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.

本文提出下列种群控制系统模型

(2.1) $ \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} $

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

(2.2) $ \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} $

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) $.

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

(3.1) $ \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} $

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

(3.2) $ \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} $

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:

(3.3) $ \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} $

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

(3.4) $ \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} $

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

(3.5) $ \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} $

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

(3.6) $ \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} $

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

(3.7) $ \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} $

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

(3.8) $ \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} $

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

(3.9) $ \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} $

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

(3.10) $ \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} $

注意到

(3.11) $ \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} $

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

(3.12) $ \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} $

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

(3.13) $ \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} $

以及

(3.14) $ \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} $

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

(3.15) $ \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} $

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

(3.16) $ \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} $

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.

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

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

(4.2) $ \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} $

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

(4.3) $ \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} $

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

(4.4) $ \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} $

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.