## Null controllability for population dynamics with age, size structuring and diffusion

### 1. Motivation and description of age, size structured model

**1.1. Motivation**

Knowing the mechanisms of tumor growth can be useful for developing treatments. To mathematically describe its evolution from a global point of view, we can use a so-called “diffusion” equation. Diffusion equations are good tools in such a context, as they allow to describe the global effects of a physical process that takes place on a much smaller scale.

In this post, we are interested of the null controllability of population dynamics with age, size structuring and diffusion.

**1.2. An example of an experimental device for the treatment of cancer**

**1.3. Description of age and size structured model**

We consider the following system

\left\lbrace \begin{array}{ll} \dfrac{\partial y}{\partial t}+\dfrac{\partial y}{\partial a}+\dfrac{\partial (g(s)y) }{\partial s}-\Delta y+\mu(a,s)y=0 &\hbox{ in }\Omega\times (0,A)\times (0,S)\times(0,+\infty) ,\\ \dfrac{\partial y}{\partial \nu}=0&\hbox{ }(x,a,s,t)\in\partial\Omega\times(0,A)\times (0,S)\times(0,\infty),\\ y\left(x,0,s,t\right) =y^{1}_{i}(x,s,t)&\hbox{ } (x,s,t)\in\Omega\times (0,S)\times(0,\infty) \\ y\left(x,a,s,0\right)=y_{0}\left(x,a,s\right)& \hbox{ }(x,a,s)\in \Omega\times(0,A)\times (0,S);\\ y(x,a,0,t)=0& \hbox{ } (x,a,t)\in\Omega\times(0,A)\times(0,\infty). \end{array} \right. (1.1)

where y(x,a,s,t) (carcinogenic cells) is a distribution of individuals of age a size s at time t and location x\in\Omega. A and S are respectively the maximal live expectancy and the maximal size, \mu(a,s) natural death rate of individuals of individual.

1. \dfrac{\partial y}{\partial a} is the aging

2. \dfrac{\partial (g(s)y) }{\partial s} is the growing

3. \mu(a,s)y is the damping

4. y^{1}_{i}(x,s,t) \hbox{, }i\in\{1,2\} is the birth the rate, here we consider two type of birth rate.

(a) y^{1}_{1}(x,s,t)=\displaystyle\int\limits_{0}^{A}\int\limits_{0}^{S}\beta_1(a,\hat{s},s)y(x,a,\hat{s},t)da d\hat{s}.
Here \beta_1 is a positive function describing the fertility rate age-size, which depends on (a,\hat{s}) and also depending on the size s of the newborns.

In probabilistic terms, fertility \beta_1 can mean the probability of an individual of age a and of size \hat{s} giving birth to an individual of size s.
and

(b)

y^{1}_{2}(x,s,t)=\displaystyle\int\limits_{0}^{A}\beta_2(a,s)y(x,a,s,t)da

(c) It is assumed that size increases in the same way for everyone in the population and is controlled by the growth function g(s), then \dfrac{\partial (g(s)y) }{\partial s} is the growth. Here g(s)=1.

Figure 1. A Brain Implant Stops Tumor Growth in Rats

**1.4. Existence and uniqueness result**

**Proposition 1.1.** Description of age and size structured model. According to Gleen Webb, if the mortality and fertility rates \mu(a,s)=\mu_1(a)+\mu_2(s) and \beta_i are such that:

(H1): \left\lbrace \begin{array}{l} \mu_1(a)\geq 0 \text{ for every } a\in (0,A)\\ \mu_1\in L^{1}\left([0,a^*]\right)\hbox{ for every }\; a^*\in [0,A) \\ \displaystyle\int\limits_{0}^{A}\mu_1(a)da=+\infty \end{array} \right.,\quad

(H2): \left\lbrace\begin{array}{l} \mu_2(s)\geq 0 \hbox{ for every } s\in (0,S)\\ \mu_2\in L^{1}\left([0,s^*]\right)\hbox{ for every }\hbox{ } s^*\in [0,A) \\ \displaystyle\int\limits_{0}^{S}\mu_2(s)ds=+\infty \end{array} \right.

(H3): \left\lbrace\begin{array}{l} \beta_i\in L^{\infty}\hbox{, }i\in\{1,2\}\cr \beta_i \ge 0 \quad {\rm \; a.e.} \quad i\in \{1,2\},\\ \end{array} \right.

for any initial condition y_0\in K=L^2\left(\Omega\times (0,A)\times (0,S)\right), the system (1) admits a unique solution.

