Local Controllability Analysis for a Tumor Dynamics Model

Puma, F. F. C.

DOI 10.5433/1679-0375.2025.v46.53579

Citation Semin., Ciênc. Exatas Tecnol. 2025, v. 46: e53579

Received: August 5, 2025 Received in revised for: October 15, 2025 Accepted: November 12, 2025 Available online: December 2, 2025

Abstract:

This study investigates the local controllability of a mathematical model describing tumor dynamics, incorporating cellular competition and external interventions represented by a control variable. The model analyzed is a modified version of the formulation proposed by , and its dynamics are explored through linearization around equilibrium points. Using the Kalman rank condition, we establish the circumstances under which the system exhibits local controllability. The results provide a theoretical foundation for understanding how external stimuli can steer the system between distinct biological states, particularly in the context of interactions between healthy and tumor cells. The work bridges concepts from control theory and mathematical oncology, aiming to support the development of more effective clinical intervention strategies. Our analysis offers insights into identifying scenarios in which treatment, modeled as an external control input, can effectively stabilize or even reverse tumor growth.

Keywords: cancer modeling, controllability theory, Kalman rank condition, mathematical oncology

Introduction

Salvador and Arenales (2022) emphasize that mathematical modeling of biological systems is a rich and active research area, with numerous models developed to describe population dynamics. Addressing a biological phenomenon involves understanding and defining it as precisely as possible, selecting the necessary information. Transposing this phenomenon into a problem using appropriate mathematical language contributes to its understanding, equation-solving, simulation, and resolution.

In particular, Lotka–Volterra-type models have received significant attention, as stability analyses within optimal control frameworks yield valuable insights into their dynamics (Zhang et al., 2022; Wang et al., 2016; El-gohary et al., 2001). One important application of these models is in the study of tumor growth.

Cancer remains the foremost public health challenge worldwide and one of the leading causes of death. Despite extensive research efforts, several fundamental aspects of the disease, including its origin, the mechanisms of progression, and its behavior in specific tissues, are still not fully understood (Bray et al., 2021; Rodrigues et al., 2011). From the perspective of mathematical modeling, this complexity is reflected in the difficulty of constructing models entirely grounded in experimentally validated biological mechanisms. As highlighted by Murray (s2002), most available tumor growth models are based on empirical curve fitting, simplified assumptions, or computational heuristics, which limits their biological and clinical applicability.

Nevertheless, mathematical modeling has proven to be an indispensable tool for understanding tumor dynamics and devising therapeutic strategies. Models based on ordinary differential equations (ODEs), in particular, have been widely used to simulate tumor progression, angiogenesis, and the effects of therapies (Byrne et al., 1995; Roose et al., 2007; Cristini et al., 2010). These models offer both qualitative and quantitative insights into the mechanisms involved in tumor development, as well as enabling the incorporation of control theory concepts to assess the impact of therapeutic interventions as highlighted by De pillis and Radunskaya (2003) and Nave (2022).

A notable example in this direction is the model introduced by (Gatenby et al., 1996b), which describes the competitive interaction between normal cells (\(N_1\)) and tumor cells (\(N_2\)) in a tissue. This formulation draws inspiration from ecological modeling principles, treating the tumor as an invasive species competing for space and resources. This approach has inspired a series of studies on tumor–host interactions and control possibilities, as discussed in Bellomo et al. (2008).

In this work, we analyze a dynamic tumor model that incorporates both cellular competition and external interventions represented by a control variable (\(u\)). In particular, we study a perturbation of the model originally proposed by Gatenby (1996), now extended with a control term representing therapeutic action. Our main objective is to investigate the local controllability of this system, understood as the ability to steer the dynamics locally between biologically meaningful states through appropriate control inputs. Puma and Henajeros (2024) present this type of analysis for a classical Lotka-Volterra model.

Based on the Kalman rank condition, we establish criteria under which local controllability is guaranteed. These results provide a theoretical foundation for understanding how therapeutic interventions, such as drug administration, can locally guide the biological system between different stages of tumor–healthy tissue interaction. Our analysis offers insights into identifying conditions under which treatment can effectively stabilize or reverse tumor growth, thus contributing to the development of mathematical approaches to support cancer therapy.

