带时滞的尺度等级结构种群系统的最优初始控制

带时滞的尺度等级结构种群系统的最优初始控制

何泽荣^,, 韩梦杰

Optimal Control of Initial Distributions in a Hierarchical Size-Structured Population System with Delay

He Zerong^,, Han Mengjie

收稿日期: 2020-06-8

基金资助:

国家自然科学基金. 11871185
浙江省自然科学基金. LY18A010010

Received: 2020-06-8

Fund supported:

the NSFC. 11871185
the NSF of Zhejiang Province. LY18A010010

作者简介 About authors

何泽荣,E-mail:zrhe@hdu.edu.cn , E-mail：zrhe@hdu.edu.cn

Abstract

This article is concerned with an optimal control problem for a hierarchical size-dependent population model with delay, the control function is the initial distribution. It is expected that the difference between the terminal state and the given target can be minimized in a least costs. The uniform continuity of states in controls is established by the method of characteristic lines and priori estimates, the minimal principle is derived by the construction of a normal cone and an adjoint system, and the existence of unique optimal strategy is proved by means of the Ekeland variational theorem and fixed-point approach. These results pave the way to applications.

Keywords： Hierarchy of size ; Integro-partial differential equations ; Optimal control ; Normal cones ; Variational principle

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本文引用格式

何泽荣, 韩梦杰. 带时滞的尺度等级结构种群系统的最优初始控制. 数学物理学报[J], 2021, 41(4): 1181-1191 doi:

He Zerong, Han Mengjie. Optimal Control of Initial Distributions in a Hierarchical Size-Structured Population System with Delay. Acta Mathematica Scientia[J], 2021, 41(4): 1181-1191 doi:

1 引言

生物种群内部个体之间的等级差异对群体演化有些什么样的影响？这个问题可以用数学建模与分析给予部分回答. 为此, 国内外学者已经建立了一些数学模型, 并给出了相应的理论和数值分析, 参见文献[1-16]及其参考文献. 这些研究工作主要关注系统的动力学行为, 比如解的存在唯一性, 平衡态的存在性与(全局)稳定性, 种群的持续生存, 种群内部抢夺竞争(Scramble competition)与对抗竞争(Contest competition)的比较, 以及密度函数的数值算法, 等. 毫无疑问, 这些成果对理解等级结构种群的演化具有积极意义. 关于种群研究的另一个重要侧面是调控问题, 包括出于生态保护的种群分布优化, 以及可再生资源开发的最优收获等问题, 还很少被研究, 参见文献[17-19]. 这类调控问题的生态学与经济学意义不言而喻, 并且作为一类新的无穷维系统控制问题, 也有其数学和控制科学价值.

本文探讨一类带有个体孕育期时滞的种群系统控制问题, 主要新意体现在: 一是个体等级由其生理尺度(如动物体重与体积、植物高度与茎杆直径)决定; 二是调控手段为种群初始分布. 由于考虑了孕期时滞, 初始分布成为尺度与时间的二元函数, 而不是通常仅含时间的一元函数. 以初始分布为控制变量的实际意义很直观: 通过精心调节当前分布, 期望种群在一段时间之后达到或者接近理想状态. 但这种调控方法极少获得关注. 本文在此方面做出一点初步努力.

2 模型与问题

本文提出下列种群控制系统模型

$\begin{eqnarray} \left\{ \begin{array}{l} \frac{{\partial p}}{{\partial t}} + \frac{{\partial (g(s)p)}}{{\partial s}} = - \left[ {{\mu _{\rm{0}}}\left( s \right) + {\mu _1}(E(p)(s, t))} \right]p(s, t), (s, t) \in {Q_T}, \\ g(0)p(0, t) = \int_0^S {\beta (s, E(p)(s, t - \tau ))p(s, t - \tau ){\rm d}s}, t \in (0, T), \\ p(s, t) = {u}(s, t), \left( {s, t} \right) \in [0, S] \times [ - \tau , 0], \\ E(p)(s, t) = \alpha \int_0^s {p(r, t){\rm d}r} + \int_s^S {p(r, t){\rm d}r}, (s, t) \in {Q_T}, \end{array} \right. \end{eqnarray}$

(2.1)

其中 $p(s, t)$ 表示 $t$ 时刻种群的个体尺度 $s$ 分布, $g(s) = {{{\rm d}s} \over {{\rm d}t}}$ 为尺度增长率; ${\mu _{\rm{0}}}\left( s \right) > 0$ 为自然死亡率, ${\mu _{\rm{1}}}$ 代表因内部竞争引起的附加死亡率, 控制变量 $u(s, t)$ 表示初始分布, $\beta$ 表示个体的平均繁殖率; $E(p)$ 表示种群内部环境, $\alpha$ 为加权系数, $0 \le \alpha < 1; {Q_T} = (0, S) \times (0, T)$ , $S$ 为个体最大尺度, $T$ 为终端时间; $\tau > 0$ 为新生个体孕育期时长.

基本假设如下

(A $_1)$ $g \in {C^1}([0, S])$ , 当 $s\in [0, S)$ 时有 $g'(s) < 0, \, g(S) = 0$ . 定义 $\Gamma (s) = \int_0^s \frac{{\rm d}x}{g(x)}$ , $\Gamma(S) < +\infty$ ;

(A $_2)$ ${\mu _0}\left( s \right) > 0, \int_0^S {{\mu _0}\left( s \right){\rm d}s = + \infty }$ . ${\mu_{1}}(x)$ 非负有界, 关于 $x$ 严格单增且满足Lipschitz条件;

(A $_3)$ $0 \le \beta( s, x) \le \bar{ \beta }, \forall ( s, x) \in [ 0, S] \times {R_ + }$ , $\beta( \cdot, x)$ 关于 $x$ 非增, 且满足Lipschitz条件;

(A $_4)$ 容许控制集为

$U = \Big\{ u \in L^{\infty }([0, S] \times [- \tau, 0]): 0 \le u(s, t) \le \bar{u} a.e. (s, t) \in [0, S] \times [- \tau, 0]\Big\},$

其中 $\bar{u}$ 为正常数.

本文主题是分析以下最优控制问题

$\begin{eqnarray} \mathop {\min}\limits_{u \in U} J\left( u \right)\mathop = \limits^\Delta \int_0^S {{{\left[ {{p^u}\left( {s, T} \right) - \bar p\left( s \right)} \right]}^2}{\rm d}s} + \sigma \int_0^S {\int_{ - \tau }^0 {{u^2}\left( {s, t} \right){\rm d}s{\rm d}t} } , \end{eqnarray}$

(2.2)

其中非负有界函数 ${\overline p \left( s \right)}$ 表示某种理想分布(例如平衡态), $\sigma > 0$ 为单位控制成本因子. 因此 $J\left( u \right)$ 表示种群终态分布与理想分布之间的均方差以及控制总成本之和.

文献[20] 对系统(2.1) 做了初步分析, 获得下列结果.

