilde{J}(u_{1},u_{2})+ \varepsilon , \end{aligned}$$
(4.9)
$$\begin{aligned} &\tilde{J} \bigl(u_{1}^{\varepsilon},u_{2}^{\varepsilon} \bigr)\leq \inf_{(u_{1},u_{2}) \in \mathcal{U}} \bigl\{ \tilde{J}(u_{1},u_{2}) +\sqrt{\varepsilon} \bigl\Vert u_{1}-u_{1}^{\varepsilon}\bigr\Vert _{L^{1}(0,T)}+\sqrt{\varepsilon} \bigl\Vert u_{2}-u_{2}^{\varepsilon}\bigr\Vert _{L^{1}(0,T)} \bigr\} . \end{aligned}$$
(4.10)

Thus, functional \(\tilde{J}_{\varepsilon}(u_{1},u_{2})=\tilde{J}(u_{1},u_{2})+\sqrt{ \varepsilon}\|u_{1}-u_{1}^{\varepsilon}\|_{L^{1}(0,T)}+\sqrt{ \varepsilon}\|u_{2}-u_{2}^{\varepsilon}\|_{L^{1}(0,T)}\) reaches its infimum at \((u_{1}^{\varepsilon},u_{2}^{\varepsilon})\). Then, in the same manner as that in Theorem 4.2, for any \(v_{i}\in \mathcal{T}_{\mathcal{U}_{i}}(u_{i}^{\varepsilon})\) (\(i=1,2\)), we have

$$ \sum_{i=1}^{2} \biggl\{ \int _{0}^{T} \bigl[\sigma _{i}u_{i}^{ \varepsilon}(t)-q_{i}^{u^{\varepsilon}}(0,t) \bigr]v_{i}(t)\,{\mathrm{d}}t + \sqrt{\varepsilon} \int _{0}^{T} \bigl\vert v_{i}(t) \bigr\vert \,{ \mathrm{d}}t \biggr\} \geq 0. $$

By Lemma 4.1, there is \(\theta _{i}\in L^{\infty}(0,T)\), \(\|\theta _{i}\|_{L^{\infty}(0,T)}\leq 1\), such that \(\sqrt{\varepsilon}\theta _{i}+q_{i}^{u^{\varepsilon}}(0,\cdot )- \sigma _{i}u_{i}^{\varepsilon}\in \mathcal{N}_{\mathcal{U}_{i}}(u_{i}^{ \varepsilon})\). Hence,

$$ u_{i}^{\varepsilon}(t) = \mathcal{F}_{i} \biggl[ \frac{q_{i}^{u^{\varepsilon}}(0,t)}{\sigma _{i}}+ \frac{\sqrt{\varepsilon}\theta _{i}(t)}{\sigma _{i}} \biggr]. $$
(4.11)

First, we show the uniqueness via the fixed-point principle. For any \(u=(u_{1},u_{2})\in \mathcal{U}\), let \((p_{1}^{u},p_{2}^{u})\) be a solution of (2.1) and (2.2) corresponding to u. Define the mapping \(\mathcal{C}: \mathcal{U}_{1}\times \mathcal{U}_{2}\rightarrow L^{ \infty}(0,T)\times L^{\infty}(0,T)\) by

$$ \mathcal{C}(u_{1},u_{2}) = \biggl(\mathcal{F}_{1} \biggl[ \frac{q_{1}^{u}(0,t)}{\sigma _{1}} \biggr],\mathcal{F}_{2} \biggl[ \frac{q_{2}^{u}(0,t)}{\sigma _{2}} \biggr] \biggr), $$
(4.12)

where \((q_{1}^{u},q_{2}^{u})\) is the solution to system (4.2) with \((p_{1}^{*},p_{2}^{*})\) replaced by \((p_{1}^{u},p_{2}^{u})\). It is easy to show that \((\mathcal{U},\|\cdot \|_{[L^{\infty}(0,T)]^{2}})\) is a Banach space. It is easy to show that \(\mathcal{C}\) maps \(\mathcal{U}\) to itself. In addition, for any \(u=(u_{1},u_{2})\), \(v=(v_{1},v_{2})\in \mathcal{U}\), it follows from Lemma 4.1 that

