针对生物种群内部个体的社会结构和行为, 生态学工作者已经作了半个多世纪的研究, 对各类种群个体的等级地位次序有了丰富的一手资料, 参见综述文献[1]及其所引用的200余份文献.若需考察具有等级差异的种群的长期演化行为, 数学模型就拥有其它手段难以比拟的优势.目前, 这方面的研究成果有一些, 虽不完善但很重要, 为我们提供了等级结构种群系统的基本认知, 可参见文献[2-17].其中, 文献[2]和[3]分析种群内部竞争, 前者运用连续模型, 后者采用离散模型.文献[4]发现非极端化的等级结构有助于系统稳定.在文献[5]中, Cushing用尺度等级结构模型分析同类自食现象, 在文献[6]中处理一类年龄等级结构模型的适定性和动力学行为.文献[7]借助尺度等级模型分析因自食导致的震荡行为.文献[8-12, 15, 16]的主题是适定性与渐近行为, 文献[13]和[17]关注等级模型的数值解法, 文献[14]研究一类等级模型的平衡态与稳定性.至于等级结构种群系统的控制问题, 相关成果报道就更少了.文献[18]以捕捞强度为控制策略, 力图以最小代价让种群分布尽可能接近某种理想状态, 这是出于优化生态角度考虑的.文献[19]从可再生种群资源的最优开发角度出发, 设计了一个最优捕捞问题, 进行了细致的理论和数值分析.这两项工作都是基于单一种群环境的.由于实际生态环境中基本上都是多种群共存, 如何调控多种群等级结构系统无疑是必须考虑的问题.文献[20]在这方面作了一点初步努力, 关于两种群竞争系统的可控性与镇定问题做了探索, 发现分布式迁移可以实现种群系统状态的有效调节.

鉴于个体尺度(如身长、体重)是决定等级的主要因素之一, 本文建模时考虑基于尺度的内部等级环境.此外, 个体孕育期时滞也将纳入繁殖过程.区别于前述3项控制问题的研究工作, 本文以幼体的动态规模作为控制策略, 目标是:经过一段指定时间后, 种群的尺度(大小)分布一方面尽可能接近预先设定的理想分布, 另一方面也要求代价尽可能的小.下一节提出模型, 设定合理的参数条件; 第3第4节分别建立状态系统解的存在唯一性、解关于控制变量的连续依赖性.第5节运用法锥和共轭系统技巧导出控制问题的最小值原理, 以此为基础, 第6节证明最优策略的存在唯一性.最后一节总结全文.

本文提出并研究下列种群模型

(2.1) $ \begin{equation} \left\{ \begin{array}{ll} { } \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} + u\left( t \right), & t \in (0, T), \\ { } p(s, t) = {p_0}(s, t), s \in [0, S], & t \in [ - \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{equation} $

其中$ p(s, t) $表示$ t $时刻种群的个体尺度分布, $ g(s) = {{{\rm d}s} \over {{\rm d}t}} $为尺度增长率; $ {\mu _{\rm{0}}}\left( s \right) $为自然死亡率, $ {\mu _{\rm{1}}} $代表因内部竞争引起的附加死亡率, $ u(t) $表示对种群幼体的投放(例如鱼苗), $ \beta $表示个体的平均繁殖率; $ {p_0}(s, t) $表示初始时间段里种群的个体尺度分布; $ 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.

本文的分析需要下列假设

$ (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_0 (s)+g'(s)\ge 0 $. $ {\mu_{1}}(y) $非负且关于$ y $严格单增; $ \beta( \cdot , y) $关于$ y $非增, $ 0 \le \beta \left( {s, y} \right) \le \overline \beta , \forall \left( {s, y} \right) \in \left[ {0, S} \right] \times {{{\Bbb R}}_ + } $, 且$ {\mu_{1}}(y) $和$ {\beta}( \cdot , y) $关于$ y $满足局部Lipschitz条件, 即:对任意常数$ M>0 $, 存在$ {L_1 (M)}, {L_2 (M)} $使得:当$ \left| {{y_1} - {y_2}} \right|\leq M $时有$ \left| {\mu_1 ( {y_1}) - \mu_1 ({y_2})} \right| \le {L_1 (M)}\left| {{y_1} - {y_2}} \right| $, $ \left| {\beta ( \cdot , {y_1}) - \beta ( \cdot , {y_2})} \right| \le {L_2 (M)}\left| {{y_1} - {y_2}} \right| $;

$ (A_3) $$ 0 \le {p_0}(s, t) \le {M_0}, $ a.e. $ \left( {s, t} \right) \in \left[ {0, S} \right] \times \left[ { - \tau , 0} \right] $; $ M_0 $为常数;

$ (A_4) $ 容许控制集为: $ U=\{ u\in L^{\infty}(0, T): 0\leq u(t)\leq \bar{u} $ a.e. $ t\in (0, T)\} $, $ \bar{u} $为常数.

本文研究的核心问题是:如何选取投放函数$ u\in U $, 使得下列泛函取值最小

(2.2) $ \begin{equation} J\left( u \right) = \frac{1}{2}\int_0^S {{{\left[ {{p^u}\left( {s, T} \right) - \overline p \left( s \right)} \right]}^2}{\rm d}s} +\frac{\sigma}{2} \int_0^T {{u^2}\left( t \right){\rm d}t} , \end{equation} $

其中$ p^u (s, t) $表示(2.1)相应于$ u $的解, $ {\overline p \left( s \right)} $代表某种给定的理想分布(例如平衡态), $ \sigma > 0 $为单位控制成本.因此$ J\left( u \right) $表示种群终态分布与理想分布的均方差与控制总成本之和.