定理2.1 在假设(A $_1)$ -(A $_4)$ 下, 系统(2.1) 存在唯一的非负有界解 $p(s, t)$ .

3 最小值原理

首先证明状态系统的解关于控制变量的连续依赖性.

引理3.1 系统(2.1) 的解 $p^u\in L^{\infty}(Q_T)$ 关于 $u\in U$ 一致连续.

证将系统演变时域 $[0, T]$ 划分为区间 $[0, \tau], [\tau, 2\tau], \cdots, [k\tau, T]$ , 逐段进行处理.

当 $t\in [0, \tau]$ 时, $t-\tau\in [-\tau, 0], p(s, t-\tau) = u(s, t-\tau)$ . 状态系统(2.1) 关于尺度的边界条件变为已知, 记为 $b(t) = g(0)p(0, t)$ . 利用定理2.1可得

$\begin{eqnarray} p(s, t) = \left\{ \begin{array}{ll} \frac{b(t-\Gamma(s))}{g(s)}\exp\left\{-\int_0^s \frac{\mu(r, t-\Gamma(s)+\Gamma(r))}{g(r)}{\rm d}r \right\}, & t>\Gamma(s);\\ u(\Gamma^{-1}(\Gamma(s)-t), 0)\exp\bigg\{-\int_0^t \left[\mu(\Gamma^{-1}(r+\Gamma(s)-t), r) \right.& \\ \left.+g'(\Gamma^{-1}(r+\Gamma(s)-t))\right]{\rm d}r\bigg\}, & t\leq \Gamma(s), \end{array} \right. \end{eqnarray}$

(3.1)

其中 $\mu(s, t) = \mu_0(s)+\mu_1 (E(p)(s, t))$ .

下面估计 $p^u$ 随 $u$ 的变差. 令 $p^i$ 为系统(2.1) 相应于 $u_i$ 的解, $i = 1, 2$ . 由此得相应函数 $b^i, \mu^i$ . 记 $b(t)$ 的上界为 $B$ ; $\mu_1$ 以及 $\beta(s, x)$ 关于 $x$ 的Lipschitz常数为 $L$ .

当 $t>\Gamma(s)$ 时, $s<\Gamma^{-1}(t)\leq \Gamma^{-1}(\tau)$ . 由(3.1) 式知

$\begin{eqnarray} && |p^1(s, t)-p^2(s, t) |\\ & = &\frac{1}{g(s)}\left |b^1(t-\Gamma(s))\exp\left\{-\int_0^s \frac{\mu^1(r, t-\Gamma(s)+\Gamma(r))}{g(r)}{\rm d}r \right\} \right. \\ &&\left. -b^2(t-\Gamma(s))\exp\left\{-\int_0^s \frac{\mu^2(r, t-\Gamma(s)+\Gamma(r))}{g(r)}{\rm d}r \right\}\right|\\ &\leq & \frac{1}{g(s)}\bigg[b^1(t-\Gamma(s))\bigg |\exp\left\{-\int_0^s \frac{\mu^1(r, t-\Gamma(s)+\Gamma(r))}{g(r)}{\rm d}r \right\} \\ &&- \exp\left\{-\int_0^s \frac{\mu^2(r, t-\Gamma(s)+\Gamma(r))}{g(r)}{\rm d}r \right\} \bigg| +\left|b^1(t-\Gamma(s))-b^2(t-\Gamma(s)) \right|\bigg]\\ &\leq & \frac{1}{g(s)}\left[B\int_0^s \frac{|\mu_1E(p^1)(r, t-\Gamma(s)+\Gamma(r))-\mu_1E(p^2)(r, t-\Gamma(s) +\Gamma(r))|}{g(r)}{\rm d}r \right.\\ & &+\left|\int_0^S \beta(r, E(u_1)(r, t-\Gamma(s)-\tau))u_1(r, t-\Gamma(s)-\tau){\rm d}r \right.\\ &&\left.\left.-\int_0^S \beta(r, E(u_2)(r, t-\Gamma(s)-\tau))u_2(r, t-\Gamma(s)-\tau){\rm d}r \right|\right]\\ &\leq & \frac{1}{g(s)}\left[BL\int_0^s \frac{|E(p^1)(r, t-\Gamma(s)+\Gamma(r))-E(p^2)(r, t-\Gamma(s)+\Gamma(r))|}{g(r)}{\rm d}r \right.\\ &&+\bar{\beta}\int_0^S |u_1(r, t-\Gamma(s)-\tau)-u_2(r, t-\Gamma(s)-\tau) |{\rm d}r\\ & &\left. +\bar{u}\int_0^S |\beta(r, E(u_1)(r, t-\Gamma(s)-\tau))-\beta(r, E(u_2)(r, t-\Gamma(s)-\tau))|{\rm d}r \right]\\ &\leq & \frac{1}{g(\Gamma^{-1}(\tau))}\bigg\{BL\int_0^s \frac{\|p^1(\cdot, t-\Gamma(s)+\Gamma(r)) -p^2(\cdot, t-\Gamma(s)+\Gamma(r))\|}{g(r)}{\rm d}r \\ && +\bar{\beta}S\|u_1 -u_2\|+\bar{u}L\int_0^S |E(u_1)(r, t-\Gamma(s)-\tau)-E(u_2)(r, t-\Gamma(s)-\tau)|{\rm d}r\bigg\}\\ &\leq & C_1 \|u_1 -u_2\|+C_2\int_0^t \|p^1(\cdot, \theta)-p^2(\cdot, \theta)\|{\rm d}\theta, \end{eqnarray}$

(3.2)

其中 $C_1, C_2$ 为正常数, $\|\cdot\|$ 均为空间 $L^{\infty}$ 中的标准范数.

当 $t\leq\Gamma(s)$ 时, 注意函数 $\mu_0, g$ 与 $u$ 无关, 由(3.1) 可得:

$\begin{eqnarray} && |p^1(s, t)-p^2(s, t) |\\ &\leq &\left| u_1(\Gamma^{-1}(\Gamma(s)-t), 0)\exp\left\{-\int_0^t \mu_1(E(p^1)(\Gamma^{-1}(r+\Gamma(s)-t), r)){\rm d}r \right\}\right.\\ &&\left. -u_2(\Gamma^{-1}(\Gamma(s)-t), 0)\exp\left\{-\int_0^t \mu_1(E(p^2)(\Gamma^{-1}(r+\Gamma(s)-t), r)){\rm d}r \right\}\right|\\ &\leq & \bar{u}\int_0^t |\mu_1(E(p^1)(\Gamma^{-1}(r+\Gamma(s)-t), r))-\mu_1(E(p^2)(\Gamma^{-1}(r+\Gamma(s)-t), r))|{\rm d}r\\ &&+|u_1(\Gamma^{-1}(\Gamma(s)-t), 0)-u_2(\Gamma^{-1}(\Gamma(s)-t), 0)|\\ &\leq & \|u_1 -u_2\|+C_3 \int_0^t \|p^1(\cdot, \theta)-p^2(\cdot, \theta)\|{\rm d}\theta, \end{eqnarray}$

(3.3)

其中 $C_3$ 为正常数.