Based on the above and the author’s knowledge, it is observed that, in the existing studies on the perturbed Gatenby model, there is no analysis from the perspective of local controllability. Therefore, the contributions of this work can be summarized as follows:

1. To formulate the local controllability problem of the control system in the neighborhood of its equilibrium points;

2. To identify specific equilibrium regions in which the dynamics of the control system ensure local controllability.

Throughout this paper, \(\mathbf{P} \in \mathbb{R}^{n \times m}\) denotes a matrix, and \(\mathbf{P}^\top\) its transpose. \(\mathbb{R}_+\) denotes the set of positive real numbers and \(\mathbb{R}_-\) the set of negative real numbers. To \(N(t)\in \mathbb{R}^{n\times 1}\), the notation \(\dot N(t)\) denotes its derivative. Moreover, it is denoted \(\sigma (\mathbf{A})\) as the eigenvalues set of matrix \(\mathbf{A}\in \mathbb{R}^{n\times n}\).

Material and methods

The methodology of this work consists of the formulation, analysis, and application of control theory tools to the study of the mathematical model that describes the competitive interaction between normal cells (\(N_1\)) and tumor cells (\(N_2\)). The competition between the cell populations \((N_1,N_2)\) is modeled analogously to the Lotka–Volterra model, incorporating resource limitation effects and interspecies interaction, (Gatenby et al., 1996a). The model is then modified to include a control function \(u(t)\), representing the therapeutic action applied to the system (e.g., chemotherapy or another pharmacological agent).

The control system can be generically expressed as:

\[ \left\{\begin{array}{ccl} \dot{N}_1 &=& f_1(N_1, N_2), \\ \dot{N}_2 &=& f_2(N_1, N_2) - u(t). \end{array}\right. \tag{1} \]

In the system described by equation (2), \(f_1\) and \(f_2\) represent the natural dynamics of the model, and \(-u(t)\) models the effect of the external intervention. For simplicity, we denote \(x=(N_1,N_2)^\top\), \(\mathbf{B}=(0,-1)^\top\), and \(f(x) = (f_1(x),f_2(x))^\top\). We now consider the equation \[ \dot x = f(x)+\mathbf{B}u. \tag{2} \] Equation (2) is a rewrite of equation (1) and from now on we will use this notation.

Definition 1. Let \(\mathcal{V}\) be an open subset of \(\mathbb{R}^n\), \(f:\mathcal{V} \rightarrow \mathbb{R}^n \in C^1\), and \(\mathbf{B} \in \mathbb{R}^{n\times m}\) a constant matrix. An Equilibrium Point or Equilibrium Solution of the system, given by equation (2), is a pair \((x^*,h) \in \mathcal{V} \times \mathbb{R}^m\) such that \[ f(x^*) + \mathbf{B}h = 0. \tag{3} \]

The notion of Local Controllability via trajectories for a system given by the equation (2) is defined in Coron (2007), since an equilibrium solution \((x^*,h)\), defined by the equation (3), is a type of constant trajectory.

Definition 2. Let \((x^*, h)\) be an equilibrium point of the control system by equation (2). Such a system is said to be locally controllable at the equilibrium \((x^*, h)\) if, for every real number \(\epsilon > 0\), there exists a real number \(\eta > 0\) such that, for every \(x_0 \in B_\eta(x^*) := \{ x \in \mathbb{R}^n \;|\; \|x - x^*\| < \eta \}\) and for every \(x_1 \in B_\eta(x^*)\), there exists a measurable function \(u : [0, \epsilon] \to \mathbb{R}^m\) such that \[ ||u(t) - h|| \leq \epsilon, \quad \forall t \in [0, \epsilon], \tag{4} \] and the solution to the Cauchy problem \[ \dot{x} = f(x) + \mathbf{B}u(t), \quad x(0) = x_0 \tag{5} \] satisfies \[ x(\epsilon) = x_1. \tag{6} \]

No checkable necessary and sufficient condition is known for small-time local controllability for general control systems, even for analytic systems. However, there are powerful sufficient conditions ([CORON:2007 - missing information]).

