Controllability for a degenerate cascade system

TL;DR

This paper investigates the null controllability of a degenerate cascade system modeling predator-prey interactions, using Carleman estimates to establish observability inequalities for the adjoint problem.

Contribution

It introduces a novel controllability analysis for a degenerate, age-structured predator-prey system with boundary degeneracy, employing advanced Carleman estimates.

Findings

Established null controllability under certain conditions.

Derived new Carleman estimates for degenerate systems.

Proved observability inequalities for the adjoint problem.

Abstract

In this paper we consider a cascade system in non divergence form which models the interaction between two different species, the first one can be seen as a predator and the other as a prey. Both of them depend on time, on age and on space. Moreover, the diffusion coefficients degenerate at the boundary of domain. We study, in particular, null controllability of the system via the observability inequality for the non homogeneous adjoint problem, which is deduced by Carleman estimates.

Equations

$$\{\begin{array}{ll}\frac{\partial y}{\partial t}+\frac{\partial y}{\partial a}-k_1(x)y_{xx}+{\mu }_{11}(t,x,a)y+{\mu }_{13}(a)y=\bar{g}(t,x,a){\chi }_{\omega },& (t,a,x)\in Q,\\ \frac{\partial w}{\partial t}+\frac{\partial w}{\partial a}-k_2(x)w_{xx}+{\mu }_{22}(t,x,a)w-\mu (t,x,a)y+{\mu }_{23}(a)w=0,& (t,a,x)\in Q,\\ y(t,a,0)=y(t,a,1)=w(t,a,0)=w(t,a,1)=0,& (t,a)\in Q_{T,A},\\ y(0,a,x)=y_0(a,x),w(0,a,x)=w_0(a,x),& (a,x)\in Q_{A,1},\\ y(t,0,x)={\int }_0^A{\gamma }_1(a,x)y(t,a,x)da,& (t,x)\in Q_{T,1},\\ w(t,0,x)={\int }_0^A{\gamma }_2(a,x)w(t,a,x)da,& (t,x)\in Q_{T,1}.\end{array}$$

$$k>0\text{ in }(0,1],k(0)=0(\text{or }k>0\text{ in }[0,1)\text{ and }k(1)=0)$$

$$k>0\text{ in }(0,1),k(0)=0(\text{or }k>0\text{ in }[0,1)\text{ and }k(1)=0)$$

$$u(t,a,x):=e^{{\int }_0^a{\mu }_{13}(\gamma )d\gamma }y(t,a,x)\text{ and }v(t,a,x):=e^{{\int }_0^a{\mu }_{23}(\gamma )d\gamma }w(t,a,x),$$

$$\{\begin{array}{ll}\frac{\partial u}{\partial t}+\frac{\partial u}{\partial a}-k_1(x)u_{xx}+{\mu }_{11}(t,x,a)u=g(t,x,a){\chi }_{\omega },& (t,a,x)\in Q,\\ \frac{\partial v}{\partial t}+\frac{\partial v}{\partial a}-k_2(x)v_{xx}+{\mu }_{22}(t,x,a)v-{\mu }_{21}(t,x,a)u=0,& (t,a,x)\in Q,\\ u(t,a,0)=u(t,a,1)=v(t,a,0)=v(t,a,1)=0,& (t,a)\in Q_{T,A},\\ u(0,a,x)=u_0(a,x),v(0,a,x)=v_0(a,x),& (a,x)\in Q_{A,1},\\ u(t,0,x)={\int }_0^A{\beta }_1(a,x)u(t,a,x)da,& (t,x)\in Q_{T,1},\\ v(t,0,x)={\int }_0^A{\beta }_2(a,x)v(t,a,x)da,& (t,x)\in Q_{T,1}.\end{array}$$

$$\bullet {\beta }_i\in C({\bar{Q}}_{A,1})\text{ and }{\beta }_i\ge 0\text{ in }{Q}_{A,1},$$

$$\bullet {\mu }_{11},{\mu }_{2i}\in C(\bar{Q})\text{ and }{\mu }_{11},{\mu }_{2i}\ge 0\text{ in }Q.$$

$$L_{\frac{1}{k_i}}^2(0,1):=\left\{u\in L^2(0,1):{\int }_0^1\frac{u^2}{k_i}dx<\infty \right\},$$

$${\mathcal{H}}_{\frac{1}{k_i}}^1(0,1):=L_{\frac{1}{k_i}}^2(0,1)\cap H_0^1(0,1)$$

$$\mathcal{H}_{\frac{1}{k_{i}}}^{2}(0,1):=\big{\{}u\in\mathcal{H}_{\frac{1}{k_{i}}}^{1}(0,1):u_{x}\in H^{1}(0,1)\big{\}},$$

$$\Vert u{\Vert }_{L_{\frac{1}{k_i}}^2(0,1)}^2:={\int }_0^1\frac{u^2}{k_i}dx,\forall u\in L_{\frac{1}{k_i}}^2(0,1),$$

$$\Vert u{\Vert }_{{\mathcal{H}}_{\frac{1}{k_i}}^1}^2:=\Vert u{\Vert }_{L_{\frac{1}{k_i}}^2(0,1)}^2+\Vert u_x{\Vert }_{L^2(0,1)}^2,\forall u\in {\mathcal{H}}_{\frac{1}{k_i}}^1(0,1)$$

$$\Vert u{\Vert }_{{\mathcal{H}}_{\frac{1}{k_i}}^2(0,1)}^2:=\Vert u{\Vert }_{{\mathcal{H}}_{\frac{1}{k_i}}^1(0,1)}^2+\Vert k_i{u}_{xx}{\Vert }_{L_{\frac{1}{k_i}}^2(0,1)}^2,\forall u\in {\mathcal{H}}_{\frac{1}{k_i}}^2(0,1).$$

$$\mathcal{H}^{2}_{\frac{1}{k_{i}}}(0,1):=\Big{\{}u\in\mathcal{H}^{1}_{\frac{1}{k_{i}}}(0,1)\,\big{|}\,u_{x}\in H^{1}(0,1)\mbox{ and }k_{i}u_{xx}\in L^{2}_{\frac{1}{k_{i}}}(0,1)\Big{\}}.$$

$${\mathcal{A}}_{i}u:={\mathcal{A}}_{a}u-A_{i}u,$$

$$u\in D({\mathcal{A}}_i)=\left\{u\in L^2(0,A;D(A_i)):\frac{\partial u}{\partial a}\in L^2(0,A;H_{\frac{1}{k_i}}^1(0,1)),u(0,x)={\int }_0^A{\beta }_i(a,x)u(a,x)da\right\},$$

$$\mathcal{U}(t)=\left(\begin{array}{c}u(t)\\ v(t)\end{array}\right),\mathcal{A}=\left(\begin{array}{cc}{\mathcal{A}}_1& 0\\ 0& {\mathcal{A}}_2\end{array}\right),D(\mathcal{A})=D({\mathcal{A}}_1)\times D({\mathcal{A}}_2),$$

$$B(t)=\left(\begin{array}{cc}{M}_{{\mu }_{11}(t)}& 0\\ -M_{{\mu }_{21}(t)}& M_{{\mu }_{22}(t)}\end{array}\right)\text{ and }f(t,\cdot ,\cdot )=\left(\begin{array}{c}g(t,\cdot ,\cdot ){\chi }_{\omega }\\ 0\end{array}\right),$$

$$\{\begin{array}{l}{\mathcal{U}}^{\prime }(t)+\mathcal{A}\mathcal{U}(t)+B(t)\mathcal{U}(t)=f(t),\\ \mathcal{U}(0)=\left(\begin{array}{c}{u}_0\\ v_0\end{array}\right).\end{array}$$