利用不等式(3.2)-(3.3) 和Bellmann引理, 可知存在正的常数 $C_4$ 使得

$\|p^1 -p^2\|_{L^{\infty}([0, S]\times [0, \tau])}\leq C_4 \|u_1 -u_2\|_{L^{\infty}([0, S]\times [-\tau, 0])}.$

当 $t\in [\tau, T]$ 时, 重复以上过程可得类似的估计结果. 引理证毕.

其次给出最优控制策略的精确刻画.

定理3.1 控制问题(2.1)-(2.2) 的任一最优对 $(u^*, p^*)$ 都有如下结构

$\begin{eqnarray} \begin{array}{l} {u^*}\left( {s, t} \right) = {\cal F}(\sigma^{-1}q(0, t+\tau)\tilde{E}(s, t)), ( s, t)\in [0, S]\times [-\tau, 0], \end{array} \end{eqnarray}$

(3.4)

其中 ${\cal F}$ 为0和 $\bar{u}$ 之间的截断函数,

$\tilde{E}(s, t) = \int_0^s \beta_{x}(r, E(p^*)(r, t))p^*(r, t){\rm d}r+\alpha \int_s^S \beta_{x}(r, E(p^*)(r, t))p^*(r, t){\rm d}r,$

$q\left( {s, t} \right)$ 为下列共轭系统的解

$\begin{eqnarray} \begin{array}{l} \left\{ \begin{array}{l} \frac{{\partial q}}{{\partial t}}{\rm{ + }}g\left( s \right)\frac{{\partial q}}{{\partial s}} = \alpha \int_s^S {\left[ {{\mu' _1}\left( {{E}(p^*)} \right)q\left( {r, t} \right) - {\beta _x}(r, {E}(p^*))q(0, t + \tau )} \right]{p^*}\left( {r, t} \right){\rm d}r} \\ + \int_0^s {\left[ {{\mu' _1}\left( {{E}(p^*)} \right)q\left( {r, t} \right) - {\beta _x}(r, {E}(p^*))q(0, t + \tau )} \right]{p^*}\left( {r, t} \right){\rm d}r} \\ + \left[ {{\mu _0}\left( s \right) + {\mu _1}\left( {{E}(p^*)} \right)} \right]q\left( {s, t} \right)- \beta (s, {E}(p^*))q(0, t + \tau ), \left( {s, t} \right) \in {Q_T}, \\ q\left( {s, T} \right) = \bar p\left( s \right) - {p^*}\left( {s, T} \right), s \in [0, S], \\ q\left( {s, t} \right) = 0, (s, t) \in [0, S] \times [T, T+\tau]. \end{array} \right. \end{array} \end{eqnarray}$

(3.5)

证令 $\left( {{u^*}, {p^*}} \right)$ 为问题(2.1)-(2.2) 的最优对. 系统(3.5) 解的存在唯一性可用时间变换和类似于文献[20] 的方法得到. 对任意 $v \in {\cal T}_U\left( u^*\right)$ (表示集 $U$ 在 ${{u^*}}$ 处的切锥) 满足 $v(s, 0) = 0$ , 当 $\varepsilon > 0$ 足够小时, ${u^\varepsilon }: = {u^*} + \varepsilon v \in U$ (参见文献[21, p21]).

令 ${p^\varepsilon }$ 为系统(2.1) 相应于 $u = u^\varepsilon$ 的解. 由 $u^*$ 的最优性知

$\begin{eqnarray*} && \int_0^S {{{\left[ {{p^*}\left( {s, T} \right) - \bar p\left( s \right)} \right]}^2}{\rm d}s} + \sigma \int_0^S {\int_{ - \tau }^0 {{{\left( {{u^*}\left( {s, t} \right)} \right)}^2}{\rm d}s{\rm d}t} } \\ &\le & \int_0^S [ p^\varepsilon ( s, T) - \bar p( s) ]^2 {\rm d}s + \sigma \int_0^S \int_{ - \tau }^0 [ u^*(s, t) +\varepsilon v(s, t)]^2{\rm d}s{\rm d}t. \end{eqnarray*}$

将上式变形, 并令 $\varepsilon \to {0^ + }$ , 可得

$\begin{eqnarray} \int_0^S {\left[ {{p^*}\left( {s, T} \right) - \bar p\left( s \right)} \right]z\left( {s, T} \right){\rm d}s} + \sigma \int_0^S {\int_{ - \tau }^0 {{u^*}\left( {s, t} \right)v\left( {s, t} \right){\rm d}s{\rm d}t} } \ge 0, \end{eqnarray}$

(3.6)

其中 $z\left( {s, t} \right) = \mathop {\lim }\limits_{\varepsilon \to {0^ + }} {\varepsilon ^{ - 1}}\left[ {{p^\varepsilon }\left( {s, t} \right) - {p^*}\left( {s, t} \right)} \right].$

上列极限的存在性由后面的引理3.2给出. 先假定该极限存在, 推导 $z\left( {s, t} \right)$ 所满足的条件.

因为 ${{p^\varepsilon }\left( {s, t} \right)}$ 是系统(2.1) 相应于 $u = {u^\varepsilon }$ 的解, 它满足

$\begin{eqnarray} \left\{ \begin{array}{l} \frac{{\partial {p^\varepsilon }}}{{\partial t}} + \frac{{\partial (g(s){p^\varepsilon })}}{{\partial s}} = - \left[ {{\mu _{\rm{0}}}\left( s \right) + {\mu _1}(E({p^\varepsilon })(s, t))} \right]{p^\varepsilon }(s, t), \\ g(0){p^\varepsilon }(0, t) = \int_0^S {\beta (s, E({p^\varepsilon })(s, t - \tau )){p^\varepsilon }(s, t - \tau ){\rm d}s}, \\ {p^\varepsilon }(s, t) = {u^\varepsilon }\left( {s, t} \right), (s, t) \in [0, S] \times [ - \tau, 0]; \\ \end{array} \right. \end{eqnarray}$

(3.7)

同理, ${{p^* }\left( {s, t} \right)}$ 是系统(2.1) 相应于 $u = {u^* }$ 的解, 它满足

$\begin{eqnarray} \left\{ \begin{array}{l} \frac{{\partial {p^*}}}{{\partial t}} + \frac{{\partial (g(s){p^*})}}{{\partial s}} = - \left[ {{\mu _{\rm{0}}}\left( s \right) + {\mu _1}(E({p^*})(s, t))} \right]{p^*}(s, t), \\ g(0){p^*}(0, t) = \int_0^S {\beta (s, E({p^*})(s, t - \tau )){p^*}(s, t - \tau ){\rm d}s}, \\ {p^*}(s, t) = {u^*}\left( {s, t} \right), (s, t) \in [0, S] \times [ - \tau, 0]; \\ \end{array} \right. \end{eqnarray}$

(3.8)

