## Optimal Boundary Control for a Hierarchical Size-Structured Population Model with Delay

He Zerong,, Dou Yimeng, Han Mengjie

Institute of Operational Research and Cybernetics. Hangzhou Dianzi University, Hangzhou 310018

 基金资助: 国家自然科学基金.  11871185

 Fund supported: the NSFC.  11871185

Abstract

In this article, we formulate a population control model, which is based upon the hierarchical size-structure and the incubation delay. For a given ideal population distribution, we investigate the optimal input problem: How to choose a inflow way such that the sum of the deviation between the terminal state and the given one and the total costs is minimal. The well-posedness is established by the method of characteristics, the existence of unique optimal policy is shown by the Ekeland variational principle, and the optimal policy is exactly described by a normal cone and an adjoint system. These results set a foundational framework for practical applications.

Keywords： Hierarchy of body size ; Delay ; Optimal control ; Normal cone ; Variational principle

He Zerong, Dou Yimeng, Han Mengjie. Optimal Boundary Control for a Hierarchical Size-Structured Population Model with Delay. Acta Mathematica Scientia[J], 2022, 42(3): 867-880 doi:

## 2 系统模型与控制问题

$$$\left\{ \begin{array}{ll} { } \frac{{\partial p}}{{\partial t}} + \frac{{\partial (g(s)p)}}{{\partial s}} = - \left[ {{\mu _{\rm{0}}}\left( s \right) + {\mu _1}(E(p)(s, t))} \right]p(s, t), &(s, t) \in {Q_T}, \\ { } g(0)p(0, t) = \int_0^S {\beta (s, E(p)(s, t - \tau ))p(s, t - \tau ){\rm d}s} + u\left( t \right), & t \in (0, T), \\ { } p(s, t) = {p_0}(s, t), s \in [0, S], & t \in [ - \tau , 0], \\ { } E(p)(s, t) = \alpha \int_0^s {p(r, t){\rm d}r} + \int_s^S {p(r, t){\rm d}r, } &(s, t) \in {Q_T}, \end{array} \right.$$$

$(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) $${\beta}( \cdot , y) 关于 y 满足局部Lipschitz条件, 即:对任意常数 M>0 , 存在 {L_1 (M)}, {L_2 (M)} 使得:当 \left| {{y_1} - {y_2}} \right|\leq M 时有 \left| {\mu_1 ( {y_1}) - \mu_1 ({y_2})} \right| \le {L_1 (M)}\left| {{y_1} - {y_2}} \right| , \left| {\beta ( \cdot , {y_1}) - \beta ( \cdot , {y_2})} \right| \le {L_2 (M)}\left| {{y_1} - {y_2}} \right| ; (A_3)$$ 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.

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

## 3 模型解的存在唯一性

$\mu_1$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

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

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

${b}(t) = g(0)p(0, t;q)$为已知.