$$(u,v)\in C([0,T];L^2(0,A;\mathbb{L}))\cap L^2(0,T;H^1(0,A;\mathbb{H}))$$

$$\{\begin{array}{ll}\frac{\partial z}{\partial t}+\frac{\partial z}{\partial a}+k_1(x)z_{xx}-{\mu }_{11}(t,a,x)z+{\mu }_{21}(t,a,x)y=-{\beta }_1(a,x)z(t,0,x),& (t,a,x)\in Q,\\ \frac{\partial y}{\partial t}+\frac{\partial y}{\partial a}+k_2(x)y_{xx}-{\mu }_{22}(t,a,x)y=-{\beta }_2(a,x)y(t,0,x),& (t,a,x)\in Q,\\ z(t,a,0)=z(t,a,1)=y(t,a,0)=y(t,a,1)=0,& (t,a)\in Q_{T,A},\\ z(t,A,x)=y(t,A,x)=0,& (t,x)\in Q_{T,1},\\ z(T,a,x)=z_T(a,x)\in L_{\frac{1}{k_1}}^2(Q_{A,1}),& (a,x)\in Q_{A,1},\\ y(T,a,x)=y_T(a,x)\in L_{\frac{1}{k_2}}^2(Q_{A,1}),& (a,x)\in Q_{A,1},\end{array}$$

$$\{\begin{array}{ll}\frac{\partial v}{\partial t}+\frac{\partial v}{\partial a}+k(x)v_{xx}-\mu (t,a,x)v=f,& (t,a,x)\in Q,\\ v(t,a,0)=v(t,a,1)=0,& (t,a)\in Q_{T,A},\\ v(t,A,x)=0,& (t,x)\in Q_{T,1}.\end{array}$$

$$\omega =(\alpha ,\beta )\subset \subset (0,1).$$

$$\phi (t,a,x):=\Theta (t,a)(p(x)-2\Vert p{\Vert }_{L^{\infty }(0,1)}),$$

$$\begin{array}{c}\Theta (t,a)=\frac{1}{t^4(T-t{)}^4{a}^4}\end{array}$$

$$\displaystyle\int_{Q}\left(s\Theta v_{x}^{2}+s^{3}\Theta^{3}\text{\small$\displaystyle\Big{(}\frac{x}{k}\Big{)}^{2}$\normalsize}v^{2}\right)e^{2s\varphi}dxdadt$$

$$\bar{\phi }(t,a,x):=\Theta (t,a)(\bar{p}(x)-2\Vert \bar{p}{\Vert }_{L^{\infty }(0,1)}),$$

$$\displaystyle\int_{Q}\left(s\Theta v_{x}^{2}+s^{3}\Theta^{3}\text{\small$\displaystyle\Big{(}\frac{x-1}{k}\Big{)}^{2}$\normalsize}v^{2}\right)e^{2s\varphi}dxdadt$$

$$\{\begin{array}{ll}\frac{\partial z}{\partial t}+\frac{\partial z}{\partial a}+k_1(x)z_{xx}-{\mu }_{11}(t,a,x)z+{\mu }_{21}(t,a,x)y=f,& (t,a,x)\in Q,\\ \frac{\partial y}{\partial t}+\frac{\partial y}{\partial a}+k_2(x)y_{xx}-{\mu }_{22}(t,a,x)y=h,& (t,a,x)\in Q\\ z(t,a,0)=z(t,a,1)=y(t,a,0)=y(t,a,1)=0,& (t,a)\in Q_{T,A},\\ z(t,A,x)=y(t,A,x)=0,& (t,x)\in Q_{T,1},\end{array}$$

$$\displaystyle\int_{Q}\left(s\Theta z_{x}^{2}+s^{3}\Theta^{3}\text{\small$\displaystyle\Big{(}\frac{x}{k_{1}}\Big{)}^{2}$\normalsize}z^{2}\right)e^{2s\varphi_{1}}dxdadt\leq C\int_{Q}(f^{2}+y^{2})\frac{1}{k_{1}}dxdadt+C\int_{Q_{T,A}}\int_{\omega}\frac{z^{2}}{k_{1}}dxdadt$$

$$\displaystyle\int_{Q}\left(s\Theta y_{x}^{2}+s^{3}\Theta^{3}\Big{(}\frac{x}{k_{2}}\Big{)}^{2}y^{2}\right)e^{2s\varphi_{2}}dxdadt$$

Full text

Keywords: Population cascade model, degenerate equations, Carleman estimates, mull controllability, observability inequality

2000AMS Subject Classification: 35Q93, 93B05, 93B07, 34H15, 35A23, 35B99

1 Introduction

This article is devoted to study the controllability properties for the following linear degenerate coupled systems in one space dimension:

[TABLE]

Here $Q:=(0,T)\times(0,A)\times(0,1)$, $Q_{T,A}:=(0,T)\times(0,A)$, $Q_{A,1}:=(0,A)\times(0,1)$ and $Q_{T,1}:=(0,T)\times(0,1)$. Moreover, $y(t,a,x)$ and $w(t,a,x)$ are the distribution of certain individuals at location $x\in(0,1)$, at time $t\in(0,T)$, where $T$ is fixed, and of age $a\in(0,A)$. $A$ is the maximal age of life, while $\gamma_{i}$, $i=1,2$, are the natural fertility. Thus, the formulas $\int_{0}^{A}\gamma_{1}yda$, $\int_{0}^{A}\gamma_{2}wda$ denote the distributions of newborn individuals at time $t$ and location $x$. Moreover, $\mu_{1i}$, $i=1,3$, $\mu_{2j},j=2,3$, are the natural death rates and are such that $\mu_{11},\mu_{22}\in C(\bar{Q},{\mathbb{R}}_{+})$.The functions $\mu_{13},\mu_{23}\in C([0,A),R_{+})$ and $\int_{0}^{A}\mu_{13}(a)da=+\infty$, $\int_{0}^{A}\mu_{23}(a)da=+\infty$; On the other hand $\mu\in C(\bar{Q},{\mathbb{R}}_{+})$ can be seen as the rate of influence of the first population on the second one. Hence the term $\mu y$ can be considered as a control in the second equation; thus, $y$ is a predator and $w$ is a prey. For more details about the modelling of such system (commonly named Lotka-McKendrick system) and the biological significance of the hypotheses, we refer to Webb [27]. Finally, $\chi_{\omega}$ is the characteristic function of the control region $\omega\subset\subset(0,1)$, the control $\bar{g}$ belongs to a suitable Lebesgue space and the functions $k_{i}$, $i=1,2,$ are the dispersion coefficients and degenerate at the boundary of the state space. In particular, we say that a function $k$ is

Definition 1.1**.**

Weakly degenerate (WD) at [math] (or at $1$)* if $k\in W^{1,1}([0,1])$,*

[TABLE]

and there exist $M_{1}\in(0,1)$ such that $xk^{\prime}(x)\leq M_{1}k(x)$ for almost every $x\in[0,1]$ (or there exist $M_{2}\in(0,1)$ such that $(x-1)k^{\prime}(x)\leq M_{2}k(x)$ for almost every $x\in[0,1]$).

or

Definition 1.2**.**

*Strongly degenerate (SD) at [math] (or at $1$) *** if $k\in W^{1,\infty}([0,1])$,

[TABLE]

and there exist $M_{1}\in[1,2)$ such that $xk^{\prime}(x)\leq M_{1}k(x)$ for almost every $x\in[0,1]$ (or there exist $M_{2}\in[1,2)$ such that $(x-1)k^{\prime}(x)\leq M_{2}k(x)$ for almost every $x\in[0,1]$).

For example, as $k$ one can consider $k(x)=x^{\alpha}$ or $k(x)=(1-x)^{\beta},$ with $\alpha,\beta>0$.

Observe that thanks to the following transformations

[TABLE]

(1.1) can be rewritten as

[TABLE]

where $u_{0}(a,x):=e^{\int_{0}^{a}\mu_{13}(\gamma)d\tau}y_{0}(a,x)$, $v_{0}(a,x):=e^{\int_{0}^{a}\mu_{23}(\gamma)d\tau}w_{0}(a,x)$, $g(t,a,x):=e^{\int_{0}^{a}\mu_{13}(\tau)d\tau}\bar{g}(t,a,x)$, $\mu_{21}(t,a,x):=e^{\int_{0}^{a}(\mu_{23}(\gamma)-\mu_{13}(\gamma))d\gamma}\mu(t,a,x)$ and $\beta_{i}(a,x):=e^{-\int_{0}^{a}\mu_{i3}(\tau)d\tau}\gamma_{i}(a,x)$, $i=1,2$. Observe that the two integrals $e^{\int_{0}^{a}\mu_{13}(\gamma)d\gamma}$ and $e^{\int_{0}^{a}\mu_{23}(\gamma)d\gamma}$ are the inverse of the probability of survival of an individual from age 0 to a. Thus, in place of (1.1), it is not restrictive to consider (1.2), as we will do from now on in the rest of the paper.