利用(3.7) 与(3.8) 式取极限, 可知 $z\left( {s, t} \right)$ 满足

$\begin{eqnarray} \left\{ \begin{array}{l} \frac{{\partial z}}{{\partial t}} + \frac{{\partial (gz)}}{{\partial s}} = - {\mu'_1}\left( {{E}(p^*)(s, t)} \right){p^*}\left( {s, t} \right)E(z)\left( {s, t} \right) \\ - \left[ {{\mu _0}\left( s \right) + {\mu _1}\left( {{E}(p^*)(s, t)} \right)} \right]z\left( {s, t} \right), \left( {s, t} \right) \in {Q_T}, \\ g(0)z(0, t) = \int_0^S {\left[ {{\beta _x}(s, {E}(p^*)(s, t - \tau )){p^*}(s, t - \tau )E\left( z \right)\left( {s, t - \tau } \right)} \right.} \\ \left. { + \beta (s, {E}(p^*)(s, t - \tau ))z(s, t - \tau )} \right]{\rm d}s, t \in (0, T), \\ z\left( {s, t} \right) = v\left( {s, t} \right), \left( {s, t} \right) \in [0, S] \times \left[ { - \tau, 0} \right]. \end{array} \right. \end{eqnarray}$

(3.9)

线性系统(3.9) 解的存在唯一性可由不动点原理确立. 将系统(3.9) 的第一式乘以 $q\left( {s, t} \right)$ , 并在 ${{Q_T}}$ 上积分, 可得

$\begin{eqnarray} &&\int_0^S {\int_0^T {\left( {\frac{{\partial z}}{{\partial t}} + \frac{{\partial (gz)}}{{\partial s}}} \right)q\left( {s, t} \right){\rm d}s{\rm d}t} } \\ & = & - \int_0^S {\int_0^T {\left\{ {{\mu' _1}\left( {{E}(p^*)} \right)E(z)\left( {s, t} \right){p^*}\left( {s, t} \right) + \left[ {{\mu _0}\left( s \right) + {\mu _1}\left( {{E}(p^*)} \right)} \right]z\left( {s, t} \right)} \right\}q\left( {s, t} \right){\rm d}s{\rm d}t} }. \end{eqnarray}$

(3.10)

注意到

$\begin{eqnarray} &&\int_0^S \int_0^T \frac{\partial z}{\partial t}q(s, t){\rm d}s{\rm d}t = \int_0^S q(s, T)z(s, T){\rm d}s - \int_0^S \int_0^T z(s, t)\frac{\partial q}{\partial t}{\rm d}s{\rm d}t; \\ &&\int_0^S {\int_0^T {\frac{{\partial \left( {gz} \right)}}{{\partial s}}q\left( {s, t} \right){\rm d}s{\rm d}t} } \\ & = & - \int_0^T {g(0)z(0, t)q(0, t){\rm d}t} - \int_0^S {\int_0^T z\left( {s, t} \right){g\left( s \right)\frac{{\partial q}}{{\partial s}}{\rm d}s{\rm d}t} } \\ & = & - \int_0^T {q(0, t)\int_0^S {\left[ {{\beta _x}(s, E({p^*})(s, t - \tau ))E\left( z \right)\left( {s, t - \tau } \right){p^*}(s, t - \tau )} \right.} } \\ && \left. { + \beta (s, E({p^*})(s, t - \tau ))z(s, t - \tau )} \right]{\rm d}s{\rm d}t - \int_0^S {\int_0^T z\left( {s, t} \right){g\left( s \right)\frac{{\partial q}}{{\partial s}}{\rm d}s{\rm d}t} }, \end{eqnarray}$

(3.11)

应用系统(3.5) 中的第2式和第3式, 以及(3.10)-(3.11)式, 可以导出

$\begin{eqnarray*} && \int_0^S {\left[ {{p^*}\left( {s, T} \right) - \bar p\left( s \right)} \right]z\left( {s, T} \right){\rm d}s} \\ & = & - \int_0^S {\int_0^T {\left( {\frac{{\partial z}}{{\partial t}} + \frac{{\partial (gz)}}{{\partial s}}} \right)q\left( {s, t} \right){\rm d}s{\rm d}t} } + \int_0^S {\int_0^T {\frac{{\partial z}}{{\partial t}}q\left( {s, t} \right){\rm d}s{\rm d}t} } \\ & &+ \int_0^S \int_0^T \frac{\partial \left( gz \right)}{\partial s}q\left( {s, t} \right){\rm d}s{\rm d}t \\ & = &\int_0^S {\int_0^T {\left\{ {{\mu' _1}\left( {{E}(p^*)} \right)E(z)\left( {s, t} \right){p^*}\left( {s, t} \right) + \left[ {{\mu _0}\left( s \right) + {\mu _1}\left( {{E}(p^*)} \right)} \right]z\left( {s, t} \right)} \right\}q\left( {s, t} \right){\rm d}s{\rm d}t} } \\ && - \int_0^T {q(0, t)\int_0^S {\left[ {{\beta _x}(s, E({p^*})(s, t - \tau ))E\left( z \right)\left( {s, t - \tau } \right){p^*}(s, t - \tau )} \right.} } \\ && \left. { + \beta (s, E({p^*})(s, t - \tau ))z(s, t - \tau )} \right]{\rm d}s{\rm d}t - \int_0^S {\int_0^T {\left( {\frac{{\partial q}}{{\partial t}}{\rm{ + }}g\left( s \right)\frac{{\partial q}}{{\partial s}}} \right)z\left( {s, t} \right){\rm d}s{\rm d}t} } \\ & = & \int_0^S {\int_0^T {\alpha \int_s^S {\left[ {{\mu' _1}\left( {{E}(p^*)} \right)q\left( {r, t} \right) - {\beta _x}(r, {E}(p^*))q(0, t + \tau )} \right]{p^*}\left( {r, t} \right){\rm d}r} z\left( {s, t} \right)} {\rm d}s{\rm d}t} \\ & & + \int_0^S {\int_0^T {\int_0^s {\left[ {{\mu' _1}\left( {{E}(p^*)} \right)q\left( {r, t} \right) - {\beta _x}(r, {E}(p^*))q(0, t + \tau )} \right]{p^*}\left( {r, t} \right){\rm d}r} z\left( {s, t} \right)} {\rm d}s{\rm d}t} \\ & & + \int_0^S {\int_0^T {\left\{ {\left[ {{\mu _0}\left( s \right) + {\mu _1}\left( {{E}(p^*)} \right)} \right]q\left( {s, t} \right) - \beta (s, {E}(p^*))q(0, t + \tau )} \right\}z\left( {s, t} \right){\rm d}s{\rm d}t} }\\ && - \int_0^S {\int_0^T {\left( {\frac{{\partial q}}{{\partial t}}{\rm{ + }}g\left( s \right)\frac{{\partial q}}{{\partial s}}} \right)z\left( {s, t} \right){\rm d}s{\rm d}t} } -\int_0^S \int_{-\tau}^0 q(0, t+\tau)\tilde{E}(s, t)v(s, t){\rm d}s{\rm d}t. \end{eqnarray*}$