On the other hand, for linear vector fields, i.e., \(f(x) = \mathbf{A}x\), there is a result that characterizes controllability, known as the Kalman Theorem. As stated in Baumeister and Leitão (2014), this theorem asserts that:

Theorem 1. The Kalman Test \[ \mathrm{rank} \ \left[ \mathbf{B} \ \ \mathbf{A}\mathbf{B} \ \ \mathbf{A}^2\mathbf{B} \ \ \cdots \ \ \mathbf{A}^{n-1}\mathbf{B}\right] = n, \tag{7} \] is a necessary and sufficient condition for the system \(\dot x=\mathbf{A}x+\mathbf{B}u\) to be controllable. That is, for every \(x_0, x_1 \in \mathbb{R}^n\), there exists a measurable and essentially bounded function \(u : [0, \epsilon] \to \mathbb{R}^m\) such that the solution to the Cauchy problem \[ \dot{x} = \mathbf{A}x+\mathbf{B}u, \quad x(0) = x_0 \tag{8} \] satisfies \[ x(\epsilon) = x_1. \tag{9} \] The system of equations, (8) and (3), is called a linear control system and \(u\) is the control function so that the trajectory starts at \(x_0\) and ends at \(x_1\).

Definition 3. Let \((x^*, h)\) be an equilibrium point of the control system by equation (2). The linearized control system at \((x^*, h)\) is the linear control system: \[ \dot{x} = {D f}(x^*) x + \mathbf{B} u, \tag{10} \] where, at time \(t\), the state is \(x(t) \in \mathbb{R}^n\) and the control is \(u(t) \in \mathbb{R}^m\).

The local controllability analysis of system given by equation (2) can be carried out by means of the linearization of the system around an equilibrium point \((x^*, h)\). Coron (2007) demonstrated this with the following theorem:

Theorem 2. Let \((x^*, h)\) be an equilibrium point of the control system by equation (2). Suppose that the linearized control system (10) at \((x^*, h)\) is controllable. Then, the nonlinear control system \(\dot{x} = f(x) + \mathbf{B}u\) is locally controllable at that equilibrium.

Theorem 2, which is straightforward and natural, is very useful. It is worth recalling that the controllability of the linearized system at \((x^*, h)\) can be easily verified using the Kalman rank condition (Theorem 1), as also employed by Puma and Henajeros (2024).

Analysis of Equilibrium and Stability

Competition between species for space and limited resources is a central phenomenon in ecology, Murray (2002). To study the interactions between populations and predict their dynamic behaviors, mathematical models such as the system Lotka (1925); Volterra (1931) are widely employed. A relevant application of these models arises in the context of cancer (Rodrigues et al., 2011). We analyze the following system:

\[ \begin{cases} \dot N_1 = \alpha N_1 \left(1 - \dfrac{N_1}{k_1} - \dfrac{a_{12} N_2}{k_1}\right) \\[0.35cm] \dot N_2 = \beta N_2 \left(1 - \dfrac{N_2}{k_2} - \dfrac{a_{21} N_1 }{k_2}\right) - u(t) \end{cases} \tag{11}\]

where \(N_1\) and \(N_2\) represent healthy and tumor cells respectively, with growth rates \(\alpha\), \(\beta\), and carrying capacities \(k_1\), \(k_2\). The parameters \(a_{12}\) and \(a_{21}\) represent interspecific competition, while the input \(u(t)\) models the effect of an external treatment.

To find the equilibria \((e,h)\in\mathbb{R}^2\times\mathbb{R}\) of (11), we solve \(\dot N_1 = 0\) and \(\dot N_2 = 0\). The equilibrium points \(e(h)=(N^*_1(h),N^*_2(h))\), with \(h\) being the harvesting parameter (\(u=h\)), are as follows:

• Total extinction: \[ e_1 \ = \ (0,0). \tag{12} \]

• Competitive exclusion: \[e_2 \ = \ (k_1,0). \tag{13} \]

