### 数学代写|数学分析代写Mathematical Analysis代考|MATH2241

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• 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|Properties of the Switching Function

An analysis of the function $L(t)$ leads to the validity of the following lemma.
Lemma 2 There is such a value $t_{0} \in[0, T)$ that on the interval $\left(t_{0}, T\right]$ the switching function $L(t)$ is negative.

Proof The functions $\psi_{1}^{}(t), \psi_{2}^{}(t), \psi_{3}^{}(t)$, as the components of the absolutely continuous solution $\psi_{}(t)$ to system (5), are absolutely continuous as well. Hence, by formula (7), the switching function $L(t)$ is also absolutely continuous, and therefore a continuous function. Due to formula (7) and the initial conditions of system (5), it takes the negative value at $t=T$ :
$$L(T)=-(\beta+\delta)<0 .$$

Then, the stability of the sign of the continuous function $L(t)$ yields the required fact. This completes the proof.
Corollary 1 From Lemma 2 and formula (6), it follows the relationship:
$$v_{}(t)=v_{\min }, \quad t \in\left(t_{0}, T\right] .$$ Now, we introduce positive constants: $$\alpha=\gamma_{2}^{-1}\left((\beta+\delta) \gamma_{1}-\delta \gamma_{2}\right), \quad \varepsilon=\alpha(\lambda-v)+\delta(\lambda-\mu)$$ and then also the following functions: \begin{aligned} g_{11}(t)=& v_{}(t)\left(\delta m_{}(t)+\beta l_{}(t)\right)+v \ g_{21}(t)=& v_{}(t) m_{}(t)\left(\gamma_{1}\left(\delta k_{}(t)-(\beta+\delta) l_{}(t)\right)+\delta(\mu-v)\right) \ g_{22}(t)=&(\lambda-\mu) \varepsilon^{-1} \gamma_{1}\left(\delta k_{}(t)-(\beta+\delta) l_{}(t)\right) \ &+\varepsilon^{-1}(\alpha(\lambda-v) \lambda+\delta(\lambda-\mu)(\lambda-v+\mu)) \ g_{31}(t)=& \gamma_{1} v_{}(t) m_{}(t), \quad g_{32}(t)=(\lambda-\mu) \gamma_{1} \varepsilon^{-1} \ g_{33}(t)=&\left(\gamma_{1} k_{}(t)-\gamma_{2} l_{}(t)\right)-(\lambda-\mu) \varepsilon^{-1} \gamma_{1}\left(\delta k_{}(t)-(\beta+\delta) l_{}(t)\right) \ &+\varepsilon^{-1}(\alpha(\lambda-v) \mu+\delta(\lambda-\mu) v) \end{aligned}

## 数学代写|数学分析代写Mathematical Analysis代考|Numerical Results

Further, only numerical investigation of optimal control $v_{*}(t)$ is possible. For the corresponding numerical calculations, the following values of the parameters and initial conditions of system (1) were used [3,11], as well as the control constraints (2):
$$\begin{array}{llll} \sigma=15.0 & \rho=3.6 & \beta=0.4 & \delta=0.005 \ \mu=0.01 & v=0.02 & \gamma_{1}=0.8 & \gamma_{2}=0.05 \ l_{0}=100.0 & k_{0}=40.0 & m_{0}=50.0 & \ v_{\min }=0.3 & T=100.0 & & \end{array}$$
The numerical calculations were carried out using the software “BOCOP 2.0.5” (see [1]), and are shown in Figs. 1 and $2 .$

In Fig. 3 the surface $\Phi(l, k)$ is presented. It can be seen that positive values of the function $\Phi(l, k)$ in formula (10) in the region of variation of the variables $l$ and $k$ provide the admissibility of singular control $v_{\text {sing }}^{*}(t)$.

Physical optimal control $\tilde{v}{}(t)$ according to Figs. 1 and 2 describes the situation when, first there is the period of the psoriasis treatment with greatest intensity. Next, it is followed by the period of the treatment with a smooth decrease in the dose of the used medication from the greatest intensity to the lower intensity. Then, there is a period of the psoriasis treatment with lower intensity, and finally the switching occurs to the period of the treatment with greatest intensity. Also, we emphasize that in all performed numerical calculations, the optimal concentration of keratinocytes $k{}(t)$ decreases to the end $T$ to the level that is the minimal for the entire period $[0, T]$ of the psoriasis treatment (see Figs. 1 and 2 ).

## 数学代写|数学分析代写Mathematical Analysis代考|Properties of the Switching Function

(7) ，切换函数 $L(t)$ 也是绝对连续的，因此是连续函数。由于公式 (7) 和系统 (5) 的初始条件，它在 $t=T$
$$L(T)=-(\beta+\delta)<0 .$$

$$v(t)=v_{\min }, \quad t \in\left(t_{0}, T\right] .$$

$$\alpha=\gamma_{2}^{-1}\left((\beta+\delta) \gamma_{1}-\delta \gamma_{2}\right), \quad \varepsilon=\alpha(\lambda-v)+\delta(\lambda-\mu)$$

$$g_{11}(t)=v(t)(\delta m(t)+\beta l(t))+v g_{21}(t)=\quad v(t) m(t)\left(\gamma_{1}(\delta k(t)-(\beta+\delta) l(t))+\delta(\mu-v)\right) g_{22}(t)$$

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