$$\begin{aligned} \bigl\Vert \mathcal{C}(u_{1},u_{2})- \mathcal{C}(v_{1},v_{2}) \bigr\Vert _{[L^{\infty}(0,T)]^{2}} &= \sum_{i=1}^{2} \biggl\Vert \mathcal{F}_{i} \biggl[ \frac{q_{i}^{u}(0,t)}{\sigma _{i}} \biggr]- \mathcal{F}_{i} \biggl[ \frac{q_{i}^{v}(0,t)}{\sigma _{i}} \biggr] \biggr\Vert _{L^{\infty}(0,T)} \\ &\leq \sum_{i=1}^{2} \biggl\Vert \frac{q_{i}^{u}(0,t)}{\sigma _{i}}- \frac{q_{i}^{v}(0,t)}{\sigma _{i}} \biggr\Vert _{L^{\infty}(0,T)} \\ &\leq \max \bigl\{ \sigma _{1}^{-1},\sigma _{2}^{-1} \bigr\} C(U,T) \Vert u-v \Vert _{[L^{\infty}(0,T)]^{2}}. \end{aligned}$$

This, together with \(\max \{\sigma _{1}^{-1},\sigma _{2}^{-1} \}C(U,T)<1\), yields that \(\mathcal{C}\) is a contraction on Banach space \((\mathcal{U},\|\cdot \|_{[L^{\infty}(0,T)]^{2}})\). Then, \(\mathcal{C}\) has a unique fixed point \(\tilde{u}=(\tilde{u}_{1},\tilde{u}_{2})\in \mathcal{U}\). Moreover, from Theorem 4.2, we know that any optimal controller, if it exists, must be the fixed point of \(\mathcal{C}\). Thus, the uniqueness is proved.

Next, we prove that \(\tilde{u}=(\tilde{u}_{1},\tilde{u}_{2})\) is the optimal control for optimization problem (2.3). That is to show \(\tilde{J}(\tilde{u}_{1},\tilde{u}_{2})=\inf \{\tilde{J}(u_{1},u_{2}): \tilde{u}_{1},\tilde{u}_{2}\in \mathcal{U} \}\). Note that \(\|\theta _{i}\|_{L^{\infty}(0,T)}\leq 1\). Thus, from (4.11) and (4.12), it follows that

$$\begin{aligned}& \bigl\Vert \mathcal{C} \bigl(u_{1}^{\varepsilon},u_{2}^{\varepsilon} \bigr)- \bigl(u_{1}^{ \varepsilon},u_{2}^{\varepsilon} \bigr) \bigr\Vert _{[L^{\infty}(0,T)]^{2}} \\& \quad \leq \sum_{i=1}^{2} \biggl\Vert \frac{q_{i}^{u^{\varepsilon}}(0,t)}{\sigma _{i}}+ \frac{\sqrt{\varepsilon}\theta _{i}(t)}{\sigma _{i}} - \frac{q_{i}^{u^{\varepsilon}}(0,t)}{\sigma _{i}} \biggr\Vert _{L^{\infty}(0,T)} \\& \quad \leq \sum_{i=1}^{2}\sigma _{i}^{-1}\sqrt{\varepsilon}. \end{aligned}$$

Note that \(\tilde{u}=(\tilde{u}_{1},\tilde{u}_{2})\in \mathcal{U}\) is the unique fixed point for the mapping \(\mathcal{P}\). Thus,

$$\begin{aligned}& \bigl\Vert u^{\varepsilon}-\tilde{u} \bigr\Vert _{[L^{\infty}(0,T)]^{2}} \\& \quad \leq \max \bigl\{ \sigma _{1}^{-1},\sigma _{2}^{-1} \bigr\} C(U,T) \bigl\Vert u^{ \varepsilon}- \tilde{u} \bigr\Vert _{[L^{\infty}(0,T)]^{2}}+\sqrt{\varepsilon} \bigl( \sigma _{1}^{-1}+ \sigma _{2}^{-2} \bigr), \end{aligned}$$

which implies \(\|u^{\varepsilon}-\tilde{u}\|_{[L^{\infty}(0,T)]^{2}}\leq [1- \max \{\sigma _{1}^{-1},\sigma _{2}^{-1} \}C(U,T) ]\sqrt{ \varepsilon}(\sigma _{1}^{-1}+\sigma _{2}^{-2})\). Hence, we have \(u^{\varepsilon}\rightarrow \tilde{u}\) as \(\varepsilon \rightarrow 0^{+}\). From Lemma 4.3 and (4.9), one has \(\tilde{J}(\tilde{u})=\inf \{\tilde{J}(u):u\in \mathcal{U} \}\). Thus, \(\tilde{u}\in \mathcal{U}\) is the optimal policy. □

Remark 4.6

In this section, using Ekeland’s variational principle and fixed-point reasoning, we show the existence and uniqueness of optimal policies for the least deviation-cost problem, which gives us a solid theoretical ground for a practical application. As for the structure of the optimal policy, we have presented a feedback strategy, in Theorem 4.2, via the normal cone technique and adjoint system. Due to the high nonlinearities in our problem, one cannot expect an explicit optimal controller. However, the feedback strategy would be helpful for a numerical computation of the optimal policy (see Sect. 6).