本节作出模型的适定性分析.为了表述简便, 记$ \mu (s, E(p)(s, t)){\rm{ = }}{\mu _{\rm{0}}}\left( s \right) + {\mu _1}(E(p)(s, t)) $.

将$ \mu_1 $中的函数$ p $固定为非负函数$ q $后, 模型(2.1)变为线性系统.在特征线$ \Gamma (s) - t = c $ ($ c $为常数)上, 解的表示形式为

(3.1) $ \begin{equation} {p}(s, t;{q}) = \left\{ \begin{array}{ll} {p_0}({\Gamma ^{ - 1}}\left( {\Gamma (s) - t} \right), 0){K}(s, t;{q}), &t \le \Gamma (s), \\ { } \frac{{{b}(t - \Gamma (s))}}{{g(s)}}{F}\left( {s, t;{q}} \right), &t > \Gamma (s), \end{array} \right. \end{equation} $

其中

(3.2) $ \begin{eqnarray} K(s, t;q)&=&{\rm exp}\Big\{-\int_0^t[\mu( {{\Gamma^{-1}}({\Gamma(s)-t+r}), E(q)({\Gamma^{-1}}(\Gamma(s)-t+r), r)}){}\\ && +g'({\Gamma^{-1}}(\Gamma(s)-t+r))]{\rm d}r \Big\}, \\ {F}\left( {s, t;{q}} \right) &=&{\rm exp}\Big\{ { - \int_0^s {\frac{{\mu \left( {y, E({q})(y, t - \Gamma (s) + \Gamma (y))} \right)}}{{g(y)}}{\rm d}y} } \Big\}, {} \end{eqnarray} $

$ {b}(t) = g(0)p(0, t;q) $为已知.

本节余下部分应用不动点原理证明系统(2.1)存在唯一的解.

首先在时间段$ (0, \tau] $上考虑系统(2.1).定义一个子集$ H $如下

$ H = \left\{ {q \in {L^\infty }\left( {0, \tau;{L^1}(0, S)} \right): q(s, t) \ge 0, {{\left\| {q\left( { \cdot , t} \right)} \right\|}_{{L^1}(0, S)}} \le {M }} \right\}, $

其中$ {M} = (SM_0 \bar{\beta}+\bar{u})\Gamma{(S)}+SM_0 $.

定义算子$ \psi : H \to {L^\infty }([0, \tau ];{L^1}(0, S)) $,

$ \left( {\psi q} \right)\left( {s, t} \right) = p(s, t;q), \forall q \in H, $

其中$ p\left( {s, t;q} \right) $由(3.1)式定义.

注意$ {\psi q} \in H $.事实上, 利用表达式(3.1)和假设(A1)–(A4)可得

$ \begin{eqnarray*} {\left\| {p(\cdot, t;q)} \right\|_{{L^1}(0, S)}}&=& \int_0^{{\Gamma ^{ - 1}}(t)} {\frac{{b(t - \Gamma (s))}}{{g(s)}}F\left( {s, t;q} \right){\rm d}s}\\ && + \int_{{\Gamma ^{ - 1}}(t)}^S {{p_0}({\Gamma ^{ - 1}}\left( {\Gamma (s) - t} \right), 0)K(s, t;q){\rm d}s}\le M. \end{eqnarray*} $

因此, $ {\psi q} \in H $.

为了证明$ \psi $有不动点, 需要做一些估计.由于$ \Gamma(S) $即为个体最大年龄, 因此可设$ \tau<\Gamma(S) $.

由(3.2)式和不等式$ \left| {{e^{ - x}} - {e^{ - y}}} \right| \le \left| {x - y} \right| $ (当$ x \ge 0, y \ge 0 $)可得

(3.3) $ \begin{eqnarray} &&\left| K(s, t;q_1) - K(s, t;q_2)\right|{}\\ &=& \Big| {\rm exp}\Big\{ -\int_0^t [ \mu (\Gamma ^{ - 1}(\Gamma (s) - t + r), E(q_1)(\Gamma ^{ - 1}(\Gamma (s) - t + r), r)){}\\ &&+ g'(\Gamma ^{ - 1}(\Gamma (s) - t + r)) ]{\rm d}r\Big\} {}\\ & &-{\rm exp}\Big\{- \int_0^t [ \mu (\Gamma ^{ - 1}(\Gamma (s) - t + r), E(q_2)(\Gamma ^{ - 1}(\Gamma (s) - t + r), r)){}\\ && + g'(\Gamma ^{ - 1}(\Gamma (s) - t + r)) ]{\rm d}r\Big\}\Big|{}\\ &\le & \left| \int_0^t \mu(\Gamma ^{ - 1}(\Gamma (s) - t + r), E(q_1)(\Gamma ^{ - 1}(\Gamma (s) - t + r), r)){\rm d}r\right.{}\\ &&\left. -\int_0^t \mu (\Gamma ^{ - 1}(\Gamma (s) - t + r), E(q_2)(\Gamma ^{ - 1}(\Gamma (s) - t + r), r)){\rm d}r\right|{}\\ &\le &{L_1}( M)\int_0^t \left| E(q_1 )(\Gamma ^{ - 1}(\Gamma (s) - t + r), r) - E(q_2 )(\Gamma ^{ - 1}(\Gamma (s) - t + r), r) \right|{\rm d}r{}\\ &\le & {L_1}( M )\int_0^t \int_0^S \left| q_1( y, r) - q_2( y, r) \right| {\rm d}y{\rm d}r{}\\ &= & {L_1}( M )\int_0^t \left\| q_1( \cdot , r) - {q_2}(\cdot , r) \right\|_{L^1 (0, S)}{\rm d}r. \end{eqnarray} $