Population dynamics models are the subject of numerous papers where they were investigated from many points of view. One among the notable questions in these models was the controllability issue for age and space structured population dynamics models which were studied in an intensive literature. In this context, we can cite the pioneering papers of V. Barbu and al. in [8], B. Ainseba and S. Anita in [1, 2, 3, 4], and recently [24, 25].

In [8], the authors proved the null controllability for a population dynamics model without diffusion, both in the cases of migration and birth control for $T\geq A$, showing directly an appropriate observability inequality for the associated adjoint system. Moreover, they concluded that, in the case of the migration control, only a classes of age was controlled in contrast with the birth control, which allows to steer all population to extinction.

In [1, 2, 3, 4], the null controllability result for the age-space structured model was established when the diffusion coefficient $k\equiv 1$ and for any space dimension, exploiting the results obtained for heat equation in [22].

In [25] the authors studied the null controllability of a linear system coming from a population dynamics model with age structuring and spatial diffusion (of Lotka-McKendrick type). They considered a control that is localized in the space variable as well as with respect to the age . In their work the first novelty was that the age interval, in which the control needs to be active, can be arbitrarily small without needing to exclude a neighborhood of zero. The second one is that the whole population can be steered into zero in a uniform time, without excluding some interval of low ages.

In [24] the authors considered a linear infinite dimensional system obtained by grafting an age structure. Such systems appear essentially in population dynamics with age structured when phenomena like spatial diffusion or transport are also taken into consideration. They asserted that if the initial system is null controllable in a time small enough, then the structured system is also null controllable in a time depending on the various involved parameters.

In [6], B. Ainseba and al. studied a more general case allowing the dispersion coefficient, $k$, to depend on the variable $x$ and to verify $k(0)=0$. Moreover, they assume the condition $T\geq A$ (as in [8]) and this constitutes a restrictiveness on the ”optimality” of the control time $T$ since it means, for example, that for a pest population, whose the maximal age $A$ may equal many days (maybe many months or years), we need much time to bring the population to the zero equilibrium. In the same trend and to overcome the condition $T\geq A$, in [9, 15] the authors used the fixed point technique (in particular the Leray-Schauder Theorem) to obtain the null controllability for an intermediate system. In [16, 17, 18], the author considered the divergence and non divergence form and used only Carleman estimates and a technique based on cut-off functions, making the proof slimmer and easier to read. However, observe that all the previous papers deal with a single equation.

Up to now, little is known about the null controllability for a population dynamics cascade system both in degenerate and nondegenerate cases. Among the papers treating this argument we recall [5, 7, 11, 13, 28] and the references therein. In particular, in [5] the authors studied a coupled reaction-diffusion equation describing interactions between a prey population and a predator population. The goal of the work was to look for a suitable control supported on a small spatial subdomain which guarantees the stabilization of the predator population to zero. In [28], the objective was different. More precisely, the authors considered an age-dependent prey-predator system and they proved the existence and uniqueness for an optimal control which gives the maximal harvest via the study of the optimal harvesting problem associated to their coupled model. However, the previous results were found in the case when the diffusion coeffcients are constants. This leads to generalize the model of [5] in [7], in [13] and in [11], where a linear or a semilinear parabolic cascade system with two different diffusion coefficients is considered. In particular, they can degenerate at the boundary or in the interior of the space domain. Finally, in [26] the author considered a system similar to the previous ones but with the addition of a singular term. However, in all the previous papers the prey and the predator depend only on time and on space. To our knowledge, this is the first paper where the two populations depend also on age and the diffusion coefficients degenerate at the boundary of the domain.

The paper is organized as follows: in Section 2 we introduce the suitable Hilbert spaces where the problem is well posed and we give, using the semigroup approach, the existence theorem; in Section 3 we prove Carleman estimates and observability inequalities for the non homogeneous associated adjoint problem via the Carleman estimates given in [16] for a single equation. Finally, in section 4 we deduce the null controllability result for an intermediate system and hence for the initial problem (1.2).

2 Assumptions and well-posedness

Through the paper, we assume that the rates $\mu_{ij}$ and $\beta_{i}$, $i,j=1,2$ satisfy the following:

Hypothesis 2.1**.**

The functions $\mu_{11}$, $\mu_{2i}$ and $\beta_{i}$, $i=1,2$, are such that

[TABLE]

To study the well posedness of (1.2), we consider four situations, namely the weakly-weakly degenerate case (WWD), i.e. the case when $k_{1}$ and $k_{2}$ are both (WD), the strongly-strongly degenerate case (SSD), i.e. the case when $k_{1}$ and $k_{2}$ are both (SD), the weakly-strongly degenerate case (WSD), i.e the case when $k_{1}$ is (WD) and $k_{2}$ is (SD), and the strongly-weakly degenerate case (SWD), i.e the case when $k_{1}$ is (SD) and $k_{2}$ is (WD). Towards this end, as in [14],[16] or in [19] we consider the following weighted spaces:

[TABLE]

[TABLE]

and

[TABLE]

$i=1,2$, with the norms

[TABLE]

[TABLE]

and

[TABLE]

Indeed, it is a trivial fact that, if $u_{x}\in H^{1}(0,1)$, then $k_{i}u_{xx}\in L^{2}_{\frac{1}{k_{i}}}(0,1)$, so that the norm for $\mathcal{H}^{2}_{\frac{1}{k_{i}}}(0,1)$ is well defined. For this reason in [21], $\mathcal{H}^{2}_{\frac{1}{k_{i}}}(0,1)$ is written in a more appealing way as

[TABLE]

As in [14, Theorem 2.3] or in [21, Theorem 2.2], we can prove that, for $i=1,2$, the operators $(A_{i},D(A_{i}))$ defined by $A_{i}u:=k_{i}u_{xx}$, with $u\in D(A_{i})=\mathcal{H}^{2}_{\frac{1}{k_{i}}}(0,1)$, are closed, self-adjoint, negative with dense domain in $L^{2}_{\frac{1}{k_{i}}}(0,1)$. Moreover, setting $\mathcal{A}_{a}u:=\displaystyle\frac{\partial u}{\partial a}$, we have that the operators

[TABLE]

for

[TABLE]

generate two strongly continuous semigroups on $L^{2}(0,A)\times L^{2}_{{\frac{1}{k_{i}}}}(0,1)$ (see also [8]). Now, consider the Hilbert spaces $\mathbb{L}=\mathrm{L}^{2}_{\frac{1}{k_{1}}}(0,1)\times\mathrm{L}^{2}_{\frac{1}{k_{2}}}(0,1)$, $\mathbb{H}=\mathcal{H}^{1}_{\frac{1}{k_{1}}}(0,1)\times\mathcal{H}^{1}_{\frac{1}{k_{2}}}(0,1)$ , $\mathbb{K}=\mathcal{H}^{2}_{\frac{1}{k_{1}}}(0,1)\times\mathcal{H}^{2}_{\frac{1}{k_{2}}}(0,1)$ and define

[TABLE]

[TABLE]

where $M_{\mu_{2i}(t)}y=\mu_{2i}(t)y$, $i=1,2$. Analogously for $M_{\mu_{11}}$. Observe that the operator $\mathcal{A}$ is a diagonal matrix and $B$ is a triangular one. Moreove, $B$ can be seen as a bounded perturbation of $(\mathcal{A},D(\mathcal{A}))$. Clealy, (1.2) can be rewritten as

[TABLE]

Setting $L^{2}_{{\frac{1}{k_{1}}}}(Q):=L^{2}(Q_{T,A};L^{2}_{\frac{1}{k_{1}}}(0,1))$ and $L^{2}_{{\frac{1}{k_{i}}}}(Q_{A,1}):=L^{2}(0,A;L^{2}_{\frac{1}{k_{i}}}(0,1))$, $i=1,2$, the following well posedness and regularity results hold (see [7] and the references therein).