### 2. Observation and null controllability

**2.1. The Null controllability problem**

We consider the following system

\left\lbrace\begin{array}{ll} \dfrac{\partial y}{\partial t}+\dfrac{\partial y}{\partial a}+\dfrac{\partial y }{\partial s}-\Delta y+\mu(a,s)y=u\chi_{\Theta} &\hbox{ in }\Omega\times (0,A)\times (0,S)\times(0,\infty) ,\\ \dfrac{\partial y}{\partial \nu}=0&\hbox{ }(x,a,s,t)\in\partial\Omega\times(0,A)\times (0,S)\times(0,\infty),\\ y\left(x,0,s,t\right) =y^{1}_{i}(x,s,t)&\hbox{ } (x,s,t)\in\Omega\times (0,S)\times(0,\infty)\hbox{, }i\in\{1,2\} \\ y\left(x,a,s,0\right)=y_{0}\left(x,a,s\right)& \hbox{ }(x,a,s)\in \Omega\times(0,A)\times (0,S);\\ y(x,a,0,t)=0& \hbox{ } (x,a,t)\in\Omega\times(0,A)\times(0,\infty). \end{array}\right. (2.1)

where

1. \Theta=\omega\times(a_1,a_2)\times (s_1,s_2);

2. u(x,a,s,t) is the control;

3. \omega\times(a_1,a_2)\times (s_1,s_2)\subset \Omega\times (0,A)\times (0,S) is the support of the control;

4. Goal: To drive the solution to equilibrium at a given final time T>0
y(.,.,.,T)\equiv 0

**2.2. The dual observation**

Since the controllability of the linear system is equivalent to an observability inequality, we consider the adjoint system:

\left\lbrace\begin{array}{ll} \dfrac{\partial q}{\partial t}-\dfrac{\partial q}{\partial a}-\dfrac{\partial q}{\partial s}-\Delta q+\mu(a,s)q=q_i(x,s,t)&\hbox{ in }\Omega\times(0,A)\times (0,S)\times(0,+\infty) i\in\{1,2\},\\ \dfrac{\partial q}{\partial \nu}=0&\hbox{ }(x,a,s,t)\in\partial\Omega\times(0,A)\times (0,S)\times(0,\infty),\\ q\left( x,A,s,t\right) =0&\hbox{ in } (x,s,t)\in\Omega\times (0,S)\times(0,\infty) \\ q(x,a,S,t)=0& \hbox{ in } (x,a,t)\in\Omega\times(0,A)\times (0,\infty)\\ q\left(x,a,s,0\right)=q_0(x,a,s)& \hbox{ in }(x,a,s)\in\partial\Omega\times(0,A)\times (0,S); \end{array}\right. (2.2)

with the following correspondence

q_1(x,s,t)=\displaystyle\int\limits_{0}^{S}\beta_1(a,s,\hat{s})q(x,0,\hat{s},t)d\hat{s}\hbox{ matches with }y^{1}_{1},
and

q_2(x,s,t)=\beta_2(a,s)q(x,0,s,t)\hbox{ matches with }y^{1}_{2}.

**2.3. The dual observation problem**

The null controllability problem therefore becomes an observability inequality problem and the question is whether:

is the inequality

\displaystyle\int\limits_{0}^{S}\int\limits_{0}^{A}\int_{\Omega}q^2(a,s,T)dxdads\leq K_T\displaystyle\int\limits_{0}^{T}\int\limits_{s_1}^{s_2}\int\limits_{a_1}^{a_2}\int_{\omega}q^2(a,s,t)dadsdt.

verified?

The observation being made in the subset

\omega\times(a_1,a_2)\times (s_1,s_2)\subset \Omega\times (0,A)\times (0,S).

**2.4. Null controllability results**

The answer to the previous question is affirmative and we have the following null controllability result.

We denote by T_1=\max\{a_1+S-s_2,s_1\}\hbox{ and }T_0=\max\{S-s_2,s_1\}.

**Theorem 2.1**

The null controllability result holds in \omega\times(a_1,a_2)\times (s_1,s_2)\subset \Omega\times (0,A)\times (0,S) with

T_0 \lt \min\{a_2-a_1,\hat{a}-a_1\}; provided the fertility rate is such that \beta_i(a,.)\equiv 0\hbox{ in }(0,a_1+\gamma), and the time T is large enough such that:

1. for the birth rate equal y^{1}_{1}, T>A-a_2+T_1+T_0 and

2. for the birth rate equal y^{1}_{2}, T>A-a_2+a_1+S-s_2+s_1.