相应地

(3.4) $ \begin{eqnarray} &&\left| {{F}\left( {s, t;{q_1}} \right) - {F}\left( {s, t;{q_2}} \right)} \right|{}\\ &=& \left| {{\rm exp}\left\{ { - \int_0^s {\frac{{\mu \left( {y, E({q_1})(y, t - \Gamma (s) + \Gamma (y))} \right)}}{{g(y)}}{\rm d}y} } \right\}} \right.{}\\ &&\left. { - {\rm exp}\left\{ { - \int_0^s {\frac{{\mu \left( {y, E({q_2})(y, t - \Gamma (s) + \Gamma (y))} \right)}}{{g(y)}}{\rm d}y} } \right\}} \right|{}\\ &\le &\int_0^s {\frac{1}{{ {g(y)}}}\left| {\left[ {\mu \left( {y, E({q_1})(y, t - \Gamma (s) + \Gamma (y))} \right) - \mu \left( {y, E({q_2})(y, t - \Gamma (s) + \Gamma (y))} \right)} \right]} \right|{\rm d}y} {}\\ &\le &{L_1}\left( M \right)\int_0^s \frac{1}{g(y)} {\left| {E({q_1})(y, t - \Gamma (s) + \Gamma (y)) - E({q_2})(y, t - \Gamma (s) + \Gamma (y))} \right|{\rm d}y} {}\\ &\le &{L_1}\left( M \right)\int_0^s \frac{1}{g(y)} {\int_0^S {\left| {{q_1}(a, t - \Gamma (s) + \Gamma (y)) - {q_2}(a, t - \Gamma (s) + \Gamma (y))} \right|{\rm d}a{\rm d}y} } {}\\ &\le &{L_1}\left( M \right)\int_0^s \frac{1}{g(y)} {{{\left\| {{q_1}(\cdot, t - \Gamma (s) + \Gamma (y)) - {q_2}(\cdot, t - \Gamma (s) + \Gamma (y))} \right\|}_{{L^1 }\left( {0, S} \right)}}{\rm d}y} {}\\ &\le &{L_1}\left( M \right)\int_{t - \Gamma (s)}^t \frac{1}{g(y)} {{{\left\| {{q_1}(\cdot, r) - {q_2}(\cdot, r)} \right\|}_{{L^1 }\left( {0, S} \right)}}{\rm d}r} {}\\ &\le &{L_1}\left( M \right)\int_0^t \frac{1}{g(y)} {{{\left\| {{q_1}( \cdot , r) - {q_2}( \cdot , r)} \right\|}_{{L^1 }\left( {0, S} \right)}}{\rm d}r} . \end{eqnarray} $

对任意的$ {q_1}, {q_2} \in H $, 利用(3.1)–(3.4)式可得($ C_1:=m_{1}e^{m_2}+\bar{u} $为$ b(t) $的上界, $ m_1 $和$ m_2 $均为由模型参数确定的常数)

(3.5) $ \begin{eqnarray} && \| \psi {q_1} (\cdot , t) - \psi {q_2} (\cdot , t)\|_{L^1(0, S)} = \| p( \cdot , t;{q_1}) - p( \cdot , t;{q_2})\|_{L^1(0, S)}{}\\ &=& \int_0^{\Gamma ^{ - 1}(t)} \frac{1}{g(s)} b(t - \Gamma (s))|F( s, t;q_1 ) - F( s, t;q_2 )|{\rm d}s {}\\ && + \int^S_{\Gamma ^{ - 1}(t)} p_0(\Gamma ^{ - 1}( \Gamma (s) - t), 0)| K(s, t;{q_1}) - K(s, t;{q_2}) |{\rm d}s{}\\ &\le & \int_0^{\Gamma ^{ - 1}(t)} \frac{1}{ g(s)}{C_2}| F( s, t;{q_1} ) - F( s, t;{q_2})| {\rm d}s {}\\ && + \int_{\Gamma ^{ - 1}(t)}^S {M_0}\left| K(s, t;{q_1}) - K(s, t;{q_2}) \right|{\rm d}s{}\\ &\le & C_1 L_1(M)\Gamma (S)\int_0^t \parallel q_1(\cdot, r)-q_2(\cdot, r)\parallel_{L^1(0, S)}{}\\ & & +SM_0 L_1(M)\int_0^t \parallel q_1(\cdot, r)-q_2(\cdot, r)\parallel_{L^1(0, S)}{}\\ & =:& C\int_0^t \parallel q_1(\cdot, r)-q_2(\cdot, r)\parallel_{L^1(0, S)}, \end{eqnarray} $

其中$ C=L_1 (M)(C_1 +SM_0) $.

定理3.1  对于任意给定的$ u\in U $, 种群系统(2.1)在$ Q_T $上存在唯一的非负有界解$ p^u $.