Most benefit-cost problem

Here, we give the corresponding results for optimization problem (2.4). A similar discussion as that in Theorem 4.5, we can show that the most benefit-cost problem (2.4) has a unique solution. Further, similar to the proof of Theorem 4.2, we have the following result.

Theorem 5.1

(Maximum principle)

Let \(u^{*}=(u_{1}^{*},u_{2}^{*})\) and \(p^{*}=(p_{1}^{*},p_{2}^{*})\). If \((u^{*},p^{*})\) is an optimal pair for the optimization problem (2.4), then

$$ u_{i}^{*}(t) = \mathcal{F}_{i} \biggl( \frac{q_{i}(0,\cdot )}{2\sigma _{i}} \biggr) (t) \doteq \textstyle\begin{cases} 0,& \textit{if } q_{i}(0,t)< 0, \\ \frac{q_{i}(0,t)}{2\sigma _{i}},& \textit{if } 0\leq \frac{q_{i}(0,t)}{2\sigma _{i}}< U_{i}, \\ U_{i},& \textit{if } \frac{q_{i}(0,t)}{2\sigma _{i}} \geq U_{i}, \end{cases} $$
(5.1)

where \(q=(q_{1},q_{2})\) is the solution of the following adjoint system

$$ \textstyle\begin{cases} \frac{\partial q_{i}}{\partial t}+\frac{\partial q_{i}}{\partial a} = \alpha _{i}\int _{a}^{A} [\sum_{j=2}^{3}\mu _{ij} (r,E(p_{1}^{*})(r,t),E(p_{2}^{*})(r,t) )q_{i}(r,t)p_{i}^{*}(r,t) \\ \hphantom{\frac{\partial q_{i}}{\partial t}+\frac{\partial q_{i}}{\partial a} ={}}{}-\beta _{i2} (r,E(p_{i}^{*})(r,t) )q_{i}(0,t)p_{i}^{*}(r,t) ]\,{\mathrm{{d}}}r \\ \hphantom{\frac{\partial q_{i}}{\partial t}+\frac{\partial q_{i}}{\partial a} ={}}{} +\int _{0}^{a} [\sum_{j=2}^{3}\mu _{ij} (r,E(p_{1}^{*})(r,t),E(p_{2}^{*})(r,t) )q_{i}(r,t)p_{i}^{*}(r,t) \\ \hphantom{\frac{\partial q_{i}}{\partial t}+\frac{\partial q_{i}}{\partial a} ={}}{} -\beta _{i2} (r,E(p_{i}^{*})(r,t) )q_{i}(0,t)p_{i}^{*}(r,t) ]\,{\mathrm{d}}r \\ \hphantom{\frac{\partial q_{i}}{\partial t}+\frac{\partial q_{i}}{\partial a} ={}}{} + [\mu _{i} (a,E(p_{1}^{*})(a,t),E(p_{2}^{*})(a,t) )q_{i}(a,t)- \beta _{i} (a,E(p_{i}^{*})(a,t) )q_{i}(0,t) ], \\ q_{i}(a,T)=g_{i}(a), \qquad q_{i}(A,t)=0, \\ E(p_{i}^{*})({a,t})=\alpha _{i}\int _{0}^{a}p_{i}^{*}(r,t)\,{\mathrm{{d}}}r + \int _{a}^{A}p_{i}^{*}(r,t)\,{\mathrm{{d}}}r, \quad (a,t)\in Q, i=1,2, \end{cases} $$
(5.2)

in which \(\mu _{i2}\) and \(\beta _{i2}\) are partial derivatives of \(\mu _{i}\) and \(\beta _{i}\) with respect to the second variable, respectively; \(\mu _{i3}\) is a partial derivative of \(\mu _{i}\) with respect to the third variable.