由共轭系统(3.5)的第一式可知

$\begin{eqnarray} \int_0^S {\left[ {{p^*}\left( {s, T} \right) - \bar p\left( s \right)} \right]z\left( {s, T} \right){\rm d}s} = -\int_0^S \int_{-\tau}^0 q(0, t+\tau)\tilde{E}(s, t)v(s, t){\rm d}s{\rm d}t. \end{eqnarray}$

(3.12)

将(3.12)式代入(3.6)式中可知: 对任意 $v \in {\cal T} _U\left( {{u^*}} \right)$ 有

$\int_0^S {\int_{ - \tau }^0 {[q(0, t + \tau )\tilde E(s, t) - \sigma {u^*}(s, t)]v(s, t){\rm d}s{\rm d}t} } \le 0.$

根据法锥的定义^{[21, p20]}知

$q(0, \tau +\cdot)\tilde{E}-\sigma {u^*} \in {\cal N}_U\left( {{u^*}} \right),$

其中 ${\cal N}_U\left( {{u^*}} \right)$ 表示集 $U$ 在 $u^*$ 处的法锥. 利用法锥元素的特征^{[21, p13]}可得定理3.1的结论.

以下证明下列极限的存在性以保证定理3.1证明的严格性.

引理3.2 极限 $\mathop {\lim }\limits_{\varepsilon \to {0^ + }} {\varepsilon ^{ - 1}}\left[ {{p^\varepsilon }\left( {s, t} \right) - {p^*}\left( {s, t} \right)} \right]$ 存在.

证记

${w_\varepsilon } = {\varepsilon ^{ - 1}}\left[ {{p^\varepsilon }\left( {s, t} \right) - {p^*}\left( {s, t} \right)} \right] - z\left( {s, t} \right),$

根据模型方程(2.1), 有

$\begin{eqnarray} &&\frac{{\partial \left\{ {\frac{1}{\varepsilon }\left[ {{p^\varepsilon }\left( {s, t} \right) - {p^*}\left( {s, t} \right)} \right]} \right\}}}{{\partial t}} + \frac{{\partial \left\{ {\frac{1}{\varepsilon }g\left( s \right)\left[ {{p^\varepsilon }\left( {s, t} \right) - {p^*}\left( {s, t} \right)} \right]} \right\}}}{{\partial s}}\\ & = & \frac{1}{\varepsilon }\left\{ {\frac{{\partial {p^\varepsilon }\left( {s, t} \right)}}{{\partial t}} + \frac{{\partial \left[ {g\left( s \right){p^\varepsilon }\left( {s, t} \right)} \right]}}{{\partial s}}} \right\} - \frac{1}{\varepsilon }\left\{ {\frac{{\partial {p^*}\left( {s, t} \right)}}{{\partial t}} + \frac{{\partial \left[ {g\left( s \right){p^*}\left( {s, t} \right)} \right]}}{{\partial s}}} \right\}\\ & = & - {\mu' _{1}}\left( {E\left( {{p^*}} \right)} \right)E\left( {\frac{1}{\varepsilon }\left[ {{p^\varepsilon } - {p^*}} \right]} \right)\left( {s, t} \right){p^\varepsilon }\left( {s, t} \right)\\ && - {\mu _1}\left( {E\left( {{p^*}} \right)\left( {s, t} \right)} \right)\frac{1}{\varepsilon }\left[ {{p^\varepsilon } - {p^*}} \right]\left( {s, t} \right) - {\mu _0}\left( s \right)\frac{1}{\varepsilon }\left[ {{p^\varepsilon } - {p^*}} \right]\left( {s, t} \right), \end{eqnarray}$

(3.13)

以及

$\begin{eqnarray} &&\frac{1}{\varepsilon }g\left( 0 \right)\left[ {{p^\varepsilon }\left( {0, t} \right) - {p^*}\left( {0, t} \right)} \right]\\ & = &\int_0^S {{\beta _x}\left( {s, {E}\left( p^* \right)\left( {s, t - \tau } \right)} \right){p^\varepsilon }(s, t - \tau )E\left( {\frac{1}{\varepsilon }\left[ {{p^\varepsilon } - {p^*}} \right]} \right)\left( {s, t - \tau } \right){\rm d}s} \\ && + \int_0^S {\beta \left( {s, {E}\left( p^* \right)\left( {s, t - \tau } \right)} \right)\frac{1}{\varepsilon }\left[ {{p^\varepsilon } - {p^*}} \right]\left( {s, t - \tau } \right){\rm d}s} +{b_0}\left( \varepsilon \right), \end{eqnarray}$

(3.14)

其中 $\mathop {\lim }\limits_{\varepsilon \to 0} {b_0}\left( \varepsilon \right) = 0$ .

由系统(3.9) 中的第一式与(3.13)式, 有

$\begin{eqnarray*} &&\frac{{\partial {w_\varepsilon }}}{{\partial t}} + \frac{{\partial \left( {g\left( s \right){w_\varepsilon }} \right)}}{{\partial s}}\\ & = & \frac{{\partial \left\{ {\frac{1}{\varepsilon }\left[ {{p^\varepsilon }\left( {s, t} \right) - {p^*}\left( {s, t} \right)} \right]} \right\}}}{{\partial t}} - \frac{{\partial z}}{{\partial t}} + \frac{{\partial \left\{ {\frac{1}{\varepsilon }g\left( s \right)\left[ {{p^\varepsilon }\left( {s, t} \right) - {p^*}\left( {s, t} \right)} \right]} \right\}}}{{\partial s}} - \frac{{\partial \left( {g\left( s \right)z} \right)}}{{\partial s}}\\ & = & - \left[ {{\mu _0}\left( s \right) + {\mu_1}\left( {E({p^*})} \right)\left( {s, t} \right)} \right]{w_\varepsilon } - {\mu'_1}\left( {E\left( {{p^*}} \right)} \right)E\left( {{w_\varepsilon }} \right)\left( {s, t} \right){p^\varepsilon }\left( {s, t} \right) \\ && - {\mu'_1}\left( {E({p^*})} \right)E(z)\left( {s, t} \right)\left( {{p^\varepsilon }\left( {s, t} \right) - {p^*}\left( {s, t} \right)} \right). \end{eqnarray*}$

由系统(3.9) 中的第二式与(3.14)式, 可得

$\begin{eqnarray*} &&g\left( 0 \right){w_\varepsilon }\left( {0, t} \right) \\ & = & \frac{1}{\varepsilon }g\left( 0 \right)\left[ {{p^\varepsilon }\left( {0, t} \right) - {p^*}\left( {0, t} \right)} \right] - g\left( 0 \right)z\left( {0, t} \right)\\ & = & \int_0^S {\beta \left( {s, {E}\left( p^* \right)\left( {s, t - \tau } \right)} \right){w_\varepsilon }\left( {s, t - \tau } \right){\rm d}s} \\ && +\int_0^S {{\beta _x}\left( {s, {E}\left( p^* \right)\left( {s, t - \tau } \right)} \right){p^\varepsilon }(s, t - \tau )E\left( {{w_\varepsilon }} \right)\left( {s, t - \tau } \right){\rm d}s} \\ &&+ \int_0^S {{\beta _x}(s, E({p^*})(s, t - \tau ))\left( {{p^\varepsilon }(s, t - \tau ) - {p^*}(s, t - \tau )} \right)E\left( z \right)\left( {s, t - \tau } \right){\rm d}s} + {b_0}\left( \varepsilon \right). \end{eqnarray*}$