证  取定$ \lambda > C $, 定义空间$ L^{\infty }( [0, \tau];{L^1}(0, S)) $上的等价范数如下

$ {\left\| q \right\|_*} = Ess{\sup\limits_{t \in \left( {0, \tau } \right)}}\left\{ {{e^{ - \lambda t}}{{\left\| {q\left( { \cdot , t} \right)} \right\|}_{{L^1}\left( {0, S} \right)}}} \right\}, \forall q \in H. $

利用(3.5)式可得

(3.6) $ \begin{eqnarray} {\left\| {\psi {q_1} - \psi {q_2}} \right\|_*} &=& Ess{\sup\limits_{t \in \left( {0, \tau} \right)}}\left\{ {{e^{ - \lambda t}}{{\left\| {\left( {\psi {q_1}} \right)\left( { \cdot , t} \right) - \left( {\psi {q_2}} \right)\left( { \cdot , t} \right)} \right\|}_{{L^1}\left( {0, S} \right)}}} \right\}{}\\ &\le & C\cdot Ess{\sup\limits_{t \in \left( {0, \tau } \right)}}\left\{ {{e^{ - \lambda t}}\int_0^t {{{\left\| {{q_1}\left( { \cdot , r} \right) - {q_2}\left( { \cdot , r} \right)} \right\|}_{{L^1}\left( {0, S} \right)}}{\rm d}r} } \right\}{}\\ & \le & C\cdot Ess{\sup\limits_{t \in \left( {0, \tau } \right)}}\left\{ {{e^{ - \lambda t}}\int_0^t {{e^{\lambda r}}{e^{ - \lambda r}}{{\left\| {{q_1}\left( { \cdot , r} \right) - {q_2}\left( { \cdot , r} \right)} \right\|}_{{L^1}\left( {0, S} \right)}}{\rm d}r} } \right\}{}\\ &\le & \frac{C}{\lambda }\left\| {{q_1} - {q_2}} \right\|_*. \end{eqnarray} $

再由$ 0 < \frac{C}{\lambda } < 1 $, 知$ \psi $是一个压缩映射, 故存在唯一的不动点$ {q^*} \in H $, 它就是系统(2.1)在$ [0, S]\times [0, \tau] $上的解.再由系统(2.1)和$ H $的定义知:系统(2.1)的解非负有界.重复以上推理可知:系统(2.1)在$ Q_T $上有唯一的非负有界解.

令$ {p_i} $为系统(2.1)相应于控制变量$ {u_i}\left( t \right) $的解($ i=1, 2 $).在特征线$ \Gamma (s) - t = c $上, 其表示形式为

(4.1) $ \begin{equation} {p_i}(s, t) = \left\{ {\begin{array}{ll} { } {p_0}({\Gamma ^{ - 1}}\left( {\Gamma (s) - t} \right), 0){K}(s, t;{p_i}), & t \le \Gamma (s), \\ { } \frac{{{b_i}(t - \Gamma (s))}}{{g(s)}}{F}\left( {s, t;{p_i}} \right), & t > \Gamma (s), \end{array}} \right. \end{equation} $

其中

(4.2) $ \begin{eqnarray} K(s, t;{p_i}) &=&{\rm exp}\Big\{-\int_0^t [\mu( {\Gamma ^{ - 1}}( {\Gamma (s) - t + r}), E({p_i})({\Gamma ^{ - 1}}(\Gamma (s) - t + r), r)){}\\ &&+ g'({\Gamma ^{ - 1}}(\Gamma (s) - t + r)) ]{\rm d}r \Big\}, {}\\ {F}( {s, t;{p_i}})& =& {\rm exp}\Big\{ { - \int_0^s {\frac{{\mu ( {y, E({p_i})(y, t - \Gamma (s) + \Gamma (y))} )}}{{g(y)}}{\rm d}y} } \Big\}, {}\\ {b_i}(t)& =& g(0){p_i}(0, t). \end{eqnarray} $

由上节分析有以下结果:对于任意给定的边界控制$ u \in U $, 系统(2.1)的解$ p^u ( {s, t}) $满足

$ 0 \le {p^u}\left( {s, t} \right) \le {M_T}, \left( {s, t} \right) \in {Q_T}, $

其中$ {M_T} $是一个独立于$ u $的常数.

下面证明$ p^u $关于$ u $的一致连续性.

定理4.1  存在与$ {u_1} $, $ {u_2} $无关的非负常数$ M_1 $, 使得

$ \left\| {{p_1} - {p_2}} \right\|_{L^{\infty}([0, T], L^1 (0, S))} \le M_1 \left\|u_1 - u_2 \right\|_{L^{\infty }(0, T)}. $

证  当$ t \le \Gamma (s) $时, 由(4.1)–(4.2)式和假设$ (A_3) $可导出

(4.3) $ \begin{eqnarray} &&\left|p_1(s, t) - p_2(s, t)\right|{}\\ &=& \left|p_0(\Gamma ^{ - 1}(\Gamma (s) - t), 0){K}(s, t;{p_1}) - p_0(\Gamma ^{ - 1}(\Gamma (s) - t), 0){K}(s, t;{p_2})\right|{}\\ &\le & p_0 (\Gamma ^{ - 1}(\Gamma (s) - t), 0) \int_0^t \left| \mu (\Gamma ^{ - 1}(\Gamma (s) - t + r), E(p_1)(\Gamma ^{ - 1}(\Gamma(s) - t + r), r))\right. {}\\ &&\left. - \mu(\Gamma ^{ - 1}(\Gamma (s) - t + r), E(p)(\Gamma ^{ - 1}(\Gamma(s) - t + r), r)) \right|{\rm d}r {}\\ &\le & {M_0}{L_1}\left( M \right)\int_0^t {\left| {E\left( {{p_1}} \right)\left( {{\Gamma ^{ - 1}}(\Gamma (s) - t + r), r} \right) - E\left( {{p_2}} \right)\left( {{\Gamma ^{ - 1}}(\Gamma (s) - t + r), r} \right)} \right|{\rm d}r}{}\\ & \le & M_0 L_1( M )\int_0^t \int_0^S \left| p_1(y, r) - p_2(y, r) \right| {\rm d}y{\rm d}r{}\\ & \le & M_0 L_1( M )\int_0^t \left\| p_1 (\cdot , r) - p_2 (\cdot , r) \right\|_{L^1 (0, S)}{\rm d}r. \end{eqnarray} $