Theorem 2.1**.**

Assume Hypothesis 2.1 and suppose that $k_{i}$, $i=1,2$, are $(SD)$ or $(WD)$ at [math] and/or at $1$. The operator $\mathcal{A}$ generates a strongly continuous semigroup $(\mathcal{T}(t))_{t\geq 0}$. Moreover, for all $g\in L^{2}_{\frac{1}{k_{1}}}(Q)$ and for all $(u_{0},v_{0})\in L^{2}(0,A;\mathbb{L})$, system (1.2) admits a unique solution

[TABLE]

of (1.2). In addition, if $g\equiv 0$, (u,v)\in C^{1}\big{(}[0,T];L^{2}(0,A;\mathbb{L})\big{)}.

3 Carleman estimates for the adjoint cascade system

In general, null controllability for a linear parabolic system is, roughly speaking, equivalent to or follows by the observability for the associated adjoint problem

[TABLE]

where $z_{T}(A,x)=0=y_{T}(A,x)$ for all $x\in[0,1]$. Thus, the crucial point is to prove such an inequality via, for example, Carleman estimates for (3.5). This is the goal of this section.

3.1 Preliminary results for Carleman estimates

In this subsection we will recall some results of [16] for a single equation, that will be crucial to prove Carleman estimates for the non homogeneous adjoint problem of (1.2). To this aim, we consider the system

[TABLE]

If the function $k$ degenerates at [math] we make the following assumptions:

Hypothesis 3.1**.**

The function $k\in C^{0}[0,1]\bigcap C^{2}(0,1]$ is such that $k(0)=0$, $k>0$ on $(0,1]$ and there exist $\varepsilon\in(0,1]$ and $M\in(0,2)$ such that the function $\displaystyle\frac{xk_{x}}{k(x)}$ $\in\>L^{\infty}(0,\varepsilon)$, $\displaystyle\frac{xk_{x}(x)}{k(x)}\leq M$, for all $x\in(0,1]$, and $\displaystyle\left(\frac{xk_{x}(x)}{k(x)}\right)_{x}\in L^{\infty}(0,\varepsilon)$. 2. 2.

$\mu\in C(\bar{Q})$* and $\mu\geq 0$ in $Q$.*

Hypothesis 3.2**.**

The control set $\omega$ is such that

[TABLE]

Now, let us introduce the weight function

[TABLE]

where

[TABLE]

and $\displaystyle p(x):=\int_{0}^{x}\frac{y}{k(y)}~{}e^{Ry^{2}}dy$, with $R>0$.

Then, defining $\mathcal{H}^{1}_{{\frac{1}{k}}}(0,1)$ and $\mathcal{H}^{2}_{{\frac{1}{k}}}(0,1)$ as $\mathcal{H}^{1}_{{\frac{1}{k_{i}}}}(0,1)$ and $\mathcal{H}^{2}_{{\frac{1}{k_{i}}}}(0,1)$, respectively, one has:

Theorem 3.1**.**

[see [16, Theorem 4.1]] Assume Hypotheses 3.1 and 3.2. Then, there exist two strictly positive constants $C$ and $s_{0}$ such that every solution v\in L^{2}\big{(}Q_{T,A};\mathcal{H}^{2}_{{\frac{1}{k}}}(0,1)\big{)}\cap H^{1}\big{(}0,T;H^{1}(0,A;\mathcal{H}^{1}_{{\frac{1}{k}}}(0,1))\big{)} of (3.6) satisfies, for all $s\geq s_{0}$,

[TABLE]

On the other hand, if $k$ degenerates at $1$, we assume in place of Hypothesis 3.1, the following one:

Hypothesis 3.3**.**

The function $k\in C^{0}[0,1]\bigcap C^{2}(0,1]$ is such that $k(1)=0$, $k>0$ on $(0,1]$ and there exist $\varepsilon_{i}\in(0,1]$ and $M\in(0,2)$ such that the function $\displaystyle\frac{(x-1)k_{x}}{k(x)}$  $\in L^{\infty}(1-\varepsilon,1)$, $\displaystyle\frac{(x-1)k_{x}(x)}{k(x)}\leq M$, for all $x\in[0,1)$, and $\displaystyle\left(\frac{(x-1)k_{x}(x)}{k(x)}\right)_{x}\in L^{\infty}(1-\varepsilon,1)$. 2. 2.

$\mu\in C(\bar{Q})$* and $\mu\geq 0$ in $Q$.*

Then define:

[TABLE]

where $\Theta$ is as before and $\displaystyle\bar{p}(x):=\int_{0}^{x}\frac{y-1}{k(y)}~{}e^{R(y-1)^{2}}dy$, with $R>0$. Hence, the analogous result of Theorem 3.1 holds:

Theorem 3.2**.**

[see [16, Theorem 4.2]] Assume Hypotheses 3.2 and 3.3. Then, there exist two strictly positive constants $C$ and $s_{0}$ such that every solution v\in L^{2}\big{(}Q_{T,A};\mathcal{H}^{2}_{{\frac{1}{k}}}(0,1)\big{)}\cap H^{1}\big{(}0,T;H^{1}(0,A;\mathcal{H}^{1}_{{\frac{1}{k}}}(0,1))\big{)} of (3.6) satisfies, for all $s\geq s_{0}$,

[TABLE]

Now, we are ready to prove Carleman estimates for the non homogeneous adjoint problem of (1.2):

[TABLE]

where $f\in L^{2}_{\frac{1}{k_{1}}}(Q)$ and $h\in L^{2}_{\frac{1}{k_{2}}}(Q)$. To this aim, we distinguish between the case when $k_{i}(0)=0$ and $k_{i}(1)=0$, $i=1,2$.

3.2 Carleman inequalities and observability inequalities when the degeneracy is at [math].

An immediate consequence of Theorem 3.1 is the next $\omega-$local Carleman estimate for (3.11). Assume

Hypothesis 3.4**.**

The functions $k_{i}\in C^{0}[0,1]\bigcap C^{2}(0,1]$, $i=1,2$, are such that $k_{i}(0)=0$, $k_{i}>0$ on $(0,1]$ and

$k_{i}$* satisfies Hypothesis 3.1.1, for $i=1,2$, * 2. 2.

there exists a positive constant $C$ such that $k_{1}(x)\geq k_{2}(x)$ for all $x\in[0,1]$.

Theorem 3.3**.**

Assume Hypotheses 2.1, 3.2 and 3.4. Take $f\in L^{2}_{\frac{1}{k_{1}}}(Q)$ and $h\in L^{2}_{\frac{1}{k_{2}}}(Q)$. Then, there exist two strictly positive constants $C$ and $s_{0}$ such that every solution (z,y)\in L^{2}\big{(}Q_{T,A};\mathbb{K}\big{)}\cap H^{1}\big{(}0,T;H^{1}(0,A;\mathbb{H})\big{)} of (3.11) satisfies, for all $s\geq s_{0}$,

[TABLE]

and

[TABLE]

Here

[TABLE]

where $\Theta$ is as in (3.9) and $\displaystyle p_{i}(x):=\int_{0}^{x}\frac{y}{k_{i}(y)}~{}e^{Ry^{2}}dy$, with $R>0$, and $i=1,2$.

Proof.

Observe that (3.11) can be rewritten as

[TABLE]

where $y$ satisfies

[TABLE]

Hence, the inequality for $y$ follows immediately by Theorem 3.1 applied to (3.14). On the other hand, the estimante on $z$ follows by Theorem 3.1 applied to (3.13). Indeed, we have

[TABLE]

Thus, the thesis follows.

∎

Remark 1**.**

Observe that the results of Theorems 3.1 , 3.2 and 3.3 still hold true if we substitute the domain $(0,T)\times(0,A)$ with a general domain $(T_{\gamma},T_{\beta})\times(\gamma,A)$ where the required assumptions are satisfied. In this case, in place of the function $\Theta$ defined in (3.9), we have to consider the weight function

[TABLE]

Using the $\omega-$local Carleman estimates given in Theorem 3.3, one can prove the next observability inequality for the adjoint system (3.5). From now on, we will make an additional assumption on the birth rates $\beta_{i}$, $i=1,2$:

Hypothesis 3.5**.**

Assume that there exist $\bar{a}_{i}<\Gamma:=\min\{T,A\}$ such that

[TABLE]

Observe that Hypothesis 3.5 is realistic, since $\bar{a}_{i}$, $i=1,2$, are the minimal age in which the female of the population become fertile, thus it is natural that before $\bar{a}_{i}$ there are no newborns.