综上, 变量 $w_\varepsilon$ 所满足的系统方程为

$\begin{eqnarray} \left\{ \begin{array}{l} \frac{{\partial {w_\varepsilon }}}{{\partial t}} + \frac{{\partial \left( {g\left( s \right){w_\varepsilon }} \right)}}{{\partial s}} = - \left[ {{\mu _0}\left( s \right) + {\mu_1}\left( {E({p^*})} \right)\left( {s, t} \right)} \right]{w_\varepsilon }\\ - {\mu'_1}\left( {E\left( {{p^*}} \right)} \right)E\left( {{w_\varepsilon }} \right)\left( {s, t} \right){p^\varepsilon }\left( {s, t} \right) \\ - {\mu'_1}\left( {E({p^*})} \right)E(z)\left( {s, t} \right)\left( {{p^\varepsilon }\left( {s, t} \right) - {p^*}\left( {s, t} \right)} \right), \left( {s, t} \right) \in {Q_{T, }}\\ g\left( 0 \right){w_\varepsilon }\left( {0, t} \right) = \int_0^S {\beta \left( {s, {E}\left( p^* \right)\left( {s, t - \tau } \right)} \right){w_\varepsilon }\left( {s, t - \tau } \right){\rm d}s} \\ + \int_0^S {{\beta _x}\left( {s, {E}\left( p^* \right)\left( {s, t - \tau } \right)} \right){p^\varepsilon }(s, t - \tau )E\left( {{w_\varepsilon }} \right)\left( {s, t - \tau } \right){\rm d}s} \\ + \int_0^S {{\beta _x}(s, E({p^*})(s, t - \tau ))\left( {{p^\varepsilon }(s, t - \tau )} \right.} \\ \left. { - {p^*}(s, t - \tau )} \right)E\left( z \right)\left( {s, t - \tau } \right){\rm d}s+ {b_0}\left( \varepsilon \right), t \in \left[ {0, T} \right], \\ {w_\varepsilon }\left( {s, t} \right) = 0, \left( {s, t} \right) \in [0, S] \times \left[ { - \tau, 0} \right]. \\ \end{array} \right. \end{eqnarray}$

(3.15)

对系统(3.15) 中的前两式右端除首项外的其余部分取极限 ${\varepsilon \to {0^ + }}$ , 可得极限系统如下

$\begin{eqnarray} \left\{ \begin{array}{l} \frac{{\partial w}}{{\partial t}} + \frac{{\partial \left( {g\left( s \right)w} \right)}}{{\partial s}} = { - \left[ {{\mu _0}\left( s \right) + {\mu _1}\left( {E({p^*})} \right)\left( {s, t} \right)} \right]w\left( {s, t} \right)} \\ -{{\mu'_1}\left( {E({p^*})} \right)E(w)\left( {s, t} \right){p^*}\left( {s, t} \right)}\\ g\left( 0 \right)w\left( {0, t} \right) = {\int_0^S {\beta \left( {s, {E}\left( p^* \right)\left( {s, t - \tau } \right)} \right)w\left( {s, t - \tau } \right){\rm d}s} } \\ + \int_0^S {{\beta _x}\left( {s, {E}\left( p^* \right)\left( {s, t - \tau } \right)} \right){{p^*}(s, t - \tau )}E\left( w \right)\left( {s, t - \tau } \right){\rm d}s}, t \in \left[ {0, T} \right], \\ {w}\left( {s, t} \right) = 0, \left( {s, t} \right) \in [0, S] \times \left[ { - \tau, 0} \right], \end{array} \right. \end{eqnarray}$

(3.16)

其中用到了 $E\left( {\frac{1}{\varepsilon }\left[ {{p^\varepsilon } - {p^*}} \right]} \right)\left( {s, t} \right)\left[ {{p^\varepsilon } - {p^*}} \right]\left( {s, t} \right) \to 0, \varepsilon \to {0^ + }.$

注意(3.16) 式是一个初始条件为零的齐次线性系统, 由其解的唯一性知 $\mathop {\lim }\limits_{\varepsilon \to {0^ + }} {w_\varepsilon } = 0$ , 这意味着

$\mathop {\lim }\limits_{\varepsilon \to {0^ + }} {\varepsilon ^{ - 1}}\left[ {{p^\varepsilon }\left( {s, t} \right) - {p^*}\left( {s, t} \right)} \right] = z\left( {s, t} \right),$

$z(s, t)$ 为系统(3.9) 的唯一解. 引理证毕.

4 最优控制的存在唯一性

定义泛函 $\varphi :{L^{\rm{1}}} ([0, S] \times [ - \tau, 0]) \to \left( { - \infty, + \infty } \right],$

$\varphi \left( u \right){\rm{ = }}\left\{ \begin{array}{ll} \int_0^S {{{\left[ {{p^u}\left( {s, T} \right) - \bar p\left( s \right)} \right]}^2}{\rm d}s} + \sigma \int_0^S {\int_{ - \tau }^0 {{u^2}\left( {s, t} \right){\rm d}s{\rm d}t} }, & u \in U, \\ + \infty, & u \notin U. \end{array} \right.$

引理4.1 泛函 $\varphi$ 下半连续.

证设 $\left\{ {{u_n}} \right\}$ 为 ${L^{\rm{1}}}([0, S] \times [ - \tau, 0])$ 中的任一序列, 当 $n \to \infty$ 时, ${{u_n} \to u}$ . 不失一般性, 可以假定 ${{u_n} \in U}$ , $\forall n \ge 1$ . 据定理2.1可知, 对任意 $s \in \left( {0, S} \right)$ , 当 $n\rightarrow \infty$ 时有

${p^{{u_n}}}\left( { s, \cdot} \right) \to {p^u}\left( { s, \cdot} \right).$

Riesz定理告诉我们, 存在一个子序列(仍记为 $\left\{ {{u_n}} \right\}$ ), 使得

${u_n}\left( {s, t} \right) \to u\left( {s, t} \right), {p^{{u_n}}}\left( {s, t} \right) \to {p^u}\left( {s, t} \right)$

在 $\left( {0, S} \right) \times \left( { - \tau, 0} \right)$ 上几乎处处成立.从而

$\begin{eqnarray*} &&\int_0^S {{{\left[ {{p^{{u_n}}}\left( {s, T} \right) - \bar p\left( s \right)} \right]}^2}{\rm d}s} + \sigma \int_0^S {\int_{ - \tau }^0 {{{\left( {{u_n}\left( {s, t} \right)} \right)}^2}{\rm d}s{\rm d}t} } \\ &\to &\int_0^S {{{\left[ {{p^u}\left( {s, T} \right) - \bar p\left( s \right)} \right]}^2}{\rm d}s} + \sigma \int_0^S {\int_{ - \tau }^0 {{u^2}\left( {s, t} \right){\rm d}s{\rm d}t} } . \end{eqnarray*}$

应用Fatou定理, 即得

$\mathop {\lim }\limits_{n \to \infty } \inf \varphi \left( {{u_n}} \right) \ge \varphi \left( u \right).$

因此知 $\varphi$ 下半连续, 引理证毕.

