### 数学代写|数学分析代写Mathematical Analysis代考|MATH2050C

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|数学分析代写Mathematical Analysis代考|Optimal Control Problem

On a given time interval $[0, T]$ we consider the nonlinear control system of differential equations:
$$\left{\begin{array}{l} l^{\prime}(t)=\sigma-\delta v(t) l(t) m(t)-\gamma_{1} l(t) k(t)-\mu l(t) \ k^{\prime}(t)=(\beta+\delta) v(t) l(t) m(t)+\gamma_{2} l(t) k(t)-\lambda k(t) \ m^{\prime}(t)=\rho-\beta v(t) l(t) m(t)-v m(t) \ l(0)=l_{0}, k(0)=k_{0}, m(0)=m_{0} ; l_{0}, k_{0}, m_{0}>0 \end{array}\right.$$

It describes the interactions of various types of cells in a human body with drug therapy of psoriasis $[3,10,11]$. In system $(1), l(t), k(t)$, and $m(t)$ are the concentrations of T-lymphocytes, keratinocytes and dendritic cells; $l_{0}, k_{0}, m_{0}$ are their initial conditions, respectively. The values $\sigma, \rho, \mu, \lambda, v, \gamma_{1}, \gamma_{2}, \delta, \beta$ are the given positive parameters of this system, which have the following meaning. The values $\sigma$ and $\rho$ are the appropriate inflow rates of T-lymphocytes and dendritic cells, $\mu$ and $v$ are the removal rates of these cells, respectively; $\lambda$ is the decay rate of keratinocytes. In addition, the rate of activation of keratinocytes due to T-lymphocytes is indicated by $\gamma_{1}$ and the rate of keratinocytes growth is denoted by $\gamma_{2}$. The value $\delta$ is the activation rate of T-lymphocytes by dendritic cells, $\beta$ is conversely the activation rate of dendritic cells due to T-lymphocytes. The interactions between T-lymphocytes and dendritic cells help to form keratinocytes through some cell biological procedures and thus the concentrations of both T-lymphocytes and dendritic cells are reduced by the terms $\delta v l m$ and $\beta v l m$, respectively. On the other hand, under mixing homogeneity, the combined interaction of T-lymphocytes and dendritic cells contributes to the growth of concentration of epidermal keratinocytes by the term $(\beta+\delta) \mathrm{vlm}$. Model (1) was kindly provided for analysis by Professor P. K. Roy (Centre for Mathematical Biology and Ecology, Department of Mathematics, Jadavpur University, Kolkata, India).
In system $(1), v(t)$ is a control function that satisfies the constraints:
$$0<v_{\min } \leq v(t) \leq 1 .$$

## 数学代写|数学分析代写Mathematical Analysis代考|Pontryagin Maximum Principle

In order to analyze the optimal control $v_{}(t)$ and the corresponding optimal solution $\left(l_{}(t), k_{}(t), m_{}(t)\right)$, we apply the Pontryagin maximum principle [9]. Firstly, we write down the Hamiltonian
\begin{aligned} H\left(l, k, m, v, \psi_{1}, \psi_{2}, \psi_{3}\right)=&\left(\sigma-\delta v l m-\gamma_{1} l k-\mu l\right) \psi_{1} \ &+\left((\beta+\delta) v l m+\gamma_{2} l k-\lambda k\right) \psi_{2}+(\rho-\beta v l m-v m) \psi_{3} \end{aligned}
where $\psi_{1}, \psi_{2}, \psi_{3}$ are adjoint variables.
Secondly, we calculate the required partial derivatives:
\begin{aligned} &H_{l}^{\prime}\left(l, k, m, v, \psi_{1}, \psi_{2}, \psi_{3}\right)=v m\left(-\delta \psi_{1}+(\beta+\delta) \psi_{2}-\beta \psi_{3}\right) \ &\quad+k\left(\gamma_{2} \psi_{2}-\gamma_{1} \psi_{1}\right)-\mu \psi_{1} \ &H_{k}^{\prime}\left(l, k, m, v, \psi_{1}, \psi_{2}, \psi_{3}\right)=l\left(\gamma_{2} \psi_{2}-\gamma_{1} \psi_{1}\right)-\lambda \psi_{2} \ &H_{m}^{\prime}\left(l, k, m, v, \psi_{1}, \psi_{2}, \psi_{3}\right)=v l\left(-\delta \psi_{1}+(\beta+\delta) \psi_{2}-\beta \psi_{3}\right)-v \psi_{3}, \ &H_{v}^{\prime}\left(l, k, m, v, \psi_{1}, \psi_{2}, \psi_{3}\right)=\operatorname{lm}\left(-\delta \psi_{1}+(\beta+\delta) \psi_{2}-\beta \psi_{3}\right) \end{aligned}