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

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

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

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

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

## 4 解关于控制变量的连续性

${p_i}$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

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

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

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

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

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

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

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

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

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

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

## 5 最小值原理

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

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

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 $$u^* 处的切锥[21, p21, Proposition 2.3]), 当 \varepsilon > 0 足够小时, {u^\varepsilon }: = {u^*} + \varepsilon v \in U . {p^\varepsilon } 为系统(2.1)相应于 u = {u^\varepsilon } 的解.由 {{u^*}} 的最优性知 将上式变形, 并令 \varepsilon \to {0^ + } , 可得 $$\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,$$ 其中 z\left( {s, t} \right) = \mathop {\lim }\limits_{\varepsilon \to {0^ + }} {\varepsilon ^{ - 1}}\left[ {{p^\varepsilon }\left( {s, t} \right) - {p^*}\left( {s, t} \right)} \right]. 上列极限的存在性由后面的引理5.1给出.先假定该极限存在, 推导 z\left( {s, t} \right) 满足的系统. 因为 p^\varepsilon$$ p^*$分别是系统(2.1)相应于$u^\varepsilon $$u^* 的解, 可知 z(s, t) 满足 $$\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.$$ 将系统(5.4)的第一式乘以 q\left( {s, t} \right) , 并在 {{Q_T}} 上积分, 可得 \begin{eqnarray} &&\int_0^S {\int_0^T {\left( {\frac{{\partial z}}{{\partial t}} + \frac{{\partial (gz)}}{{\partial s}}} \right)q\left( {s, t} \right){\rm d}s{\rm d}t} } {}\\ &=& - \int_0^S {\int_0^T {\left\{ {{\mu' _1}\left( {{E}(p^*)} \right)E(z)\left( {s, t} \right){p^*}\left( {s, t} \right) + \left[ {{\mu _0}\left( s \right) + {\mu _1}\left( {{E}(p^*)} \right)} \right]z\left( {s, t} \right)} \right\}q\left( {s, t} \right){\rm d}s{\rm d}t} }.{\qquad} \end{eqnarray} 利用(5.2)式可知 \begin{eqnarray} &&\int_0^S {\int_0^T {\frac{{\partial z}}{{\partial t}}q\left( {s, t} \right){\rm d}s{\rm d}t} } = \int_0^S {\left[ {\bar p\left( s \right) - {p^*}\left( {s, T} \right)} \right]z\left( {s, T} \right){\rm d}s} - \int_0^S {\int_0^T {\frac{{\partial q}}{{\partial t}}z\left( {s, t} \right){\rm d}s{\rm d}t} } , {}\\ &&\int_0^S {\int_0^T {\frac{{\partial \left( {gz} \right)}}{{\partial s}}q\left( {s, t} \right){\rm d}s{\rm d}t} } {}\\ &=& - \int_0^T {g(0)z(0, t)q(0, t){\rm d}t} - \int_0^S {\int_0^T {g\left( s \right)z\left( {s, t} \right)\frac{{\partial q}}{{\partial s}}{\rm d}s{\rm d}t} } {}\\ &=& - \int_0^T q(0, t)\Big\{ \int_0^S \left[ \beta _x(s, E(p^*)(s, t - \tau ))E( z )(s, t - \tau)p^*(s, t - \tau ) \right. {}\\ && \left. + \beta(s, E(p^*)(s, t-\tau ))z(s, t-\tau) \right]{\rm d}s + v( t) \Big\}{\rm d}t - \int_0^S \int_0^T g( s)z(s, t)\frac{\partial q}{\partial s}{\rm d}s{\rm d}t. \end{eqnarray} 然后, 由(5.5)–(5.6)式推知 由共轭系统(5.2)中的第一个方程可得 $$\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}.$$ 将(5.7)式代入(5.3)式中, 可知:对任意的 v \in {T _U}\left( {{u^*}} \right) 根据法锥的定义[21, p20, (2.8)式] 其中 {N_U}\left( {{u^*}} \right) 表示集 U$$ {{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.

记

$\begin{eqnarray} & &\frac{{\partial \left\{ {\frac{1}{\varepsilon }\left[ {{p^\varepsilon }\left( {s, t} \right) - {p^*}\left( {s, t} \right)} \right]} \right\}}}{{\partial t}} + \frac{{\partial \left\{ {\frac{1}{\varepsilon }g\left( s \right)\left[ {{p^\varepsilon }\left( {s, t} \right) - {p^*}\left( {s, t} \right)} \right]} \right\}}}{{\partial s}}{}\\ & =& \frac{1}{\varepsilon }\left\{ {\frac{{\partial {p^\varepsilon }\left( {s, t} \right)}}{{\partial t}} + \frac{{\partial \left[ {g\left( s \right){p^\varepsilon }\left( {s, t} \right)} \right]}}{{\partial s}}} \right\} - \frac{1}{\varepsilon }\left\{ {\frac{{\partial {p^*}\left( {s, t} \right)}}{{\partial t}} + \frac{{\partial \left[ {g\left( s \right){p^*}\left( {s, t} \right)} \right]}}{{\partial s}}} \right\}{}\\ & =& - {\mu _{1, x}}\left( {E\left( {{p^*}} \right)} \right)E\left( {\frac{1}{\varepsilon }\left[ {{p^\varepsilon } - {p^*}} \right]} \right)\left( {s, t} \right){p^\varepsilon }\left( {s, t} \right){}\\ && - {\mu _1}\left( {E\left( {{p^*}} \right)\left( {s, t} \right)} \right)\frac{1}{\varepsilon }\left[ {{p^\varepsilon } - {p^*}} \right]\left( {s, t} \right) - {\mu _0}\left( s \right)\frac{1}{\varepsilon }\left[ {{p^\varepsilon } - {p^*}} \right]\left( {s, t} \right), \end{eqnarray}$