Theorem 3.4**.**

Assume Hypotheses $\ref{ratesAss},$ 3.2, 3.4 and 3.5. Then, for every $\delta\in(0,A)$, there exists a strictly positive constant $C=C(\delta)$ such that every solution (z,y)\in L^{2}\big{(}Q_{T,A};\mathbb{K}\big{)}\cap H^{1}\big{(}0,T;H^{1}(0,A;\mathbb{H})\big{)} of (3.5) satisfies

[TABLE]

and

[TABLE]

Moreover, if $y_{T}(a,x)=0$ for all $(a,x)\in(0,\bar{a}_{2})\times(0,1)$ and $z_{T}(a,x)=0$ for all $(a,x)\in(0,\bar{a}_{1})\times(0,1)$, one has

[TABLE]

and

[TABLE]

Proof.

As before, (3.5) can be rewritten as

[TABLE]

where $y$ satisfies

[TABLE]

Setting,

[TABLE]

using the method of characteristic lines, the assumption on $\beta_{i}$, $i=1,2$, and the fact that $z(t,A,x)=y(t,A,x)=0$ for all $(t,x)\in Q_{T,1}$, one has, as in [16, Theorem 4.4], the following implicit formulas for every solution $z$ and $y$ of (3.19) and (3.20), respectively:

[TABLE]

where $G(s,t,a,x):=-\mu_{21}(s+t-a,s,x)y(s+t-a,s,x)$, if $t\geq T_{1}+a$ (observe that in this case $T+a-t\leq\bar{a}_{1}$) and, setting $\Gamma:=T-A$,

[TABLE]

otherwise. Moreover

[TABLE]

if $t\geq T_{2}+a$ (observe that in this case $T+a-t\leq\bar{a}_{2}$) and

[TABLE]

otherwise. Here $(S_{i}(t))_{t\geq 0}$ are the semigroups generated by the operators $A_{i}-\mu_{ii}Id$, $i=1,2$, ($Id$ is the identity operator). In particular, it results

[TABLE]

if $t\geq T-\bar{a}_{1}$ and

[TABLE]

if $t\geq T-\bar{a}_{2}$.

Now, we distinguish between the two cases $\bar{a}_{1}\leq\bar{a}_{2}$ and $\bar{a}_{1}>\bar{a}_{2}$.

$\bar{a}_{1}\leq\bar{a}_{2}$: Using the same idea of [16], we now prove that there exists a positive constant $C$ such that:

[TABLE]

and

[TABLE]

Indeed, define, for $\varsigma>0$, the functions $w=e^{\varsigma t}y$ and $W=e^{\varsigma t}z$. Then, $w$ and $W$ satisfy, respectively, the problems

[TABLE]

and

[TABLE]

where $\tilde{Q}_{i}:=(T_{i},T)\times Q_{A,1}$, $\tilde{Q}_{i,T,A}:=(T_{i},T)\times(0,A)$ and $\tilde{Q}_{i,T,1}:=(T_{i},T)\times(0,1)$, $i=1,2$.

Multiplying the equations of (3.30) and (3.31) by $\displaystyle-\frac{w}{k_{2}}$ and $\displaystyle-\frac{W}{k_{1}}$ and integrating by parts on $Q_{2,t}:=(T_{2},t)\times(0,A)\times(0,1)$ and $Q_{1,t}:=(T_{1},t)\times(0,A)\times(0,1)$, respectively, it results

[TABLE]

and

[TABLE]

for $\epsilon>0$. Choosing $\displaystyle\epsilon=\min\left\{\frac{1}{2\|\beta_{1}\|_{L^{\infty}(Q)}A},\frac{1}{2\|\beta_{2}\|_{L^{\infty}(Q)}A}\right\}$ and $\displaystyle\varsigma=\max\left\{\frac{\|\beta_{1}\|_{L^{\infty}(Q)}+1}{\epsilon},\frac{\|\beta_{2}\|_{L^{\infty}(Q)}}{\epsilon}\right\}$, we have

[TABLE]

and

[TABLE]

Then, integrating (3.32) over $\displaystyle\left[T-\frac{\bar{a}_{2}}{2},T-\frac{\bar{a}_{2}}{4}\right]$ and (3.33) over $\displaystyle\left[T-\frac{\bar{a}_{1}}{2},T-\frac{\bar{a}_{1}}{4}\right]$, we have (3.28) and (3.29), respectively.

Now, take $\delta\in(0,A)$. By (3.28), we have

[TABLE]

Consider the term $\displaystyle\int_{T-\frac{\bar{a}_{2}}{2}}^{T-\frac{\bar{a}_{2}}{4}}\int_{\delta}^{A}\int_{0}^{1}\frac{1}{k_{2}}y^{2}(t,a,x)dxdadt$. By the Hardy-Poincaré inequality (see, for example, [12]), one has

[TABLE]

for a strictly positive constant $C.$ Hence,

[TABLE]

where $\tilde{\Theta}_{2}$ is defined in (3.15) with $T_{\gamma}:=T-\bar{a}_{2}$, $T_{\beta}:=T$, $\gamma=0$ and $\tilde{\varphi}_{2}$ is the function $\varphi_{2}$ associated to $\tilde{\Theta}_{2}$ according to (3.8). Thus, by Theorem 3.1 (or Theorem 3.3) applied to $\bar{Q}_{2}:=(T-\bar{a}_{2},T)\times(0,A)\times(0,1)$ and Remark 1,

[TABLE]

where, in this case, $h(t,a,x):=-\beta_{2}(a,x)y(t,0,x)$. Hence, by (3.27), since $t\geq T-\bar{a}_{2}$,

[TABLE]

for a strictly positive constant $C$. By (3.34) and (3.37), (3.17) follows.

Now, we will prove (3.18). Again, by (3.29),

[TABLE]

Proceeding as before, one can prove

[TABLE]

Now, consider $\displaystyle\int_{T-\frac{\bar{a}_{1}}{2}}^{T-\frac{\bar{a}_{1}}{4}}\int_{\delta}^{A}\int_{0}^{1}\frac{1}{k_{1}}z^{2}dxdadt$. As before, by Theorem 3.3 applied to $\bar{Q}_{1}:=\left(T-\displaystyle\frac{3}{4}\bar{a}_{1},T-\displaystyle\frac{\bar{a}_{1}}{8}\right)\times(0,A)\times(0,1)$ and Remark 1, replacing $\tilde{\varphi}_{2}$, $\tilde{\Theta}_{2}$ by $\tilde{\varphi}_{1}$, $\tilde{\Theta}_{1}$, respectively, where $\tilde{\varphi}_{1}$ is the function $\varphi_{1}$ associated to $\tilde{\Theta}_{1}$ according to (3.8), and $\tilde{\Theta}_{1}$ is defined in (3.15) with $T_{\gamma}:=T-\displaystyle\frac{3}{4}\bar{a}_{1}$, $T_{\beta}:=T-\displaystyle\frac{\bar{a}_{1}}{8}$, $\gamma=0$, one has

[TABLE]

where, in this case, $f(t,a,x):=-\beta_{1}(a,x)z(t,0,x)$. Hence, by (3.26),

[TABLE]

It remains to estimate $\displaystyle\int_{\bar{Q}_{1}}\frac{y^{2}}{k_{2}}dxdadt$ and $\displaystyle\int_{T-\bar{a}_{1}}^{T}\int_{0}^{1}\left(\int_{0}^{T-t}\frac{1}{k_{1}}y^{2}(s+t,s,x)ds\right)dxdt$. First of all, we prove

[TABLE]

Again we have

[TABLE]

By (3.35) and proceeding as before,

[TABLE]

where $\bar{\Theta}_{2}$ is defined in (3.15) with $T_{\gamma}:=T-\bar{a}_{1}$, $T_{\beta}:=T$, $\gamma=0$ and $\bar{\varphi}_{2}$ is the function $\varphi_{2}$ associated to $\bar{\Theta}_{2}$ according to (3.8). Thus, by Theorem 3.1 (or Theorem 3.3) applied to $Q_{2}:=\left(T-\bar{a}_{1},T\right)\times(0,A)\times(0,1)$ ,

[TABLE]

where, in this case, $g(t,a,x):=-\beta_{2}(a,x)y(t,0,x)$. Hence, by (3.27),

[TABLE]

for a strictly positive constant $C$. Thus (3.39) follows. Finally, we prove that there exists a positive constant $C$, such that