Then, in accordance with the Pontryagin maximum principle, for the optimal control $v_{}(t)$ and the optimal solution $\left(l_{}(t), k_{}(t), m_{}(t)\right)$ there exists a vector-function $\psi_{}(t)=\left(\psi_{1}^{}(t), \psi_{2}^{}(t), \psi_{3}^{}(t)\right)$ such that:

• $\psi_{}(t)$ is a nontrivial solution of the adjoint system: \left{\begin{aligned} \psi_{1}^{ \prime}(t)=&-v_{}(t) m_{}(t)\left(-\delta \psi_{1}^{}(t)+(\beta+\delta) \psi_{2}^{}(t)-\beta \psi_{3}^{}(t)\right) \ &-k_{}(t)\left(\gamma_{2} \psi_{2}^{}(t)-\gamma_{1} \psi_{1}^{}(t)\right)+\mu \psi_{1}^{}(t), \ \psi_{2}^{ \prime}(t)=&-l_{}(t)\left(\gamma_{2} \psi_{2}^{}(t)-\gamma_{1} \psi_{1}^{}(t)\right)+\lambda \psi_{2}^{}(t), \ \psi_{3}^{* \prime}(t)=&-v_{}(t) l_{}(t)\left(-\delta \psi_{1}^{}(t)+(\beta+\delta) \psi_{2}^{}(t)-\beta \psi_{3}^{}(t)\right)+v \psi_{3}^{}(t), \ \psi_{1}^{}(T)=0, \psi_{2}^{}(T)=-1, \psi_{3}^{*}(T)=0 \end{aligned}\right.

## 数学代写|数学分析代写Mathematical Analysis代考|Optimal Control Problem

$\$ \$$Veft$$
l^{\prime}(t)=\sigma-\delta v(t) l(t) m(t)-\gamma_{1} l(t) k(t)-\mu l(t) k^{\prime}(t)=(\beta+\delta) v(t) l(t) m(t)+\gamma_{2} l(t) k(t)-\lambda k(t) m^{\prime}(t)
$$【正确的。 \ \$$

$$0<v_{\min } \leq v(t) \leq 1$$

## 数学代写|数学分析代写Mathematical Analysis代考|Pontryagin Maximum Principle

$$H\left(l, k, m, v, \psi_{1}, \psi_{2}, \psi_{3}\right)=\left(\sigma-\delta v l m-\gamma_{1} l k-\mu l\right) \psi_{1} \quad+\left((\beta+\delta) v l m+\gamma_{2} l k-\lambda k\right) \psi_{2}+(\rho-\beta v$$

$$H_{l}^{\prime}\left(l, k, m, v, \psi_{1}, \psi_{2}, \psi_{3}\right)=v m\left(-\delta \psi_{1}+(\beta+\delta) \psi_{2}-\beta \psi_{3}\right) \quad+k\left(\gamma_{2} \psi_{2}-\gamma_{1} \psi_{1}\right)-\mu \psi_{1} H_{k}^{\prime}(l,$$

• $\psi(t)$ 是伴随系统的非平凡解: $\$ \$V$ left {
$$\psi_{1}^{\prime}(t)=-v(t) m(t)\left(-\delta \psi_{1}(t)+(\beta+\delta) \psi_{2}(t)-\beta \psi_{3}(t)\right) \quad-k(t)\left(\gamma_{2} \psi_{2}(t)-\gamma_{1} \psi_{1}(t)\right)+\mu \psi_{1}$$
【正确的。
$\$ \

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