当$ t > \Gamma (s) $时, 由假设$ (A_2) $, (4.1)与(4.2)式可得

(4.4) $ \begin{eqnarray} &&\left| {{p_1}\left( {s, t} \right) - {p_2}\left( {s, t} \right)} \right|{}\\ &\le & \frac{1}{g\left( s \right)} \left| {{b_1}(t - \Gamma (s)){F}\left( {s, t;{p_1}} \right) - {b_2}(t - \Gamma (s)){F}\left( {s, t;{p_2}} \right)} \right|{}\\ &=& \frac{1} {g\left( s \right)}\left[ {\left| {{b_1}(t - \Gamma (s)) - {b_2}(t - \Gamma (s))} \right|{F}\left( {s, t;{p_1}} \right) + {b_2}(t - \Gamma (s))\left| {{F}\left( {s, t;{p_1}} \right) - {F}\left( {s, t;{p_2}} \right)} \right|} \right]{}\\ &\le & \frac{1} {g\left( s \right)}{M_g}\left[ {\left| {{b_1}(t - \Gamma (s)) - {b_2}(t - \Gamma (s))} \right| + {C_2}\left| {{F}\left( {s, t;{p_1}} \right) - {F}\left( {s, t;{p_2}} \right)} \right|} \right]; \end{eqnarray} $

根据(4.2)式可知

(4.5) $ \begin{eqnarray} &&\left| {{b_1}\left( x \right) - {b_2}\left( x \right)} \right|{}\\ &= &\left| {\int_0^S {\beta \left( {s, E\left( {{p_1}} \right)\left( {s, x - \tau } \right)} \right){p_1}\left( {s, x - \tau } \right){\rm d}s} + {u_1}\left( x \right)} \right.{}\\ &&\left. { - \int_0^S {\beta \left( {s, E\left( {{p_2}} \right)\left( {s, x - \tau } \right)} \right){p_2}\left( {s, x - \tau } \right){\rm d}s} - {u_2}\left( x \right)} \right|{}\\ &\le & \int_0^S {\left| {\beta \left( {s, E\left( {{p_1}} \right)\left( {s, x - \tau } \right)} \right) - \beta \left( {s, E\left( {{p_2}} \right)\left( {s, x - \tau } \right)} \right)} \right|{p_1}\left( {s, x - \tau } \right){\rm d}s} {}\\ && + \int_0^S {\beta \left( {s, E\left( {{p_2}} \right)\left( {s, x - \tau } \right)} \right)\left| {{p_1}\left( {s, x - \tau } \right) - {p_2}\left( {s, x - \tau } \right)} \right|{\rm d}s} + \left| {{u_1}\left( x \right) - {u_2}\left( x \right)} \right|{}\\ &\le & {M_T}{L_2}\left( M \right)\int_0^S {\left| {E\left( {{p_1}} \right)\left( {s, x - \tau } \right) - E\left( {{p_2}} \right)\left( {s, x - \tau } \right)} \right|{\rm d}s} {}\\ && + \bar \beta \int_0^S {\left| {{p_1}\left( {s, x - \tau } \right) - {p_2}\left( {s, x - \tau } \right)} \right|{\rm d}s} + \left| {{u_1}\left( x \right) - {u_2}\left( x \right)} \right|{}\\ &\le & [{M_T}{L_2}( M )+\bar{\beta}]\int_0^S \left| p_1(s, x-\tau) - p_2(s, x-\tau) \right|{\rm d}s+\|u_1 -u_2\|_{\infty}. \end{eqnarray} $

若令$ x = t - \Gamma (s) $, 则由(4.5)式知

(4.6) $ \begin{eqnarray} &&\left| {{b_1}\left(t-\Gamma(s) \right) - {b_2}\left(t-\Gamma(s) \right)} \right|{}\\ &\le & [\bar{\beta}+ {M_T}{L_2}\left( M \right)]\int_0^t {{{\left\| {{p_1}\left( {\cdot, t-\Gamma(s)-\tau} \right) - {p_2}\left( {\cdot, t-\Gamma(s)-\tau} \right)} \right\|}_{{L^1 }\left( {0, S} \right)}}{\rm d}r}. \end{eqnarray} $