Numerical simulations

In this section, we provide some examples to verify the effectiveness of the obtained results and find other dynamic properties of the system. From Theorem 4.2, optimal strategy \(u_{i}^{*}\) (\(i=1,2\)) is a fixed point of mapping \(\mathcal{F}_{i}\). Clearly, the fixed point can be approximated by an iteration scheme.

First, we discuss the effects of ideal distribution \(\bar{p}(a)=(\bar{p}_{1}(a), \bar{p}_{2}(a))\) on optimal policy \(u^{*}(t)=(u_{1}^{*}(t),u_{2}^{*}(t))\) and then on optimal functional \(J(u^{*})\) for optimization problem (2.3).

Example 1

(The effects of on \(J(u^{*})\)). Let \(\alpha _{1}=0.4\), \(\alpha _{2}=0.5\), \(A=10\), \(T=20\), \(\sigma _{1}=3\), \(\sigma _{2}=4\), and the error bound \(\varepsilon =0.005\). The mortality and fertility are, respectively,

$$\begin{aligned} &\mu _{1}(a,x_{1},x_{2}) = \textstyle\begin{cases} 0.01\cos ^{2}(4a)+0.008x_{1}+0.007x_{2}+\frac{2-a}{50}, & 0\leq a< 2; \\ 0.01\cos ^{2}(4a)+0.008x_{1}+0.007x_{2}, & 2\leq a< 8; \\ 0.01\cos ^{2}(4a)+0.008x_{1}+0.007x_{2}+\frac{a-8}{10-a},& 8 \leq a< 10; \end{cases}\displaystyle \\ &\mu _{2}(a,x_{1},x_{2}) = \textstyle\begin{cases} 0.02\cos ^{2}(4a)+0.005x_{1}+0.006x_{2}+\frac{2-a}{50},& 0\leq a< 2; \\ 0.02\cos ^{2}(4a)+0.005x_{1}+0.006x_{2},& 2\leq a< 8; \\ 0.02\cos ^{2}(4a)+0.005x_{1}+0.006x_{2}+\frac{a-8}{10-a},& 8 \leq a< 10; \end{cases}\displaystyle \end{aligned}$$

and

$$\begin{aligned} &\beta _{1}(a,x_{1}) = \textstyle\begin{cases} 0.4\cos ^{2}(a-2), & 1\leq a< 2; \\ 0.4, & 2\leq a< 9; \\ 0, & \text{otherwise}; \end{cases}\displaystyle \\ &\beta _{2}(a,x_{2})= \textstyle\begin{cases} 0.8(a-1)\cos ^{2}(a), &1\leq a< 2; \\ 0.8\cos ^{2}(a), &2\leq a< 9; \\ 0, &\text{otherwise}. \end{cases}\displaystyle \end{aligned}$$

The initial age distributions of species \(p_{1}(a,t)\) and \(p_{2}(a,t)\) are, respectively, \(p_{1}^{0}(a)=8(10-a)(3+\cos (2a))\) and \(p_{2}^{0}(a)=10(10-a)\cos ^{2}(a)\). Also, the initial controls are

$$ u_{1}^{0}(t)=u_{2}^{0}(t) = \textstyle\begin{cases} 0, & t< 4; \\ 1, & t\geq 4. \end{cases} $$

Moreover, the ideal distributions we hope to reach are (with the coefficient s)

$$ \bar{p}_{1}(a)=8s(10-a) \bigl(3+\cos (2a) \bigr)\quad \text{and}\quad \bar{p}_{2}(a)=10s(10-a) \cos ^{2}(a). $$

Calculating in MATLAB with the above parameters, we obtain the numerical results for the optimal policy \(u^{*}\) and optimal state \(p^{*}\) corresponding to \(s=2\) (see Fig. 1). Moreover, the indexes \(J(u^{*})=j\times 10^{5}\) corresponding to \(s=2,3,4,\ldots,106\) are listed in Table 1.

Figure 1
figure 1

Optimal controls \(u_{1}^{*}\), \(u_{2}^{*}\) and optimal densities \(p_{1}^{*}\), \(p_{2}^{*}\) with \(s=2\)

Table 1 The effects of s on \(J(u^{*})\)

Figure 1 shows the optimal strategy \(u^{*}(t)\) and the optimal population density \(p^{*}(a,t)\) with \(s=2\) within the ideal distribution \(\bar{p}(a)\). Moreover, as can be seen from Table 1, optimal objective functional \(J(u^{*})\) is increasing with respect to the coefficient s. This means that as the ideal distribution increases, the more baby individuals (i.e., the greater the control cost) should be input to make the final distribution of the species sufficiently close to the ideal distribution.