利用特征线法可得下列结果(略去证明).

引理4.2 记共轭系统(3.5) 相应于初始控制 $u, v$ 的解为 $q^u, q^v$ , 则对任意 $\left( {s, t} \right) \in \left( {0, S} \right) \times \left( { 0, T} \right)$ , 都有

$|q^v (s, t)| + | q^u (0, t) - q^v (0, t)| \le C(\bar{u}, T)\| u - v \|_{L^{\infty }( [0, S] \times [ - \tau, 0])},$

其中 $C\left( {\bar{u}, T} \right)$ 为 $\bar{u}$ 和 $T$ 所构成的正的常数.

定理4.1 如果 $\sigma^{-1}{C}\left( {\bar{u}, T} \right)$ 充分小, 那么控制问题(2.1)-(2.2) 存在唯一的解.

证根据Ekeland变分原理^{[21, p29, Th3.2]}知, 对任意 $\varepsilon > 0$ , 存在 ${u_\varepsilon } \in U$ , 使得

$\begin{equation} \varphi \left( {{u_\varepsilon }} \right) \le \inf \varphi + \varepsilon, \end{equation}$

(4.1)

$\begin{equation} \varphi (u_{\varepsilon }) \le \inf \left\{\varphi( u) + \sqrt{\varepsilon} \| u - u_{\varepsilon}\|_{L^1 ([0, S] \times [- \tau, 0])}: u \in U \right\}. \end{equation}$

(4.2)

由(4.2) 式知: $u_{\varepsilon}$ 必为泛函 $\varphi(u)+\sqrt{\varepsilon}\|u-u_{\varepsilon}\|_{L^1}$ 的最小值点.

利用定理3.1的证明方法可知: 对任一 $v \in {\cal T}_U(u_{\varepsilon})$ , 都有

$\int_0^S {\int_{ - \tau }^0 {\left[ {\sigma {u_\varepsilon }(s, t) - {q^{{u_\varepsilon }}}(0, t + \tau )\tilde E_{\varepsilon}(s, t)} \right]v\left( {s, t} \right){\rm d}s{\rm d}t} } + \sqrt \varepsilon \int_0^S {\int_{ - \tau }^0 {\left| {v\left( {s, t} \right)} \right|{\rm d}s{\rm d}t} } \ge 0,$

其中 $q^{\varepsilon}$ 表示系统(3.5) 相应于 $u^* = u_{\varepsilon}$ 的解, $\tilde{E}_{\varepsilon}(s, t)$ 表示 $\tilde{E}(s, t)$ 定义中的 $p^*$ 应理解为系统(2.1) 相应于 $u^* = u_{\varepsilon}$ 的解. 利用文献[22] 的结果知: 存在函数

$f_{\varepsilon} \in L^{\infty }( [0, S] \times [- \tau, 0]), \|f_{\varepsilon } \| \le 1,$

使得

$- \sigma {u_\varepsilon } + {q^{{u_\varepsilon }}}\left( {0, \tau + \cdot} \right)\tilde E_{\varepsilon} + \sqrt \varepsilon {f_\varepsilon } \in {\cal N}_U\left( {{u_\varepsilon }} \right),$

因此

$\begin{equation} {u_\varepsilon }\left( {s, t} \right) = {\cal F}\left[ {{\sigma ^{ - 1}}{q^{{u_\varepsilon }}}\left( {0, t + \tau } \right)\tilde E_{\varepsilon}(s, t) + {\sigma ^{ - 1}}\sqrt \varepsilon {f_\varepsilon }\left( {s, t} \right)} \right]. \end{equation}$

(4.3)

首先证明最优控制的唯一性. 定义映射 $\psi : U \to U$ , 有

$\begin{equation} \left( {\psi u} \right)\left(s, t \right) = {\cal F}\left[ {{\sigma ^{ - 1}}q^u (0, t + \tau )\tilde E_u (s, t)} \right], \ {\rm a.e.}\ \left( {s, t} \right) \in \left( {0, S} \right) \times \left( { - \tau, 0} \right), \end{equation}$

(4.4)

其中 $q^u$ 表示系统(3.5)相应于 $u^* = u$ 的解, $\tilde{E}_u (s, t)$ 表示 $\tilde{E}(s, t)$ 定义中的 $p^*$ 应理解为系统(2.1) 相应于 $u^* = u$ 的解. 结合(4.4) 式与引理4.2可得

$\begin{eqnarray*} & &|(\psi u)(s, t) - (\psi v)(s, t) |\\ & = &| {\cal F}(\sigma ^{ - 1}q^u (0, t + \tau )\tilde{E}_u (s, t)) -{\cal F}(\sigma ^{ - 1}q^v (0, t + \tau)\tilde{E}_v(s, t) ) |\\ &\le & \sigma ^{ - 1}\left\{ |\tilde E_u(s, t) |\cdot |q^u (0, t+\tau) - q^v (0, t+\tau)|+|q^u (0, t+\tau)|\cdot |\tilde E_u(s, t)-\tilde E_v(s, t) |\right\}\\ & \le & \sigma ^{ - 1}C(\bar{u}, T)\| u - v\|_{L^{\infty }([0, S] \times [- \tau, 0])}. \end{eqnarray*}$

显然, 由定理的条件可知 $\psi$ 是压缩的, 从而 $\psi$ 存在唯一不动点 ${u_0} \in U$ . 定理4.1意味着问题(2.1)-(2.2) 的任一解都是 $\psi$ 的不动点. 所以该问题至多有一个最优解.