• Competitive exclusion: \(e_3^+(h)\) and \(e_3^-(h)\) where

\[e_3^+(h) \ = (0,k_2^+(h))\ = \ \left(0\ ,\frac{k_2}{2}+ \sqrt{\left( \frac{k_2}{2} \right)^2-\frac{k_2}{\beta}h}\right) \ \; \ \ \ h\in \left[0,\frac{k_2\beta}{4}\right]. \tag{14} \] \[e_3^-(h) \ = \ (0,k_2^-(h))\ = \ \left(0\ ,\frac{k_2}{2}- \sqrt{\left( \frac{k_2}{2} \right)^2-\frac{k_2}{\beta}h}\right) \ \; \ \ \ h\in \left(0,\frac{k_2\beta}{4}\right). \tag{15} \]

• Coexistence: \(e_4^+(h)\) and \(e_4^-(h)\), represent positive equilibria.

\[ e_4^{+}(h) \ = \ (N_1^{+}(h),N_2^+(h))\ , \ \ \ a_{12}<\frac{k_1}{k_2}<\frac{1}{a_{21}} \ \text{ and } \ \ h\in \left( 0, \frac{\beta(1-a_{12}a_{21})}{4k_2}R_2^2\right] \tag{16} \]

\[ e_4^{-}(h) \ = \ (N_1^{-}(h),N_2^-(h))\ , \ \ \ a_{12}<\frac{k_1}{k_2}<\frac{1}{a_{21}} \ \text{ and } \ \ h\in \left( 0, \frac{\beta(1-a_{12}a_{21})}{4k_2}R_2^2\right) \tag{17} \]

where \(N^\pm_1(h) = k_1 - a_{12}N^\pm_2(h)\),  \(N^\pm_2 (h)= \dfrac{R_2}{2}\pm \sqrt{\left(\dfrac{R_2}{2}\right)^2-\dfrac{k_2 h}{\beta(1 - a_{12}a_{21})}}\) and \(R_2= \dfrac{k_2 - a_{21}k_1}{1 - a_{12}a_{21}}\). \[ \] Let \[I = \left(0, \frac{\beta(1-a_{12}a_{21})}{4k_2} R_2^2 \right) \mbox{ and } \quad R_1 = \frac{k_1 - a_{12} k_2}{1 - a_{12} a_{21}}. \] If \(a_{12} < \dfrac{k_1}{k_2} < \dfrac{1}{a_{21}}\), then:

1. \(\left( N_1^+(0), N_2^+(0) \right) = (R_1, R_2) \in (0,k_1) \times (0,k_2)\).

2. \(\left( N_1^\pm(h), N_2^\pm(h) \right) \in (0,k_1) \times (0,k_2)\) for all \(h \in I\).

We will prove the statements in parts.

1. First, we show that \(\left( N_1^+(0), N_2^+(0) \right) = (R_1, R_2) \in (0,k_1) \times (0,k_2)\). Indeed, \[R_1 = \frac{k_1 - a_{12} k_2}{1 - a_{12} a_{21}},\] and by the condition \(a_{12} < \dfrac{k_1}{k_2}\), we have \(k_1 - a_{12} k_2 > 0\). Moreover, \(1 - a_{12} a_{21} > 0\) since \(a_{12} a_{21}~<~\dfrac{k_1}{k_2} a_{21} < 1\). Thus, \(R_1 > 0\).

   On the other hand, \[R_1 < k_1 \iff k_1 - a_{12} k_2 < k_1 (1 - a_{12} a_{21}) \iff \frac{k_1}{k_2} < \frac{1}{a_{21}},\] which is exactly one of the assumptions of the lemma. Hence, \(0 < R_1 < k_1\).

   Similarly, \(R_2 \in (0, k_2)\).