Now, we investigate the effects of price function \(g(a)=(g_{1}(a),g_{2}(a))\) on optimal policy \(u^{*}(t)=(u_{1}^{*}(t),u_{2}^{*}(t))\) and then on optimal functional \(J(u^{*})\) for optimization problem (2.4) in the numerical approach.

Example 2

(The effects of \(g_{i}\) on \(J(u^{*})\))

Take \(\alpha _{1}=0.3\), \(\alpha _{2}=0.4\), \(A=10\), \(T=20\), \(\sigma _{1}=3\), \(\sigma _{2}=2\), and the error bound \(\varepsilon =0.005\). The mortality and fertility of species \(p_{1}(a,t)\) and \(p_{2}(a,t)\) are

$$\begin{aligned} &\mu _{1}(a,x_{1},x_{2}) = \textstyle\begin{cases} 0.02(1-\sin ^{2}(4a))+0.02x_{1}+0.01x_{2}+\frac{2-a}{50}, & 0\leq a< 2; \\ 0.02(1-\sin ^{2}(4a))+0.02x_{1}+0.01x_{2}, & 2\leq a< 8; \\ 0.02(1-\sin ^{2}(4a))+0.02x_{1}+0.01x_{2}+\frac{a-8}{10-a}, & 8\leq a< 10; \end{cases}\displaystyle \\ &\mu _{2}(a,x_{1},x_{2}) = \textstyle\begin{cases} 0.05(1-\sin ^{2}(4a))+0.01x_{1}+0.02x_{2}+\frac{2-a}{50}, & 0\leq a< 2; \\ 0.05(1-\sin ^{2}(4a))+0.01x_{1}+0.02x_{2}, & 2\leq a< 8; \\ 0.05(1-\sin ^{2}(4a))+0.01x_{1}+0.02x_{2}+\frac{a-8}{10-a}, & 8\leq a< 10; \end{cases}\displaystyle \\ &\beta _{1}(a,x_{1}) = \textstyle\begin{cases} 0.473(\cos (0.4a)+1)(a-1), & 1\leq a< 2; \\ 0.473(\cos (0.4a)+1), & 2\leq a< 7; \\ 0.473(\cos (0.4a)+1)(8-a), & 7\leq a< 8; \\ 0, & \text{otherwise}; \end{cases}\displaystyle \\ &\beta _{2}(a,x_{2}) = \textstyle\begin{cases} 0.998\sin ^{2}(2a)(a-1), & 1\leq a< 2; \\ 0.998\sin ^{2}(2a), & 2\leq a< 7; \\ 0.998\sin ^{2}(2a)(8-a), & 7\leq a< 8; \\ 0, & \text{otherwise}. \end{cases}\displaystyle \end{aligned}$$

The initial age distributions are \(p_{1}^{0}(a)=5(10-a)(\cos (2a)+2)\) and \(p_{2}^{0}(a)=4(10-a)\cos ^{2} (a+\frac{\pi}{6} )\). Also, the initial controls are

$$\begin{aligned} u_{1}^{0}(t)=u_{2}^{0}(t) = \textstyle\begin{cases} 0, & t< 5; \\ 1, & t\geq 5. \end{cases}\displaystyle \end{aligned}$$

Moreover, the price functions are (with the coefficient s)

$$\begin{aligned} &g_{1}(a)= \textstyle\begin{cases} 0.5s, & 0\leq a< 2; \\ a(\sin(a)+4)+0.5s, & 2\leq a< 8; \\ 2+0.5s, & 8\leq a< 10; \end{cases}\displaystyle \\ &g_{2}(a)= \textstyle\begin{cases} 0.5s, & 0\leq a< 2; \\ a(\cos (a)+4)+0.5s, & 2\leq a< 8; \\ 2+0.5s, & 8\leq a< 10. \end{cases}\displaystyle \end{aligned}$$

The optimal strategies \(u^{*}(t)\) and optimal densities \(p^{*}(a,t)\) corresponding to \(s=0\) are illustrated in Fig. 2. Table 2 shows the benefits corresponding to \(s=0,1,2,\ldots,103\).