[TABLE]

Indeed, since $t\in(T-\bar{a}_{1},T)$ and $\bar{a}_{1}\leq\bar{a}_{2}$, it follows $t\geq T-\bar{a}_{1}\geq T-\bar{a}_{2}$. Thus, by the implicit formula of $y$, it follows that

[TABLE]

Hence

[TABLE]

$\bar{a}_{1}>\bar{a}_{2}$: As before, one can prove (3.32), (3.33), (3.37) and hence (3.17). Now, we prove (3.18). As for the case $\bar{a}_{1}\leq\bar{a}_{2}$, we multiply the equation of (3.31) by $\displaystyle-\frac{W}{k_{1}}$ and integrate by parts on $\bar{Q}_{2,t}:=(T_{1},t)\times(0,A)\times(0,1)$. Hence, one can obtain

[TABLE]

Now, take $\delta\in(0,A)$. Then, integrating (3.41) over $\displaystyle\left[T-\frac{\bar{a}_{2}}{4},T-\frac{\bar{a}_{2}}{8}\right]$, we have

[TABLE]

Proceeding as in (3.36), we have

[TABLE]

where $\tilde{\Theta}_{2}$ and $\tilde{\varphi}_{2}$ are as in (3.36), hence $T_{\gamma}:=T-\bar{a}_{2}$, $T_{\beta}:=T$ and $\gamma=0$. Observe that we have $\displaystyle t\geq T-\frac{\bar{a}_{2}}{4}\geq T-\bar{a}_{2}$, hence we can apply the implicit formula (3.27).

Now, consider $\displaystyle\int_{T-\frac{\bar{a}_{2}}{4}}^{T-\frac{\bar{a}_{2}}{8}}\int_{\delta}^{A}\int_{0}^{1}\frac{1}{k_{1}}z^{2}dxdadt$. As before, by Theorem 3.3 applied to $\bar{Q}_{1}:=\left(T-\displaystyle\frac{\bar{a}_{2}}{2},T-\displaystyle\frac{\bar{a}_{2}}{10}\right)\times(0,A)\times(0,1)$, replacing $\tilde{\varphi}_{2}$, $\tilde{\Theta}_{2}$ by $\tilde{\varphi}_{1}$, $\tilde{\Theta}_{1}$, respectively, where $\tilde{\varphi}_{1}$ is the function $\varphi_{1}$ associated to $\tilde{\Theta}_{1}$ according to (3.8), and $\tilde{\Theta}_{1}$ is defined in (3.15) with $T_{\gamma}:=T-\displaystyle\frac{\bar{a}_{2}}{2}$, $T_{\beta}:=T-\displaystyle\frac{\bar{a}_{2}}{10}$, $\gamma=0$, one has

[TABLE]

where again $f(t,a,x):=-\beta_{1}(a,x)z(t,0,x)$. Hence

[TABLE]

Again we have

[TABLE]

Now, it remains to estimate

[TABLE]

Since $t\in\left[T-\frac{\bar{a}_{2}}{2},T-\frac{\bar{a}_{2}}{10}\right]$, it holds $\frac{\bar{a}_{2}}{10}\leq T-t$, thus one can divide the integral in the following way

[TABLE]

Now, by (3.35) and Theorem 3.1 (or Theorem 3.3) applied to $y$ in $\bar{Q}_{3}:=\left(T-\bar{a}_{2},T-\frac{\bar{a}_{2}}{12}\right)\times(0,T-t)\times(0,1)$, replacing $\varphi_{2}$, $\Theta_{2}$ by $\tilde{\varphi}_{3}$, $\tilde{\Theta}_{3}$, respectively, where $\tilde{\varphi}_{3}$ is the function $\varphi_{2}$ associated to $\tilde{\Theta}_{3}$ according to (3.8), and $\tilde{\Theta}_{3}$ is defined in (3.15) with $T_{\gamma}:=T-\bar{a}_{2}$, $T_{\beta}:=T-\frac{\bar{a}_{2}}{12}$, $\gamma=0$, one has

[TABLE]

where, in this case, $h(s+t,s,x):=-\beta_{2}(s,x)y(s+t,0,x)$. Moreover, $s+t\geq t\geq T-\bar{a}_{2}$; thus, by (3.27), one has

[TABLE]

Analogously,

[TABLE]

Finally, since $t\geq T-\frac{\bar{a}_{2}}{2}\geq T-\bar{a}_{2}$, one has $s+t\geq T-\bar{a}_{2}+s$; thus, by (3.24) obtaining

[TABLE]

Hence, (3.18) holds also in the case $\bar{a}_{1}>\bar{a}_{2}$. ∎

We underline that in this paper we improve (3.17) given in [16] for a single equation. Indeed here we do not require that $T<A$ and the proof is simpler. The same improvement holds also for the next result which generalizes the previous theorem.

Theorem 3.5**.**

Assume Hypotheses 2.1, 3.2, 3.4 and 3.5. Then, for every $\delta\in(\gamma,A)$, where $\gamma=\max\{\bar{a}_{1},\bar{a}_{2}\}$, there exists a strictly positive constant $C=C(\delta)$ such that every solution (z,y)\in L^{2}\big{(}Q_{T,A};\mathbb{K}\big{)}\cap H^{1}\big{(}0,T;H^{1}(0,A;\mathbb{H})\big{)} of (3.5) satisfies

[TABLE]

and

[TABLE]

Proof.

As in the previous theorem, we distinguish between the case $\bar{a}_{1}\leq\bar{a}_{2}$ and $\bar{a}_{1}>\bar{a}_{2}$.

First of all, assume $\bar{a}_{1}\leq\bar{a}_{2}$. Again (3.32) and (3.33) hold. Then, integrating (3.32) over $\left[T-\displaystyle\frac{\bar{a}_{2}}{2},T-\displaystyle\frac{\bar{a}_{2}}{4}\right]$ and (3.33) over $\left[T-\displaystyle\frac{\bar{a}_{1}}{2},T-\displaystyle\frac{\bar{a}_{1}}{4}\right]$, we have for all $\delta>\gamma$:

[TABLE]

and

[TABLE]

Proceeding as before, one can prove (3.37). Thus, using Theorem 3.1, we can prove

[TABLE]

It remains to estimate

[TABLE]

Observe that $t\geq T-\displaystyle\frac{\bar{a}_{2}}{2}\geq T-\bar{a}_{2}$ and $a\in(0,\delta-\bar{a}_{2})$. Thus $T-t\leq\bar{a}_{2}\leq\delta-a\leq A-a$ and, by the first formula in (3.25), we have

[TABLE]

It follows:

[TABLE]

[TABLE]

By (3.48)-(3.51), (3.46) follows. Now, we estimate (3.49). Proceeding as in the previous theorem and in (3.51), one has

[TABLE]

and

[TABLE]

Moreover, by (3.44),

[TABLE]

Now, we consider the integral

[TABLE]

Observe that $t\geq T-\displaystyle\frac{\bar{a}_{1}}{2}\geq T-\bar{a}_{1}$ and $a\in(0,\delta-\bar{a}_{1})$. Thus $T-t\leq\bar{a}_{1}\leq\delta-a\leq A-a$ and, by the first formula in (3.23), the first formula in (3.25) and by (3.26), it follows

[TABLE]

[TABLE]

Now, we assume $\bar{a}_{1}>\bar{a}_{2}$:

As before, one can prove (3.48), (3.50), (3.51) and hence (3.46). Now, we prove (3.47). Again (3.41) holds with $\bar{Q}_{2,t}:=\left(\displaystyle T-\frac{\bar{a}_{2}}{4},t\right)\times(0,A)\times(0,1)$. Now, take $\delta\in(\gamma,A)$. Then, integrating (3.41) over $\displaystyle\left[T-\frac{\bar{a}_{2}}{4},T-\frac{\bar{a}_{2}}{8}\right]$, we have

[TABLE]

Proceeding as in (3.37), we have

[TABLE]

(observe that (3.36) holds with $\tilde{\Theta}_{2}$ and $\tilde{\varphi}_{2}$ as in (3.36), and $T_{\gamma}:=T-\bar{a}_{2}$, $T_{\beta}:=T$ and $\gamma=0$). Moreover, we have $\displaystyle t\geq T-\frac{\bar{a}_{2}}{4}\geq T-\bar{a}_{2}$, hence we can apply the implicit formula (3.27). Now, consider the term