2. Next, we show that \(\left( N_1^\pm(h), N_2^\pm(h) \right) \in (0,k_1) \times (0,k_2)\) for all \(h \in I\). The choice of the interval \[I = \left(0, \frac{\beta(1 - a_{12} a_{21})}{4 k_2} R_2^2 \right)\] ensures that \[0 < \left( \frac{R_2}{2} \right)^2 - \frac{k_2 h}{\beta (1 - a_{12} a_{21})} < \left( \frac{R_2}{2} \right)^2, \quad \forall h \in I,\] or equivalently, \[0 < \sqrt{ \left( \frac{R_2}{2} \right)^2 - \frac{k_2 h}{\beta (1 - a_{12} a_{21})} } < \frac{R_2}{2}.\] Thus, \[- \frac{R_2}{2} < \pm \sqrt{ \left( \frac{R_2}{2} \right)^2 - \frac{k_2 h}{\beta (1 - a_{12} a_{21})} } < \frac{R_2}{2},\] and consequently, \[0 < \frac{R_2}{2} \pm \sqrt{ \left( \frac{R_2}{2} \right)^2 - \frac{k_2 h}{\beta (1 - a_{12} a_{21})} } < R_2.\] Therefore, \(N_2^{\pm}(h) \in (0, R_2)\).

   On the other hand, \(a_{12} < \dfrac{k_1}{k_2}\) implies \(R_2 = \dfrac{k_2 - a_{21} k_1}{1 - a_{12} a_{21}} < \dfrac{k_1}{a_{12}}\). Using \(N_2^{\pm}(h) < R_2 < \dfrac{k_1}{a_{12}}\), we obtain \[0 < N_2^{\pm}(h) < \frac{k_1}{a_{12}} \Longrightarrow 0 < a_{12} N_2^{\pm}(h) < k_1 \Longrightarrow 0 < k_1 - a_{12} N_2^{\pm}(h) < k_1.\] Hence, \(N_1^{\pm}(h) \in (0,k_1)\) for all \(h \in I\).

In this way we complete the proof of Lemma 1. Furthermore, Figure 1 illustrates the result of the lemma, that is, the control equilibrium invariance property for \(h\in I\).

The linearized system by equation (11) around \((e,h)=((N_1^*,N_2^*),h)\) is defined as: \[\left[\begin{array}{c} \dot N_1 \\ \dot N_2 \end{array}\right]= \underbrace{\left[\begin{array}{cc} \alpha \left(1- \dfrac{2N^*_1}{k_1}- \dfrac{a_{12}N^*_2}{k_1} \right) & - \dfrac{\alpha a_{12}N^*_1}{k_1} \\[0.45cm] - \dfrac{\beta a_{21}N^*_2}{k_2} & \beta \left(1- \dfrac{2N^*_2}{k_2}- \dfrac{a_{21}N^*_1}{k_2} \right) \end{array}\right]}_{\ { \ Df(e)}} \left[\begin{array}{c} N_1 \\ N_2 \end{array}\right]+\left[\begin{array}{c} 0 \\ -1 \end{array}\right]u \tag{18} \]

By direct calculation, we obtain the Jacobian matrices \(\mathbf{J}_i=Df(e_i)\) for each equilibrium point \(e_i\), defined by equations (12) to (17). \[ \mathbf{J}_1 = \left[\displaystyle \begin{array}{cc} \alpha & 0 \\ 0 & \beta \end{array}\right]; \tag{19} \]

\[\mathbf{J}_2 = \left[\displaystyle \begin{array}{cc} - \alpha & - a_{12} \alpha \\[0.2cm] 0 & -\beta \left(a_{21}\dfrac{k_1}{k_2}-1 \right) \end{array}\right]; \tag{20} \]

\[\mathbf{J}^+_{3}(h) = \left[\displaystyle \begin{array}{cc} -\alpha \left(a_{12}\dfrac{k^+_2(h)}{k_1} -1\right) & 0 \\[0.35cm] - a_{21} \beta\dfrac{k_2^+(h)}{k_2} & - \beta \left(\dfrac{2k^+_2(h)}{k_2} -1\right) \end{array}\right]\ , \ \ h\in \left[0,\dfrac{k_2\beta}{4}\right] \; \tag{21} \]

\[\mathbf{J}^-_{3}(h) = \left[\displaystyle \begin{array}{cc} - \alpha \left(a_{12}\frac{k^-_2(h)}{k_1} -1\right) & 0 \\[0.35cm] - a_{21} \beta\dfrac{k_2^-(h)}{k_2} & - \beta \left(\dfrac{2k^-_2(h)}{k_2} -1\right) \end{array}\right]\ , \ \ h\in \left(0,\dfrac{k_2\beta}{4}\right) \; \tag{22} \]