此外, 由(4.2)式知

(4.7) $ \begin{eqnarray} &&\left| {{F}\left( {s, t;{p_1}} \right) - {F}\left( {s, t;{p_2}} \right)} \right|{}\\ &= &\left| {{\rm exp}\left\{ { - \int_0^s {\frac{{\mu \left( {y, E({p_1})(y, t - \Gamma (s) + \Gamma (y))} \right)}}{{g(y)}}{\rm d}y} } \right\}} \right.{}\\ &&\left. { - {\rm exp}\left\{ { - \int_0^s {\frac{{\mu \left( {y, E({p_2})(y, t - \Gamma (s) + \Gamma (y))} \right)}}{{g(y)}}{\rm d}y} } \right\}} \right|{}\\ &\le & \int_0^s {\frac{1}{g(y)} \left| {\mu \left( {y, E({p_1})(y, t - \Gamma (s) + \Gamma (y))} \right) - \mu \left( {y, E({p_2})(y, t - \Gamma (s) + \Gamma (y))} \right)} \right|{\rm d}y} {}\\ &\le & {L_1}\left( M \right)\int_0^s \frac{1}{g(y)} {\left| {E({p_1})(y, t - \Gamma (s) + \Gamma (y)) - E({p_2})(y, t - \Gamma (s) + \Gamma (y))} \right|{\rm d}y} {}\\ &\le & {L_1}\left( M \right)\int_0^s \frac{1}{g(y)} {\int_0^S {\left| {{p_1}(a, t - \Gamma (s) + \Gamma (y)) - {p_2}(a, t - \Gamma (s) + \Gamma (y))} \right|{\rm d}a{\rm d}y} } {}\\ &\le & {L_1}\left( M \right)S\int_0^s \frac{1}{g(y)} {{{\left\| {{p_1}(\cdot, t - \Gamma (s) + \Gamma (y)) - {p_2}(\cdot, t - \Gamma (s) + \Gamma (y))} \right\|}_{{L^1 }\left( {0, S} \right)}}{\rm d}y} . \end{eqnarray} $

再由(4.3)–(4.4)式和(4.6)–(4.7)式导出

(4.8) $ \begin{equation} {\left\| {{p_1}\left( {\cdot, t} \right) - {p_2}\left( {\cdot, t} \right)} \right\|_{{L^\infty }\left( {0, S} \right)}} \le {C_2}{\left\| {{u_1} - {u_2}} \right\|_{{L^1}\left( {0, T} \right)}} + {C_3}\int_0^t {{{\left\| {{p_1}\left( {\cdot, y} \right) - {p_2}\left( {\cdot, y} \right)} \right\|}_{{L^1 }\left( {0, S} \right)}}{\rm d}y} , \end{equation} $

其中$ C_2, C_3 $为与$ u $无关的常数.

最后, 由(4.8)式和Bellman不等式知:存在与$ {u_1} $, $ {u_2} $无关的常数$ M_1>0 $, 使得

$ \| p_1(\cdot, t) - p_2(\cdot, t)\|_{L^1 (0, S)} \le M_1 \| u_1 - u_2\|_{L^{\infty}(0, T)}, $

从而

$ \|p_1 - p_2 \|_{L^1 ( Q_T)} \le M_1\| u_1 - u_2\|_{L^{\infty}(0, T)}. $

证毕.

定理5.1  最优控制问题(2.1)–(2.2)的任一最优策略$ {u^*} $都具有如下结构

(5.1) $ \begin{equation} {u^*}\left( t \right) ={\cal F}\left( {\frac{{q(0, t)}}{\sigma }} \right) := \left\{ \begin{array}{ll} 0, & q(0, t) \le 0, \\ { } \frac{{q(0, t)}}{\sigma }, {\quad} & 0 < q(0, t) < \sigma \bar{u}, \\ \bar{u}, & q(0, t) \ge \sigma \bar{u}, \end{array} \right. \end{equation} $

其中$ q\left( {s, t} \right) $为下列共轭系统的解

(5.2) $ \begin{equation} \left\{ \begin{array}{ll} { }\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} \\ { } {\qquad}{\qquad} {\qquad}{\quad} + \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} \\ { } {\qquad}{\qquad}{\qquad} {\quad} + \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), \left( {s, t} \right) \in \left( {0, S} \right) \times \left[ {T, T + \tau } \right], \\ { } q\left( {S, t} \right) = 0, t \in [0, T+\tau], \end{array} \right. \end{equation} $

其中$ {{\beta _x}} $表示函数$ \beta $对第二个变量的导数.

证  令$ \left( {{u^*}, {p^*}} \right) $为控制问题(2.1)–(2.2)的最优对.利用反向时间变换, 系统(5.2)解的存在唯一性可以类似于第3节得到.对任意固定的$ v \in {{T}_U}\left( {{u^*}} \right) $ (表示集$ U $在$ u^* $处的切锥^{[21, p21, Proposition 2.3]}), 当$ \varepsilon > 0 $足够小时, $ {u^\varepsilon }: = {u^*} + \varepsilon v \in U $.

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*} &&\frac{1}{2}\int_0^S {{{\left[ {{p^\varepsilon }\left( {s, T} \right) - \overline p \left( s \right)} \right]}^2}{\rm d}s} +\frac{\sigma}{2} \int_0^T {{{\left( {{u^\varepsilon }\left( t \right)} \right)}^2}{\rm d}t}\\ & \ge &\frac{1}{2} \int_0^S {{{\left[ {{p^*}\left( {s, T} \right) - \overline p \left( s \right)} \right]}^2}{\rm d}s} +\frac{\sigma}{2} \int_0^T {{{\left( {{u^*}\left( t \right)} \right)}^2}{\rm d}t} , \end{eqnarray*} $

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

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

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5.1给出.先假定该极限存在, 推导$ z\left( {s, t} \right) $满足的系统.