[TABLE]

Clearly, as in (3.51),

[TABLE]

Now, consider $\displaystyle\int_{T-\frac{\bar{a}_{2}}{4}}^{T-\frac{\bar{a}_{2}}{8}}\int_{\delta-\bar{a}_{1}}^{A}\int_{0}^{1}\frac{1}{k_{1}}z^{2}dxdadt$. As before, by Theorem 3.1 applied to $\bar{Q}_{1}:=\left(T-\displaystyle\frac{\bar{a}_{2}}{2},T-\displaystyle\frac{\bar{a}_{2}}{10}\right)\times(0,A)\times(0,1)$, replacing $\tilde{\varphi}_{2}$, $\tilde{\Theta}_{2}$ by $\tilde{\varphi}_{1}$, $\tilde{\Theta}_{1}$, respectively, where $\tilde{\varphi}_{1}$ is the function $\varphi_{1}$ associated to $\tilde{\Theta}_{1}$ according to (3.8), and $\tilde{\Theta}_{1}$ is defined in (3.15) with $T_{\gamma}:=T-\displaystyle\frac{\bar{a}_{2}}{2}$, $T_{\beta}:=T-\displaystyle\frac{\bar{a}_{2}}{10}$, $\gamma=0$, one has

[TABLE]

where again $f(t,a,x):=\mu_{21}(t,a,x)y-\beta_{1}(a,x)z(t,0,x)$. Hence, proceeding as in the previous theorem and as in (3.55), one has

[TABLE]

Now, it remains to estimate

[TABLE]

As in (3.45), one can prove

[TABLE]

Hence, (3.47) holds.

∎

Remark 2**.**

Using a density argument one can prove (3.46) and (3.47) for every solution $(z,y)\in C([0,T];L^{2}(0,A;\mathbb{L}))\cap L^{2}(0,T;H^{1}(0,A;\mathbb{H}))$ of (3.5).

3.3 Carleman inequalities and observability inequalities when the degeneracy is at $1$.

In this case, in order to obtain $\omega-$local Carleman estimates and $\omega-$local observability inequality, in place of $\varphi_{i}$, we consider the weight functions

[TABLE]

where $\Theta$ is as in (3.9) and $\displaystyle\bar{p}_{i}(x):=\int_{0}^{x}\frac{y-1}{k_{i}(y)}~{}e^{R(y-1)^{2}}dy$, with $R>0$ and $i=1,2$. Assume

Hypothesis 3.6**.**

The functions $k_{i}\in C^{0}[0,1]\bigcap C^{2}[0,1)$ are such that $k_{i}(1)=0$, $k_{i}>0$ on $[0,1)$ and

$k_{i}$* satisfies Hypothesis 3.3.1, for $i=1,2$, * 2. 2.

there exists a positive constant $C$ such that $k_{1}(x)\geq k_{2}(x)$ for all $x\in[0,1]$.

Proceeding as in the proof of Theorem 3.3, one can prove the next estimate:

Theorem 3.6**.**

Assume Hypotheses 2.1, 3.2 and 3.6. Take $f\in L^{2}_{\frac{1}{k_{1}}}(Q)$ and $g\in L^{2}_{\frac{1}{k_{2}}}(Q)$. Then, there exist two strictly positive constants $C$ and $s_{0}$ such that every solution (z,y)\in L^{2}\big{(}Q_{T,A};\mathbb{K}\big{)}\cap H^{1}\big{(}0,T;H^{1}(0,A;\mathbb{H})\big{)} of (3.11) satisfies, for all $s\geq s_{0}$,

[TABLE]

and

[TABLE]

As a consequence of the previous result, it holds:

Theorem 3.7**.**

Assume Hypotheses 2.1, 3.2, 3.6 and 3.5. Then, for every $\delta\in(0,A)$, there exists a strictly positive constant $C=C(\delta)$ such that every solution (z,y)\in L^{2}\big{(}Q_{T,A};\mathbb{K}\big{)}\cap H^{1}\big{(}0,T;H^{1}(0,A;\mathbb{H})\big{)} of (3.5) satisfies (3.17) and (3.18). Moreover, if $y_{T}(a,x)=0$ for all $(a,x)\in(0,\bar{a}_{2})\times(0,1)$ and $z_{T}(a,x)=0$ for all $(a,x)\in(0,\bar{a}_{1})\times(0,1)$, one has

[TABLE]

and

[TABLE]

Moreover,

Theorem 3.8**.**

Assume Hypotheses 2.1, 3.2, 3.6 and 3.5. Then, for every $\delta\in(\gamma,A)$, where $\gamma=\max\{\bar{a}_{1},\bar{a}_{2}\}$, there exists a strictly positive constant $C=C(\delta)$ such that every solution (z,y)\in L^{2}\big{(}Q_{T,A};\mathbb{K}\big{)}\cap H^{1}\big{(}0,T;H^{1}(0,A;\mathbb{H})\big{)} of (3.5) satisfies (3.46) and (3.47).

Again, using a density argument one can prove (3.46) and (3.47) even if $(z,y)\in C([0,T];L^{2}(0,A;\mathbb{L}))\cap L^{2}(0,T;H^{1}(0,A;\mathbb{H}))$; hence, Remark 2 still holds if $k_{i}(1)=0$, for $i=1,2$.

4 Null controllability via observability inequality

In this section we will deduce a null controllability result for (1.2) by Theorems 3.5, 3.8 and Remark 2. Actually, thanks to Theorems 3.5 and 3.8, we can obtain a null controllability result for the following intermediate problem:

[TABLE]

where, we recall, $\tilde{Q}_{i}=(T-\bar{a}_{i},T)\times Q_{A,1}$, $\tilde{Q}_{i,T,1}=(T-\bar{a}_{i},T)\times(0,1)$ and $\tilde{Q}_{i,T,A}=(T-\bar{a}_{i},T)\times(0,A)$, $i=1,2$. In particular, the following result holds:

Theorem 4.1**.**

Assume Hypotheses 2.1, 3.2, 3.5 and 3.4 or 3.6 and suppose $\bar{a}_{1}=\bar{a}_{2}$. Fix $T>0$ and $(u_{0},v_{0})\in L^{2}(0,A;\mathbb{L})$. Then for every $\delta\in(\gamma,A)$, there exist a contros $g_{\delta}\in L^{2}_{\frac{1}{k_{1}}}(\tilde{Q}_{1})$ such that the solution $(u_{\delta},v_{\delta})\in C([T-\bar{a}_{1},T];L^{2}(0,A;\mathbb{L}))\cap L^{2}(T-\bar{a}_{1},T;H^{1}(0,A;\mathbb{H}))$ of (4.57) satisfies

[TABLE]

Moreover, there exists $C=C(\delta)>0$, such that

[TABLE]

Before proving the previous result observe that, if $\bar{a}_{1}=\bar{a}_{2}$, then $(u,v)$ solves (4.57) if and only if $v$ satisfies

[TABLE]

where $u$ solves

[TABLE]

where $\tilde{Q}:=(T-\bar{a}_{1},T)\times Q_{A,1}$, $\tilde{Q}_{T,A}=(T-\bar{a}_{1},T)\times(0,A)$ and $\tilde{Q}_{T,1}:=(T-\bar{a}_{1},T)\times(0,1)$.

Proof of Theorem 4.1.

Take $(g_{1},g_{2})\in L^{2}(0,A;\mathbb{L})$ such that $g_{i}(A,x)=0$ in $(0,1)$, $i=1,2$, and fix $\delta\in(\gamma,A)$. Let $y$ be the solution of

[TABLE]

Now, fixed $v_{0}\in L^{2}(0,A;L^{2}_{\frac{1}{k_{2}}}(0,1))$, define

[TABLE]

The functional $J$ is strictly convex, continuous and coercive over the Hilbert space $\mathcal{H}$ defined by the completion of $L^{2}(\delta,A;L^{2}_{\frac{1}{k_{2}}}(0,1))$ with respect to the norms $\|v\|_{L^{2}(\tilde{Q}_{T,A}\times\omega)}$ (see [2]). Thus, there exists a unique minimum, $\hat{g}_{2}$, of $J$ and $\hat{g}_{2}(A,x)=0$ in $(0,1)$. Let $\hat{y}$ be the solution of(4.62) associated to $\hat{g}_{2}$.