再证 ${u_0}$ 是该问题的解. 利用(4.3)-(4.4)式, 可得

$\begin{eqnarray*} &&\left| {{u_\varepsilon }\left( {s, t} \right) - \left( {\psi {u_\varepsilon }} \right)\left( {s, t} \right)} \right|\\ & = & \left| {{\cal F}\left[ {{\sigma ^{ - 1}}{q^{{u_\varepsilon }}}\left( {0, t + \tau } \right)\tilde E_{\varepsilon}(s, t) + {\sigma ^{ - 1}}\sqrt \varepsilon {f_\varepsilon }\left( {s, t} \right)} \right] - {\cal F}\left[ {{\sigma ^{ - 1}}{q^{{u_\varepsilon }}}(0, t + \tau )\tilde E_{\varepsilon}(s, t)} \right]} \right|\\ &\le & {\sigma ^{ - 1}}\sqrt \varepsilon \left| {{f_\varepsilon }\left( {s, t} \right)} \right| \le {\sigma ^{ - 1}}\sqrt \varepsilon . \end{eqnarray*}$

因此

$\begin{eqnarray*} & &\| u_{\varepsilon} - u_0 \|_{L^{\infty }([0, S] \times [ -\tau, 0])} = \|u_{\varepsilon} - \psi u_0 \|_{L^{\infty }([0, S] \times [ - \tau, 0])}\\ &\le& \|u_{\varepsilon } - \psi u_{\varepsilon } \|_{L^{\infty }([0, S] \times [ - \tau, 0])} + \| \psi u_{\varepsilon } - \psi u_0\|_{L^{\infty }([0, S] \times [ - \tau, 0])}\\ &\le& \sigma ^{ - 1}C(\bar{u}, T)\| u_{\varepsilon } - u_0\|_{L^{\infty }([0, S] \times [ - \tau, 0])} + \sigma ^{ - 1}\sqrt{\varepsilon}. \end{eqnarray*}$

从而当 $\varepsilon \to {{\rm{0}}^{\rm{ + }}}$ 时, ${{u_\varepsilon } \to {u_0}}$ .

对不等式(3.1) 取极限 $\varepsilon\rightarrow 0^+$ 并利用引理3.1, 得到

$\varphi ({u_{\rm{0}}}) \le \mathop {\lim }\limits_{\varepsilon \to {0^ + }} \inf \varphi \left( {{u^\varepsilon }} \right) \le \inf \varphi,$

从而 $\varphi ({u_{\rm{0}}}) = \inf \{ \varphi (u):u \in U\}.$ 定理4.1证毕.

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原文顺序

文献年度倒序

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Individual differences between animals and the natural regulation of their numbers

1978

... 生物种群内部个体之间的等级差异对群体演化有些什么样的影响？这个问题可以用数学建模与分析给予部分回答. 为此, 国内外学者已经建立了一些数学模型, 并给出了相应的理论和数值分析, 参见文献[1-16]及其参考文献. 这些研究工作主要关注系统的动力学行为, 比如解的存在唯一性, 平衡态的存在性与(全局)稳定性, 种群的持续生存, 种群内部抢夺竞争(Scramble competition)与对抗竞争(Contest competition)的比较, 以及密度函数的数值算法, 等. 毫无疑问, 这些成果对理解等级结构种群的演化具有积极意义. 关于种群研究的另一个重要侧面是调控问题, 包括出于生态保护的种群分布优化, 以及可再生资源开发的最优收获等问题, 还很少被研究, 参见文献[17-19]. 这类调控问题的生态学与经济学意义不言而喻, 并且作为一类新的无穷维系统控制问题, 也有其数学和控制科学价值. ...

Dominance rank, copulatory behavior, and differential reproduction

1982

Ecological stability and social hierarchy

1979

The dynamics of hierarchical age-structured populations

1994

Hierarchical models of intra-specific competition: scramble versus contest

1996

Monotone approximation for a hierarchical age-structured population model

2005

Oscilations caused by cannibalism in a size-structured population model

1995

Asymptotic behaviour of a model of hierarchically structured population dynamics

1997

Existence and uniqueness for height structured hierarchical population models

2001

A quasilinear hierarchical size-structured model: Well-posedness and approximation

2005

Basic theory for a class of models of hierarchically structured population dynamics with distributed states in the recruitment

2006

A monotone approximation for a size-structured population model with a generalized environment

2007

A high order WENO scheme for a hierarchical size-structured population model

2007

Asymptotic analysis of a size-structured cannibalism modfel with infinite dimensional environmental feedback

2009

A discrete hierarchical model of intra-specific competition

2003

一类非线性年龄等级结构种群模型的数值解法

2020

一类非线性年龄等级结构种群模型的数值解法

2020

Optimal harvesting of a hierarchical age-structured population system

2019

一类非线性年龄等级结构种群系统模型的可控性

2020

一类非线性年龄等级结构种群系统模型的可控性

2020

带有离散个体等级结构的种群系统的控制问题

2020

带有离散个体等级结构的种群系统的控制问题

2020

具有时滞的尺度等级结构种群模型

2019

... 文献[20] 对系统(2.1) 做了初步分析, 获得下列结果. ...

... 证令

$\left( {{u^*}, {p^*}} \right)$

为问题(2.1)-(2.2) 的最优对. 系统(3.5) 解的存在唯一性可用时间变换和类似于文献[20] 的方法得到. 对任意

$v \in {\cal T}_U\left( u^*\right)$

(表示集

$U$

在

${{u^*}}$

处的切锥) 满足

$v(s, 0) = 0$

, 当

$\varepsilon > 0$

足够小时,

${u^\varepsilon }: = {u^*} + \varepsilon v \in U$

(参见文献[21, p21]). ...

具有时滞的尺度等级结构种群模型

2019

... 文献[20] 对系统(2.1) 做了初步分析, 获得下列结果. ...

... 证令

$\left( {{u^*}, {p^*}} \right)$

为问题(2.1)-(2.2) 的最优对. 系统(3.5) 解的存在唯一性可用时间变换和类似于文献[20] 的方法得到. 对任意

$v \in {\cal T}_U\left( u^*\right)$

(表示集

$U$

在

${{u^*}}$

处的切锥) 满足

$v(s, 0) = 0$

, 当

$\varepsilon > 0$

足够小时,

${u^\varepsilon }: = {u^*} + \varepsilon v \in U$

(参见文献[21, p21]). ...

... 证令

$\left( {{u^*}, {p^*}} \right)$

为问题(2.1)-(2.2) 的最优对. 系统(3.5) 解的存在唯一性可用时间变换和类似于文献[20] 的方法得到. 对任意

$v \in {\cal T}_U\left( u^*\right)$

(表示集

$U$

在

${{u^*}}$

处的切锥) 满足

$v(s, 0) = 0$

, 当

$\varepsilon > 0$

足够小时,

${u^\varepsilon }: = {u^*} + \varepsilon v \in U$

(参见文献[21, p21]). ...

... 根据法锥的定义^{[21, p20]}知 ...

... 其中

${\cal N}_U\left( {{u^*}} \right)$

表示集

$U$

在

$u^*$

处的法锥. 利用法锥元素的特征^{[21, p13]}可得定理3.1的结论. ...

... 证根据Ekeland变分原理^{[21, p29, Th3.2]}知, 对任意

$\varepsilon > 0$

, 存在

${u_\varepsilon } \in U$

, 使得 ...

Optimal control of population dynamics

1999

... 其中

$q^{\varepsilon}$

表示系统(3.5) 相应于

$u^* = u_{\varepsilon}$

的解,

$\tilde{E}_{\varepsilon}(s, t)$

表示

$\tilde{E}(s, t)$

定义中的

$p^*$

应理解为系统(2.1) 相应于

$u^* = u_{\varepsilon}$

的解. 利用文献[22] 的结果知: 存在函数 ...

〈

〉