因为$ p^\varepsilon $和$ p^* $分别是系统(2.1)相应于$ u^\varepsilon $和$ u^* $的解, 可知$ z(s, t) $满足

(5.4) $ \begin{equation} \left\{ \begin{array}{ll} { } \frac{{\partial z}}{{\partial t}} + \frac{{\partial (gz)}}{{\partial s}} = - {\mu'_1}\left( {{E}(p^*)} \right){p^*}\left( {s, t} \right)E(z)\left( {s, t} \right) - \left[ {{\mu _0}\left( s \right) + {\mu _1}\left( {{E}(p^*)} \right)} \right]z\left( {s, t} \right), \\ {\quad}{\qquad}{\qquad}{\qquad} \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.} \\ { } {\qquad}{\qquad}{\qquad}\left. { + \beta (s, {E}(p^*)(s, t - \tau ))z(s, t - \tau )} \right]{\rm d}s + v\left( t \right), t \in (0, T), \\ { } z\left( {s, t} \right) = 0, \left( {s, t} \right) \in \left( {0, S} \right) \times \left[ { - \tau , 0} \right]. \end{array} \right. \end{equation} $

将系统(5.4)的第一式乘以$ q\left( {s, t} \right) $, 并在$ {{Q_T}} $上积分, 可得

(5.5) $ \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} }.{\qquad} \end{eqnarray} $

利用(5.2)式可知

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

然后, 由(5.5)–(5.6)式推知

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

由共轭系统(5.2)中的第一个方程可得

(5.7) $ \begin{equation} \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^T {q(0, t)v\left( t \right){\rm d}t}. \end{equation} $

将(5.7)式代入(5.3)式中, 可知:对任意的$ v \in {T _U}\left( {{u^*}} \right) $有

$ \int_0^T {\left[ {q(0, t) - \sigma {u^*}\left( t \right)} \right]v\left( t \right){\rm d}t} \le 0. $

根据法锥的定义^{[21, p20, (2.8)式]}知

$ q(0, \cdot ) - \sigma {u^*} \in {N_U}\left( {{u^*}} \right), $

其中$ {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, p14, (1.38)式]}, 即得定理5.1的结论.

引理5.1  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

(5.8) $ \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, x}}\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} $

其中$ E\left( {\frac{1}{\varepsilon }\left[ {{p^\varepsilon } - {p^*}} \right]} \right)\left( {s, t} \right) = \alpha \int_0^s {\frac{1}{\varepsilon }\left[ {{p^\varepsilon } - {p^*}} \right]} \left( {r, t} \right){\rm d}r + \int_s^S {\frac{1}{\varepsilon }\left[ {{p^\varepsilon } - {p^*}} \right]} \left( {r, t} \right){\rm d}r $;

此外

(5.9) $ \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} - v\left( t \right)+{b_0}\left( \varepsilon \right), \end{eqnarray} $

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

应用(2.1), (5.4)和(5.8)–(5.9)式导出

(5.10) $ \begin{equation} \left\{ \begin{array}{ll} { } \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) \\ {\qquad}{\quad}\ \, {\qquad}{\qquad}{\qquad} - {{\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} \\ {\quad} {\qquad}{\qquad}{\qquad} { } + \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} \\ {\qquad}{\qquad}{\qquad} {\quad} { } + \int_0^S {\left[ {{\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)} \right.} \\ {\qquad}{\qquad}{\qquad}{\quad} + {b_0}\left( \varepsilon \right), t \in (0, T), \\ {w_\varepsilon }\left( {s, t} \right) = 0, \left( {s, t} \right) \in \left( {0, S} \right) \times \left[ { - \tau , 0} \right]. \end{array} \right. \end{equation} $

对系统(5.10)中的前两式右端除首项外的其余部分取极限$ {\varepsilon \to {0^ + }} $, 应用定理4.1获得极限系统如下

(5.11) $ \begin{equation} \left\{ \begin{array}{ll} { } \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)} \\ {\qquad}{\qquad}{\qquad} {\qquad}{\quad} { } -{{{\mu '}_1}\left( {E({p^*})} \right)E(w)\left( {s, t} \right){p^*}\left( {s, t} \right)}, (s, t)\in Q_T\\ 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} } \\ {\qquad}{\qquad}{\qquad}\ \ { } + \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 (0, T), \\ {w}\left( {s, t} \right) = 0, \left( {s, t} \right) \in \left( {0, S} \right) \times \left[ { - \tau , 0} \right], \end{array} \right. \end{equation} $

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^ + }. $

注意系统(5.11)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 $, 这意味着下列极限存在

$ \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). $

引理证毕.

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

$ \varphi \left( u \right) = \left\{ \begin{array}{ll} { } \frac{1}{2}\int_0^S {{{\left[ {{p^u}\left( {s, T} \right) - \bar p\left( s \right)} \right]}^2}{\rm d}s} +\frac{\sigma}{2} \int_0^T {{u^2}\left( t \right){\rm d}t} , & u \in U, \\ + \infty , & u \notin U. \end{array} \right. $

引理6.1  泛函$ \varphi $下半连续.

证  设$ \left\{ {{u_n}} \right\} \subset U $为$ {L^{\rm{1}}}\left( {0, T} \right) $中的任一序列, 当$ 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 $.据定理4.1知, 对任意$ t \in \left( {0, T} \right) $, 当$ n\rightarrow \infty $时有

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

由强收敛与逐点收敛的关系知:存在$ \{u_n\} $的一个子序列(仍记为$ \left\{ {{u_n}} \right\} $), 使得:当$ n\to \infty $时

$ {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$ Q_T $上几乎处处成立.从而

$ \int_0^S {{{\left[ {{p^{{u_n}}}\left( {s, T} \right) - \bar p\left( s \right)} \right]}^2}{\rm d}s} + \sigma \int_0^T {{{\left( {{u_n}\left( t \right)} \right)}^2}{\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^T {{u^2}\left( t \right){\rm d}t}. $

利用Fatou引理^{[22, p146, Theorem 3.20]}, 得到

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

由此知$ \varphi $下半连续, 引理证毕.