**2.5. Idea of Proof**

**2.5.1. Change of variables**

The proof of the observability inequality is based on the estimation of the non-local term q_i\hbox{, }i\in\{1,2\}.\\

As \beta(a,.)\equiv 0\hbox{ in }(0,a_1+\gamma), the first equation of the adjoint system become

\dfrac{\partial q}{\partial t}-\dfrac{\partial q}{\partial a}-\dfrac{\partial q}{\partial s}-\Delta q+\mu(a,s)q=0\hbox{ in }\Omega\times(0,\hat{a})\times (0,S)\times(0,+\infty).
We denote by \tilde{q}(x,a,s,t)=q(x,a,s,t)\exp\left(-\int\limits_{0}^{a}\mu_1(\alpha)d\alpha-\int\limits_{0}^{s}\mu_2(r)dr\right).w(x,\lambda)=\tilde{q}(x,t-\lambda,s+t-\lambda,\lambda) \text{ ; }x\in\Omega\hbox{, }\lambda\in (0,t).

**2.5.2. Estimation of the non local term**

Then w satisfies:

• \dfrac{\partial w(x,\lambda)}{\partial \lambda}-\Delta w(x,\lambda)=0\text{ in } \Omega\times (0,t).

• The q(x,0,s,t) can be estimate from the observation; according to the variables s and t. Indeed, for s\in (0, S_2-a_1), we can estimate q(x,0,s,t) from the observation for all t \gt \max\{a_1,s_1\} and for s\in (S_2-a_1,S), we can estimate q(x,0,s,t) from the observation for all t \gt a_1+S-s_2.

• Then the sum q_{1} can be estimate for all time t \gt T_1.

### 3. Illustration of the observability inequality

**3.1. Illustration of the estimate of q(x,0,s,t)**

Figure 2. Here T_0 \lt \min\{a_2-a_1,\hat{a}-a_1\} and we choose a_2=\hat{a}. Since t \gt T_1 all the backward characteristics starting from (0,s,t) enters the observation domain (the green and blue lines), or without the domain by the boundary s=S (red line).

**3.2. Illustration of cases where we are not able to estimate q(x,0,s,t)**

**3.2.1. Graphical proof of the condition T_0=S-s_2 \lt \min\{a_2-a_1,\hat{a}-a_1\}**

FIGURE_3

Figure 3. For S-s_2 \gt \hat{a}-a_1, we can not estimate q (x,0,s,t) for s\in(c_1,c_2) by the characteristic method. Indeed for even t \gt a_1+S-s_2 the characteristics starting at (0,s,t) without the domain by the boundary t=0 \hbox{ or enter in the region } a \gt \hat{a}, without going through the observation domain.

**3.3. Graphical proof of the condition T_0=s_1 \lt \min\{a_2-a_1,\hat{a}-a_1\}**

Figure 4. For the second case if s_1 \gt \hat{a}-a_1, we can not estimate q (x,0,s,t) for s\in (c_1,c_2) by the characteristic method. Indeed for even t>a_1+S-s_2 the characteristics starting at (0,s,t) without the domain by the boundary t=0 \hbox{ or enter in the region } a>\hat{a}, without going through the observation domain.

**3.4. Minimal time**

**3.4.1. Graphical proof of minimal time**

Figure 5. \Lambda=s_2-a_1. The backward characteristics starting from (a,s,T) with a\in (a_0,A) (red lines or green lines) hits the boundary (a=A), gets renewed by the renewal condition q_1 and then enters the observation domain (green lines) or without by the boundary s=S.

**3.4.2. Graphical proof of minimal time**

Figure 6. Here \hat{a}=a_2. The backward characteristics starting from (a,s,T) hits the boundary (a=A), gets renewed by the renewal condition q_2 and then enters the observation domain (red line) or without and leave the domain \Sigma by the boundary s=S (green line)

### References

[1] Y. Simpore, U. Biccari. Controllability and Positivity Constraints in Population Dynamics with age, size Structuring and Diffusion (2022) arxiv: 10.48550/arXiv.2209.04018[2] Y. Simpore, Y. El gantouh, U. Biccari. Null Controllability for a Degenerate Structured Population Model (2022). arxiv: 10.48550/arXiv.2209.03645

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