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

He Zerong,, Han Mengjie

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

 Fund supported: the NSFC.  11871185the NSF of Zhejiang Province.  LY18A010010

Abstract

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

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

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

## 2 模型与问题

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

(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) $$0 \le \beta( s, x) \le \bar{ \beta }, \forall ( s, x) \in [ 0, S] \times {R_ + } , \beta( \cdot, x) 关于 x 非增, 且满足Lipschitz条件; (A _4) 容许控制集为 其中 \bar{u} 为正常数. 本文主题是分析以下最优控制问题 \begin{eqnarray} \mathop {\min}\limits_{u \in U} J\left( u \right)\mathop = \limits^\Delta \int_0^S {{{\left[ {{p^u}\left( {s, T} \right) - \bar p\left( s \right)} \right]}^2}{\rm d}s} + \sigma \int_0^S {\int_{ - \tau }^0 {{u^2}\left( {s, t} \right){\rm d}s{\rm d}t} } , \end{eqnarray} 其中非负有界函数 {\overline p \left( s \right)} 表示某种理想分布(例如平衡态), \sigma > 0 为单位控制成本因子. 因此 J\left( u \right) 表示种群终态分布与理想分布之间的均方差以及控制总成本之和. 文献[20] 对系统(2.1) 做了初步分析, 获得下列结果. 定理2.1 在假设(A _1) -(A _4) 下, 系统(2.1) 存在唯一的非负有界解 p(s, t) . ## 3 最小值原理 首先证明状态系统的解关于控制变量的连续依赖性. 引理3.1 系统(2.1) 的解 p^u\in L^{\infty}(Q_T) 关于 u\in U 一致连续. 将系统演变时域 [0, T] 划分为区间 [0, \tau], [\tau, 2\tau], \cdots, [k\tau, T] , 逐段进行处理. t\in [0, \tau] 时, t-\tau\in [-\tau, 0], p(s, t-\tau) = u(s, t-\tau) . 状态系统(2.1) 关于尺度的边界条件变为已知, 记为 b(t) = g(0)p(0, t) . 利用定理2.1可得 \begin{eqnarray} p(s, t) = \left\{ \begin{array}{ll} \frac{b(t-\Gamma(s))}{g(s)}\exp\left\{-\int_0^s \frac{\mu(r, t-\Gamma(s)+\Gamma(r))}{g(r)}{\rm d}r \right\}, & t>\Gamma(s);\\ u(\Gamma^{-1}(\Gamma(s)-t), 0)\exp\bigg\{-\int_0^t \left[\mu(\Gamma^{-1}(r+\Gamma(s)-t), r) \right.& \\ \left.+g'(\Gamma^{-1}(r+\Gamma(s)-t))\right]{\rm d}r\bigg\}, & t\leq \Gamma(s), \end{array} \right. \end{eqnarray} 其中 \mu(s, t) = \mu_0(s)+\mu_1 (E(p)(s, t)) . 下面估计 p^u$$ 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$b^i, \mu^i$.$b(t)$的上界为$B$; $\mu_1$以及$\beta(s, x)$关于$x$的Lipschitz常数为$L$.

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

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

$t\leq\Gamma(s)$时, 注意函数$\mu_0, g $$u 无关, 由(3.1) 可得: \begin{eqnarray} && |p^1(s, t)-p^2(s, t) |\\ &\leq &\left| u_1(\Gamma^{-1}(\Gamma(s)-t), 0)\exp\left\{-\int_0^t \mu_1(E(p^1)(\Gamma^{-1}(r+\Gamma(s)-t), r)){\rm d}r \right\}\right.\\ &&\left. -u_2(\Gamma^{-1}(\Gamma(s)-t), 0)\exp\left\{-\int_0^t \mu_1(E(p^2)(\Gamma^{-1}(r+\Gamma(s)-t), r)){\rm d}r \right\}\right|\\ &\leq & \bar{u}\int_0^t |\mu_1(E(p^1)(\Gamma^{-1}(r+\Gamma(s)-t), r))-\mu_1(E(p^2)(\Gamma^{-1}(r+\Gamma(s)-t), r))|{\rm d}r\\ &&+|u_1(\Gamma^{-1}(\Gamma(s)-t), 0)-u_2(\Gamma^{-1}(\Gamma(s)-t), 0)|\\ &\leq & \|u_1 -u_2\|+C_3 \int_0^t \|p^1(\cdot, \theta)-p^2(\cdot, \theta)\|{\rm d}\theta, \end{eqnarray} 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 t\in [\tau, T] 时, 重复以上过程可得类似的估计结果. 引理证毕. 其次给出最优控制策略的精确刻画. 定理3.1 控制问题(2.1)-(2.2) 的任一最优对 (u^*, p^*) 都有如下结构 \begin{eqnarray} \begin{array}{l} {u^*}\left( {s, t} \right) = {\cal F}(\sigma^{-1}q(0, t+\tau)\tilde{E}(s, t)), ( s, t)\in [0, S]\times [-\tau, 0], \end{array} \end{eqnarray} 其中 {\cal F} 为0和 \bar{u} 之间的截断函数, q\left( {s, t} \right) 为下列共轭系统的解 \begin{eqnarray} \begin{array}{l} \left\{ \begin{array}{l} \frac{{\partial q}}{{\partial t}}{\rm{ + }}g\left( s \right)\frac{{\partial q}}{{\partial s}} = \alpha \int_s^S {\left[ {{\mu' _1}\left( {{E}(p^*)} \right)q\left( {r, t} \right) - {\beta _x}(r, {E}(p^*))q(0, t + \tau )} \right]{p^*}\left( {r, t} \right){\rm d}r} \\ + \int_0^s {\left[ {{\mu' _1}\left( {{E}(p^*)} \right)q\left( {r, t} \right) - {\beta _x}(r, {E}(p^*))q(0, t + \tau )} \right]{p^*}\left( {r, t} \right){\rm d}r} \\ + \left[ {{\mu _0}\left( s \right) + {\mu _1}\left( {{E}(p^*)} \right)} \right]q\left( {s, t} \right)- \beta (s, {E}(p^*))q(0, t + \tau ), \left( {s, t} \right) \in {Q_T}, \\ q\left( {s, T} \right) = \bar p\left( s \right) - {p^*}\left( {s, T} \right), s \in [0, S], \\ q\left( {s, t} \right) = 0, (s, t) \in [0, S] \times [T, T+\tau]. \end{array} \right. \end{array} \end{eqnarray} 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$$ {{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]).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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