Figure 2
figure 2

Optimal controls \(u_{1}^{*}\), \(u_{2}^{*}\) and optimal densities \(p_{1}^{*}\), \(p_{2}^{*}\) with \(s=2\)

Table 2 Effects of \(g_{i}\) on \(J(u^{*})\)

Figure 2 presents the optimal policy \(u^{*}(t)\) and the optimal population density \(p^{*}(a,t)\) with \(s=2\) in the price function \(g(a)=(g_{1}(a),g_{2}(a))\) for the most benefit-cost problem (2.4). Further, from Table 2, we can see that optimal objective functional \(J(u^{*})\) is increasing with respect to s. Thus, one can say that the total net economic benefit yielded from harvesting the species in the final state will increase as the price of the individual rises.

Finally, we will discuss the effects of the costs factor \(\sigma _{i}\) on the optimal functional \(J(u^{*})\) for optimization problem (2.4).

Example 3

(The effects of \(\sigma _{i}\) on \(J(u^{*})\)). Take \(\sigma _{1}=\sigma _{2}=k\), \(k=1,2,3,\ldots,104\), and the price functions are

$$\begin{aligned} &g_{1}(a)= \textstyle\begin{cases} 0, & 0\leq a< 2; \\ a(\sin(a)+4), & 2\leq a< 8; \\ 2, & 8\leq a< 10; \end{cases}\displaystyle \\ &g_{2}(a)= \textstyle\begin{cases} 0, & 0\leq a< 2; \\ a(\cos (a)+4), & 2\leq a< 8; \\ 2, & 8\leq a< 10. \end{cases}\displaystyle \end{aligned}$$

The other parameters are the same as those in Example 2. Table 3 demonstrates the changes of optimal benefits \(J(u^{*})\) with respect to the control cost factor \(\sigma _{i}\). From the data in Table 3 alone, the objective optimal benefit \(J(u^{*})\) decreases as the cost factor \(\sigma _{i}\) increases. This means that as the cost factor increases, the total net economic benefit yielded from harvesting the species in the final state will decrease.

Table 3 \(J(u^{*})\) corresponding to \(\sigma _{i}\)

Conclusion and discussion

This paper is concerned with the optimal boundary control problems for a hierarchical age-structured two-species model. In the previous sections, we have established the well-posedness of the state system provided the control variable (the influx rate of baby for each species) is put in. Meanwhile, the continuous dependence of density of each species on the control parameters is established. More importantly, the least deviation-cost problem and the most benefit-cost problem are discussed. Here, the least deviation-cost problem means that the system with any initial distribution is manipulated to be as close to an ideal distribution as possible at the least cost. The most benefit-cost problem refers to maximizing the benefit achieved by developing the final population distribution with the least cost of control (i.e., the cost of inputting baby individuals). For the least deviation-cost problem, Theorem 4.2 establishes the minimum principle by using an adjoint system, and Theorem 4.5 shows the existence of a unique optimal strategy by means of Ekeland’s variational principle. Meanwhile, the corresponding results, the existence of a unique optimal control policy and the maximum principle (see Theorem 5.1), for the most benefit-cost problem are also given.

From Theorems 4.2 and 5.1, the optimal strategies can be regarded as fixed points of contraction mappings about the adjoint variables. Thus, we can approximate the optimal policies by the standard iterative paradigm. Moreover, some examples and numerical results are presented to verify the effectiveness of the obtained results and find other dynamic properties of the system. Although the phenomena observed in Tables 1, 2, and 3 have certain biological significance, they are intuitive in the general sense and do not have universal significance. However, the theoretical analysis of these phenomena is very difficult or even impossible to deal with.

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Acknowledgements

The authors thank the anonymous reviewers for their valuable suggestions.

Funding

This research was supported by the National Natural Science Foundation of China (No. 12001341, 11871185), the Natural Science Foundation of Shanxi Province (No. 201901D211410) and the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (No. 2020L0258).

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Contributions

RL analyzed the well-posedness of the system and discussed the optimal control problem. NZ carried out the numerical analysis. ZH reviewed and edited the manuscript, and was a major contributor in writing the manuscript. All the authors read and approved the final manuscript.

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Correspondence to Ze-Rong He.

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Liu, R., Zhou, N. & He, ZR. Optimal boundary control problems for a hierarchical age-structured two-species model. J Inequal Appl 2022, 76 (2022). https://doi.org/10.1186/s13660-022-02812-1

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MSC

  • 92D05
  • 47D06
  • 35B35

Keywords

  • Hierarchical structure
  • Optimal control
  • Least deviation-cost problem
  • Most benefit-cost problem