\[ \mathbf{J}^+_{4}(h) = \left[\displaystyle \begin{array}{cc} - \dfrac{\alpha}{k_1} N_1^+(h) & - \dfrac{\alpha \, a_{12}}{k_1} N_1^+(h) \\ & \\ - \dfrac{\beta \, a_{21}}{k_2} N_2^+(h) & - \dfrac{\beta}{k_2} N_2^+(h) + \dfrac{h}{N_2^+(h)} \end{array}\right] \ , \ \ \ h\in {I} \ \text{ and } \ a_{12}<\frac{k_1}{k_2}<\frac{1}{a_{21}} \; \tag{23} \]

\[ \mathbf{J}^-_{4} (h)= \left[\displaystyle \begin{array}{cc} - \dfrac{\alpha}{k_1} N_1^-(h) & - \dfrac{\alpha \, a_{12}}{k_1} N_1^-(h) \\ & \\ - \dfrac{\beta \, a_{21}}{k_2} N_2^-(h) & - \dfrac{\beta}{k_2} N_2^-(h) + \dfrac{h}{N_2^-(h)} \end{array}\right] \ , \ \ \ h\in I \ \text{ and } \ a_{12}<\frac{k_1}{k_2}<\frac{1}{a_{21}} \. \tag{24} \]

For example, in equation (23), it is straightforward to verify that \(e_4^+(0)=(R_1,R_2)\) yields the following matrix:

\[ \mathbf{J}_{4}^+(0) = \left[\displaystyle \begin{array}{cc} - \dfrac{\alpha}{k_1} R_1 & - a_{12}\dfrac{\alpha }{k_1} R_1 \\ & \\ - a_{21}\dfrac{\beta }{k_2} R_2 & - \dfrac{\beta}{k_2} R_2 \end{array}\right]. \tag{25} \]

The Jacobian matrices given for (12) to (17) define linearized systems and it is important to know their stability properties to deduce the stability of the original system by equations (26). This analysis will be based on definitions and results in the following section.

Stability analysis of the equilibrium of the uncontrolled system   xxxxxxxxxxxxxxx

We analyze the following system:

\[ \begin{cases} \dot N_1 = \alpha N_1 \left(1 - \dfrac{N_1}{k_1} - \dfrac{a_{12} N_2}{k_1}\right) \\[0.3cm] \dot N_2 = \beta N_2 \left(1 - \dfrac{N_2}{k_2} - \dfrac{a_{21} N_1 }{k_2}\right) \end{cases} \tag{26} \]

Let \(\mathcal{V}\subset \mathbb{R}^2\) be open, \(f:\mathcal{V}\rightarrow\mathbb{R}^2\) of class \(C^1\), and \(e^*\) an equilibrium of \(\dot N=f(N)\). Then

1. We say that \(e^*\) is a stable equilibrium if, given \(\epsilon>0\), there exists \(\delta=\delta(\epsilon)>0\) such that for any initial condition \(N(0)\) satisfying \(|| N(0)-e^*|| <\delta\), the trajectory \(N(t)\) satisfies: \[ || N(t)-e^*|| < \epsilon \ , \ \forall \ t\geq 0. \tag{27} \] An equilibrium \(e^*\) is said to be unstable if it is not stable.

2. We say that \(e^*\) is an asymptotically stable equilibrium if, in addition to being stable, it satisfies the following condition: \[\lim_{t\rightarrow \infty} ||N(t)-e^*||=0. \tag{28} \]

The local stability around an equilibrium \(e^*\) can be classified by analyzing the nature of the eigenvalues of the Jacobian matrix \(Df(e^*)\) associated with it. Below, we present results from stability theory (Doering et al., 2016).

Theorem 3. Let \(\mathcal{V}\subset \mathbb{R}^2\) be open, \(f:\mathcal{V}\rightarrow\mathbb{R}^2\) of class \(C^1\), \(e^*\) an equilibrium state of the system \(\dot N=f(N)\), and \(Df(e^*)=\mathbf{J}\in\mathbb{R}^{2\times 2}\) the associated Jacobian matrix.