Now, fixed $u_{0}\in L^{2}(0,A;L^{2}_{\frac{1}{k_{1}}}(0,1))$, let $z$ be the solution of

[TABLE]

and define

[TABLE]

As before, there exists a unique minimum, $\hat{g}_{1}$, of $F$ and $\hat{g}_{1}(A,x)=0$ in $(0,1)$. Let $\hat{z}$ be the solution of(4.63) associated to $\hat{g}_{1}$.

Since $\hat{g}_{2}$ is the minimum of $J$, it results

[TABLE]

for all $g_{2}\in L^{2}(Q_{A,1})$ such that $g_{2}(A,x)=0$ in $(0,1)$. Analogously, we have

[TABLE]

for all $g_{1}\in L^{2}(Q_{A,1})$ such that $g_{1}(A,x)=0$ in $(0,1)$. In particular, for $g_{2}=\hat{g}_{2}$ and $g_{1}=\hat{g}_{1}$, it results

[TABLE]

and

[TABLE]

By Hölder’s inequality, Remark 2, Theorems 3.5 and 3.8 applied to $\hat{y}$ in $\tilde{Q}_{2}$ and using the fact that $y_{T}(a,x)=0$ for all $(a,x)\in(0,\delta)\times(0,1)$, one has

[TABLE]

Thus, by (4.66) and (4.68), it follows

[TABLE]

Hence

[TABLE]

Analogously, by Hölder’s inequality, Remark 2, Theorems 3.5 and 3.8 applied to $\hat{z}$ in $\tilde{Q}$ and using the fact that $z_{T}(a,x)=0$ for all $(a,x)\in(0,\delta)\times(0,1)$, one has

[TABLE]

In the last inequality we have used the Cauchy’s inequality $\displaystyle ab\leq\epsilon a^{2}+\frac{1}{\epsilon}b^{2}$ with $\displaystyle\epsilon=\frac{1}{2}$. Thus, by (4.67), (4.70) and (4.71)

[TABLE]

Hence

[TABLE]

Now, define $g:=\hat{z}\chi_{\omega}$ and let $u$ be the solution of (4.61) in $\tilde{Q}_{1}$ associated to $g$ and $u_{0}$ and let $v$ be the solution of (4.60) in $\tilde{Q}_{2}$ associated to $u$ and $v_{0}$. By (4.72), it follows

[TABLE]

Moreover, multiplying the equation of (4.63) by $\displaystyle\frac{u}{k_{1}}$ and integrating over $\tilde{Q}_{1}$, one has:

[TABLE]

Recall that $u(t,0,x)=\int_{0}^{A}\beta_{1}(a,x)u(t,a,x)da$ and $\displaystyle\frac{\partial u}{\partial t}+\frac{\partial u}{\partial a}-k_{1}(x)u_{xx}+\mu_{11}(t,a,x)u=g\chi_{\omega}$; hence

[TABLE]

Thus, being by (4.65),

[TABLE]

it follows

[TABLE]

for all $g_{1}\in L^{2}_{\frac{1}{k_{1}}}(Q_{A,1})$ with $g_{1}(A,x)=0$ in $(0,1)$, $i=1,2$. In particular, for $g_{1}\equiv 0$, it results

[TABLE]

Hence, by (4.73),

[TABLE]

for all $g_{1}\in L^{2}_{\frac{1}{k_{1}}}(Q_{A,1})$ with $g_{1}(A,x)=0$ in $(0,1)$, $i=1,2$. Thus,

[TABLE]

Now, multiplying the equation of (4.62) by $\displaystyle\frac{v}{k_{2}}$ and integrating over $\tilde{Q}_{2}$, one has:

[TABLE]

Recall that $v(t,0,x)=\int_{0}^{A}\beta_{2}(a,x)v(t,a,x)da$ and $\displaystyle\frac{\partial v}{\partial t}+\frac{\partial v}{\partial a}-k_{2}(x)v_{xx}+\mu_{22}(t,a,x)v=\mu_{21}u$; hence

[TABLE]

Thus, being by (4.64), it follows

[TABLE]

for all $g_{2}\in L^{2}_{\frac{1}{k_{2}}}(Q_{A,1})$ with $g_{2}(A,x)=0$ in $(0,1)$. In particular, for $g_{2}\equiv 0$, it follows

[TABLE]

Proceeding as before, we can conclude that

[TABLE]

Hence, the thesis follows. ∎

Theorem 4.2**.**

Assume Hypotheses 2.1, 3.2, 3.5 and 3.4 or 3.6, suppose $k_{1}(x)=k_{2}(x)$ for all $x\in(0,1)$ and $\bar{a}_{1}=\bar{a}_{2}$. Fix $T>0$ and $(u_{0},v_{0})\in L^{2}(0,A;\mathbb{L})$. Then for every $\delta\in(\gamma,A)$, there exist a control $g_{\delta}\in L^{2}_{\frac{1}{k_{1}}}(Q)$ such that the solution $(u_{\delta},v_{\delta})\in C([0,T];L^{2}(0,A;\mathbb{L}))\cap L^{2}(0,T;H^{1}(0,A;\mathbb{H}))$ of (1.2) satisfies

[TABLE]

Moreover, there exists $C=C(\delta)>0$ such that

[TABLE]

Proof.

Set $T_{1}=T-\bar{a}_{1}$ (clearly $T_{2}=T_{1}$). By Theorem 2.1, there exists a unique solution $(U,V)$ of

[TABLE]

where $\bar{Q}=(0,T_{1})\times Q_{A,1}$, $\bar{Q}_{T_{1},1}=(0,T-\bar{a}_{1})\times(0,1)$ and $\bar{Q}_{T_{1},A}=(0,T-\bar{a}_{1})\times(0,A)$, $i=1,2$.

Set $\tilde{u}_{0}(a,x):=U(T_{1},a,x)$, $\tilde{v}_{0}(a,x):=V(T_{1},a,x)$; clearly $\tilde{u}_{0},\tilde{v}_{0}\in L^{2}_{\frac{1}{k_{1}}}(Q_{A,1})$. Now, consider

[TABLE]

Again, by Theorem 2.1, there exists a unique solution $(W,Y)$ of (4.78) and, by the previous theorem, for every $\delta\in(\gamma,A)$ there exist a control $G_{\delta}\in L^{2}_{\frac{1}{k_{1}}}(Q)$ such that

[TABLE]

Now, define $(u,v)$ and $g_{\delta}$ by

[TABLE]

and

[TABLE]

Then $(u,v)$ satisfies (1.2) and $g_{\delta}\in L^{2}_{\frac{1}{k_{1}}}(Q)$ is such that

[TABLE]

It remains to prove (4.76). To this aim, observe that, by (4.59)

[TABLE]

for a strictly positive constant $C=C_{\delta}$. Thus, it is sufficient to estimate the last integral in (4.79). To do this, we multiply the two equations of (4.77) by $\displaystyle\frac{U}{k_{1}}$ and $\displaystyle\frac{V}{k_{1}}$, respectively; then integrating over $Q_{A,1}$, we obtain

[TABLE]

and

[TABLE]

Hence, using the fact that $U(t,0,x)=\int_{0}^{A}\beta_{1}(a,x)U(t,a,x)da$ and $V(t,0,x)=\int_{0}^{A}\beta_{2}(a,x)V(t,a,x)da$, one has

[TABLE]

and

[TABLE]

Thus,

[TABLE]

and

[TABLE]

Setting $F_{1}(t):=\|U(t)\|^{2}_{L^{2}_{\frac{1}{k_{1}}}(Q_{A,1})}$ and multiplying (4.80) by $e^{-Ct}$, it results

[TABLE]

Integrating over $(0,t)$, for all $t\in(0,T),$ we have

[TABLE]

In particular,

[TABLE]

Now, set $F_{2}(t):=\|V(t)\|^{2}_{L^{2}_{\frac{1}{k_{1}}}(Q_{A,1})}$ and multiply (4.81) by $e^{-Ct}$, it results, by (4.82),

[TABLE]

Integrating over $(0,t)$, for all $t\in(0,T),$ we have

[TABLE]

In particular,

[TABLE]

Hence, by (4.79),

[TABLE]

Thus (4.76) follows. ∎