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

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

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

## 6 最优边界控制的存在唯一性

设$\left\{ {{u_n}} \right\} \subset U $${L^{\rm{1}}}\left( {0, T} \right) 中的任一序列, 当 n \to \infty 时, {{u_n} \to u} .不失一般性, 可以假定 {{u_n} \in U} , \forall n \ge 1 .据定理4.1知, 对任意 t \in \left( {0, T} \right) , 当 n\rightarrow \infty 时有 由强收敛与逐点收敛的关系知:存在 \{u_n\} 的一个子序列(仍记为 \left\{ {{u_n}} \right\} ), 使得:当 n\to \infty Q_T 上几乎处处成立.从而 利用Fatou引理[22, p146, Theorem 3.20], 得到 由此知 \varphi 下半连续, 引理证毕. 引理6.2[23] 如果 {w} \in {L^{\rm{1}}}\left( {0, T} \right) 满足 则存在 \theta \in {L^\infty }\left( {0, T} \right) , 使得 {\left\| \theta \right\|_{{L^\infty }\left( {0, T} \right)}} \le 1 , 且 - w + \alpha \theta \in {N_U}\left( u \right) . 运用特征线方法可得下列结果(细节略去). 引理6.3 对于共轭系统(5.2), 存在一个固定的常数 M_2 > 0 , 使得:若 \left\| p_0 \right\|_{L^{1}((0, S)\times [-\tau, 0])} \le M_2 , 则对任意的 t \in \left( {0, T} \right), u, v\in U , 都有 其中 C\left( {M_2, T} \right)$$ M_2$$T$所构成的正的常数, 它与$u, v$无关.

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

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

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

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

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

## 参考文献 原文顺序 文献年度倒序 文中引用次数倒序 被引期刊影响因子

Dewsbury D A .

Dominance rank, copulatory behavior, and differential reproduction

The Quarterly Review of Biology, 1982, 57 (2): 135- 159

Henson S M , Cushing J M .

Hierarchical models of intra-specific competition: scramble versus contest

J Math Biol, 1996, 34 (7): 755- 772

Jang R J , Cushing J M .

A discrete hierarchical model of intra-specific competition

J Math Anal Appl, 2003, 280 (1): 102- 122

Gurney W S C , Nisbet R M .

Ecological stability and social hierarchy

Theoretical Population Biology, 1979, 16, 48- 80

Cushing J M .

A size-structured model for cannibalism

Theoretical Population Biology, 1992, 42, 347- 361

Cushing J M .

The dynamics of hierarchical age-structured populations

J Math Biol, 1994, 32, 705- 729

Cushing J M , Li J .

Oscilations caused by cannibalism in a size-structured population model

Canadian Applied Mathematics Quarterly, 1995, 3 (2): 155- 172

Calsina Á , Salda$\ddot{\rm n}$a J .

Asymptotic behavior of a model of hierarchically structured population dynamics

J Math Biol, 1997, 35, 967- 987

Kraev E A .

Existence and uniqueness for height structured hierarchical population models

Natural Resources Modeling, 2001, 14 (1): 45- 70

Ackleh A S , Deng K .

Monotone approximation for a hierarchical age-structured population model

Dynamics of Continuous, Discrete and Impulsive Systems, 2005, 12, 203- 214

Ackleh A S , Deng K , Thibodeaux J J .

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

Journal of Biological Dynamics, 2007, 1 (4): 305- 319

Ackleh A S , Deng K , Hu S .

A quasilinear hierarchical size-structured model: well-posedness and application

Appl Math Optim, 2005, 51, 35- 59

Shen J , Shu C W , Zhang M .

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

J Sci Comput, 2007, 33, 279- 291

Farkas J Z , Hinow P .

Steady states in hierarchical structured populations with distributed states at birth

Discrete and Continuous Dynamical Systems, 2012, 17 (8): 2671- 2689

Liu Y , He Z .

On the well-posedness of a nonlinear hierachical size-structured population model

The ANZIAM Journal, 2017, 58 (3/4): 482- 490

He Z , Ni D , Liu Y .

Theory and approximation of solutions to a harvested hierarchical age-structured population model

Journal of Applied Analysis and Computation, 2018, 8 (5): 1326- 1341

He Z , Zhang Z , Qiu Z .

Numerical method of a nonlinear hierarchical age-structured population model

Acta Mathematica Scientia, 2020, 40A (2): 515- 526

He Z , Ni D , Wang S .

Control problem for a class of hierarchical population system

Journal of Systems Science and Mathematical Sciences, 2018, 38 (10): 1140- 1148

He Z , Ni D , Wang S .

Optimal harvesting of a hierarchical age-structured population system

International Journal of Biomathematics, 2019, 12 (8): 1950091

He Z , Zhou N .

Controllability and stabilization of a nonlinear hierarchical age-structured competing system

Electronic Journal of Differential Equations, 2020, 2020 (58): 1- 16

Barbu V . Mathematical Methods in Optimization of Differential Systems. Boston: Kluwer Academic Publishers, 1994

McDonald J N , Weiss N A . A Course in Real Analysis. Singapore: Elsevier, 2004

Barbu V , Iannelli M .

Optimal control of population dynamics

J Optim Theo Appl, 102, 1- 14

Aniţa S . Analysis and Control of Age-Dependent Population Dynamics. Dordrecht: Kluwer Academic Publishers, 2000

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