1. If all eigenvalues of \(\mathbf{J}\) have negative real parts, then \(e^*\) is an asymptotically stable equilibrium.

2. If \(\mathbf{J}\) has at least one eigenvalue with positive real part, then \(e^*\) is an unstable equilibrium.

The equilibrium point is now classified by analyzing the nature of the eigenvalues of the corresponding matrix \(\mathbf{J}\). For this, it is sufficient to use the matrices defined by equations (12) to (17): \[ \text{Eigenvalues of } \mathbf{J}_1\ =\ \{\alpha,\beta\}, \tag{29} \]

\[\text{Eigenvalues of } \mathbf{J}_2\ =\ \left\{-\alpha,-\beta\left( a_{21}\frac{k_1}{k_2}-1\right)\right\}, \tag{30} \]

\[\text{Eigenvalues of } \mathbf{J}^+_3(0)\ =\ \left\{-\alpha\left(a_{12}\frac{k_2}{k_1} -1 \right),-\beta\right\}, \tag{31} \]

\[\text{Eigenvalues of } \mathbf{J}^+_4(0)\ =\ \left\{ \lambda\in \mathbb{C} \ \bigg| \ \lambda^2+\left(\frac{\alpha R_1}{k_1}+\frac{\beta R_2}{k_2}\right)\lambda+\frac{\alpha R_1}{k_1}\frac{\beta R_2}{k_2}(1-a_{12}a_{21})=0\right\}. \tag{32} \]

From now on, we denote \(\mathbf{J}_3^+(0)\) and \(\mathbf{J}_4^+(0)\) by \(\mathbf{J}_3\) and \(\mathbf{J}_4\), respectively. In the following section we present the stability results of these equilibria and from the perspective of controllability we compare both notions.

Results and discussion

We present the main results of the work. Firstly, the stability results corresponding to the uncontrolled system given by equations (26).

Theorem 4. Consider the equilibria of the system given by equations (26): \( e_1 = (0,0) \), \( e_2 = (k_1,0) \), \( e_3 = (0,k_2) \) and \( e_4 = (R_1,R_2) \). Then:

1. \( e_1 \) is unstable (local source).

2. If \( \dfrac{1}{a_{21}} < \dfrac{k_1}{k_2} \) then \( e_2 \) is asymptotically stable (local attractor).

3. If \( \dfrac{k_1}{k_2} < a_{12} \) then \( e_3 \) is asymptotically stable (local attractor).

4. If \( \dfrac{k_1}{k_2} > a_{12} \) then \( e_3 \) is unstable (saddle).

5. If \( \dfrac{1}{a_{21}} < \dfrac{k_1}{k_2} < a_{12} \) then \( e_4 \) is unstable (saddle).

Applying the results of local stability (Theorem 3), we obtain the following. A simple observation of the sign of the eigenvalues of the matrices defined in (30)–(33) allows us to identify that:

1. \( \sigma(\mathbf{J}_1) \subset \mathbb{R}_+ \).

2. \( \sigma(\mathbf{J}_2) \subset \mathbb{R}_- \) since \( \dfrac{1}{a_{21}} < \dfrac{k_1}{k_2} \).

3. \( \sigma(\mathbf{J}_3) \subset \mathbb{R}_- \) since \( \dfrac{k_1}{k_2} < a_{12} \).

4. \( \mathbf{J}_3 \) has one positive and one negative eigenvalue if \( \dfrac{k_1}{k_2} > a_{12} \). Therefore, \( e_3 \) is unstable (saddle) in this case.

5. If \( \dfrac{1}{a_{21}} < \dfrac{k_1}{k_2} < a_{12} \) then \( \mathbf{J}_4 \) has a positive eigenvalue, \( \lambda_1 \in \sigma(\mathbf{J}_4) \cap \mathbb{R}_+ \) . Therefore, \( e_4 \) is unstable (saddle).