引理6.2^[23]  如果$ {w} \in {L^{\rm{1}}}\left( {0, T} \right) $满足

$ \int_0^T {\left[ {w\left( t \right)v\left( t \right) + \alpha \left| {v\left( t \right)} \right|} \right]{\rm d}t \ge 0, \forall v \in {T_U}\left( u \right)} , $

则存在$ \theta \in {L^\infty }\left( {0, T} \right) $, 使得$ {\left\| \theta \right\|_{{L^\infty }\left( {0, T} \right)}} \le 1 $, 且$ - w + \alpha \theta \in {N_U}\left( u \right) $.

运用特征线方法可得下列结果(细节略去).

引理6.3  对于共轭系统(5.2), 存在一个固定的常数$ M_2 > 0 $, 使得:若$ \left\| p_0 \right\|_{L^{1}((0, S)\times [-\tau, 0])} \le M_2 $, 则对任意的$ t \in \left( {0, T} \right), u, v\in U $, 都有

$ \left\|q^u( \cdot , t) - q^v( \cdot , t) \right\|_{L^{\infty }(0, S)} + \left|q^u ( {0, t} ) - {q^v}\left( {0, t} \right) \right| \le C\left( {M_2, T} \right)\left\| {u - v} \right\|_{\infty }, $

其中$ C\left( {M_2, T} \right) $为$ M_2 $和$ T $所构成的正的常数, 它与$ u, v $无关.

定理6.1  若$ \sigma^{-1}{C}\left( {M_2, T} \right)<1 $, 则控制问题(2.1)–(2.2)存在唯一的解.

证  根据Ekeland变分原理^{[24, p180, Theorem A1.3.1]}知, 对任意给定的$ \varepsilon > 0 $, 存在$ {u_\varepsilon } \in U $, 使得

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

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

利用不等式(6.2)和定理5.1的证明方法可知:对任一$ v \in {{T}_U}\left( {{u_{\varepsilon}}} \right) $, 都有

$ \int_0^T {\left[ {\sigma {u_\varepsilon }\left( t \right) - {q^{{u_\varepsilon }}}\left( {0, t} \right)} \right]v\left( t \right){\rm d}t} + \sqrt \varepsilon \int_0^T {\left| {v\left( t \right)} \right|{\rm d}t} \ge 0. $

利用引理6.2导出:存在函数$ {f_\varepsilon } \in {L^\infty }\left( {0, T} \right), {\left\| {{f_\varepsilon }} \right\|_{{L^\infty }\left( {0, T} \right)}} \le 1 $, 使得

$ - \sigma {u_\varepsilon } + {q^{{u_\varepsilon }}}\left( {0, \cdot} \right) + \sqrt \varepsilon {f_\varepsilon } \in {N_U}\left( {{u_\varepsilon }} \right). $

因此

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

首先证明最优控制的唯一性.定义映射

(6.4) $ \begin{equation} \psi :{L^\infty }\left( {0, T} \right) \to U, \left( {\psi u} \right)\left( t \right) = {\cal F}\left[ {{\sigma ^{ - 1}}{q^u}\left( {0, t} \right)} \right], t \in \left( {0, T} \right). \end{equation} $

由(6.4)式, 引理6.2与引理6.3可知

$ \begin{eqnarray*} \left| {\left( {\psi u} \right)\left( t \right) - \left( {\psi v} \right)\left( t \right)} \right| &=& \left| { {\cal F}\left[ {{\sigma ^{ - 1}}{q^u}\left( {0, t} \right)} \right] - {\cal F}\left[ {{\sigma ^{ - 1}}{q^v}\left( {0, t} \right)} \right]} \right|\\ & \le& {\sigma ^{ - 1}}\left| {{q^u}\left( {0, t} \right) - {q^v}\left( {0, t} \right)} \right|\\ &=&\sigma^{-1} C( {M_2, T})\left\| {u - v} \right\|_{\infty }. \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 $.定理5.1意味着控制问题(2.1)–(2.2)的任一解都是$ \psi $的不动点.所以最优解至多有一个.

再证$ {u_0} $必为问题(2.1)–(2.2)的解.应用(6.3)–(6.4)式, 可得

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

因此

$ \begin{eqnarray*} \left\|u_{\varepsilon } -u_0 \right\|_{\infty } & =&\left\| u_{\varepsilon } - \psi {u_{0}} \right\|_{\infty }\\ &\le & \left\|u_{\varepsilon } - \psi {u_\varepsilon } \right\|_{\infty } + \left\| \psi {u_\varepsilon } - \psi {u_0} \right\|_{\infty }\\ &\le & \sigma^{-1}{C}(M, T)\left\| {u_\varepsilon } - {u_0} \right\|_{\infty } + \sigma ^{ - 1}\sqrt{ \varepsilon}. \end{eqnarray*} $

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

在(6.1)式中取极限并利用引理6.1, 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\}. $

定理证毕.

从控制理论观点看, 问题(2.1)–(2.2)是一类新的无穷维系统最优控制问题.定理6.1表明:最优解有且只有一个.定理5.1则对该解作了精细刻画, 利用共轭变量建立了反馈控制律.从生态实践观点看, 问题(2.1)–(2.2)旨在通过幼体投放优化种群分布.运用前文所得结果(5.1)–(5.2)和数值方法可以计算出具体的投放策略.