\[\lambda_1 = \dfrac{ -\left(\dfrac{\alpha R_1}{k_1}+\dfrac{\beta R_2}{k_2}\right) + \sqrt{ \left(\dfrac{\alpha R_1}{k_1}+\dfrac{\beta R_2}{k_2}\right)^2 + 4\dfrac{\alpha R_1}{k_1}\dfrac{\beta R_2}{k_2}(a_{12}a_{21}-1) } }{2} > 0 \tag{33} \]

\[\lambda_2 = \dfrac{ -\left(\dfrac{\alpha R_1}{k_1}+\dfrac{\beta R_2}{k_2}\right) - \sqrt{ \left(\dfrac{\alpha R_1}{k_1}+\dfrac{\beta R_2}{k_2}\right)^2 + 4\dfrac{\alpha R_1}{k_1}\dfrac{\beta R_2}{k_2}(a_{12}a_{21}-1) } }{2} < 0 \tag{34} \]

Consider the equilibria of system by equation (11) with \(a_{12}<\dfrac{k_1}{k_2}<\dfrac{1}{a_{21}}\). Then

1. The system is locally controllable around \((e_2,0)=((k_1, 0), 0).\)

2. It is locally controllable around \((e_4^+(h),h)=(N_1^+(h), N_2^+(h), h)\), for \(h \in \left[0, \dfrac{\beta(1 - a_{12}a_{21})}{4k_2} R_2^2 \right]\).

3. It is locally controllable around \((e_4^-(h),h)=(N_1^-(h), N_2^-(h), h)\), for \(h \in \left(0, \dfrac{\beta(1 - a_{12}a_{21})}{4k_2} R_2^2 \right)\).

Applying the results of Local Controllability (Theorem 2) for the system by equation (11) and consider that the Test Kalman (7) is equivalent to \[\det \left[\mathbf{B} \ \ \ \mathbf{J}_i\mathbf{B} \right]\neq 0,\] for \(\mathbf{B}=(0,-1)^\top\) and \(\mathbf{J}_i=Df(e_i)\), we obtain the result.

Moreover, for \(u(t) \equiv 0\), representing the uncontrolled tumor case, we recall the stability analysis of the equilibrium \(e_i\) presented in Rodrigues et al. (2011), and compare it with the corresponding controllable equilibrium \((e_i, 0)\). A summary of this comparison is provided in Table 1.

In Figure 2, we observe the dynamics predicted in Table 1. Specifically, we consider the synthetic parameters \(\alpha = 2.2\), \(\beta = 1.1\), \(k_1 = 60\), \(k_2 = 10\), \(a_{12} = 6.5\), and \(a_{21} = 0.25\), which were assumed for the purpose of generating the corresponding numerical simulation and graphical representation, without any a priori biological interpretation.

With these values, we obtain the dynamics around the equilibria: \((0,0)\) (unstable), \((60,0)\) (asymptotically stable), \((0,10)\) (asymptotically stable), and \((8,8)\) (unstable). Also shown in Figure 3 is the corresponding trajectory \((N_1(t),N_2(t))\) (blue curve and orange curve, respectively).

Conclusions

We investigated the local controllability of a nonlinear tumor-normal cell interaction model under therapeutic control. By analyzing the linearized dynamics around multiple equilibrium configurations and applying the Kalman rank condition, we rigorously established conditions under which the system is locally controllable. These results confirm the ability of control actions to influence the state trajectory in the vicinity of equilibria, enabling a precise theoretical characterization of treatment effectiveness.

For instance, in Figure 3, when \(u \equiv 0\), the numerical solution converges to the equilibrium corresponding to the exclusion of species \(N_1\). However, under a constant harvesting rate \(u = 0.1\), the system transitions to the equilibrium of extinction of species \(N_2\). These findings underscore the potential of control strategies in modulating tumor–healthy cell interactions and point to the need for further studies on sustainable harvesting thresholds that could effectively reduce tumor cell populations while preserving the persistence of healthy cells \(N_1\), with direct relevance to biomedical treatment planning.

Acknowledgments

The author would like to thank the Department of Mathematics (MAT) at UFSC Campus Blumenau and the National Professional Master’s Program in Mathematics (PROFMAT) for their support of this work.

Author Contributions

F. F. C. Puma participated in: Conceptualization, Project managements, Formal Analysis, Data Curation, Investigation, Methodology, Supervision, Validation, Visualization and Writing – preparation of the original draft, revision and editing.

Conflicts of Interest

The author declare no conflict of interest.

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