Differential equations are crucial in solving practical problems in many scientific fields, such as physics, chemistry, biology, economics, and engineering, etc., modeling many real-world processes. However, the complexity of these phenomena often introduces various parameters that can significantly influence the outcome. A particularly intriguing problem in this context is identifying the parameters that ensure a specific quantity (e.g., density, energy) related to the solution of the differential equation reaches a desired value. This challenge naturally leads to a control problem.

Our study has two strong motivations.

(a):

The first motivation concerns differential equations with moving singularities, which frequently appear in nonlinear models from applied sciences, such as physics and mathematical biology [1].

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(b):

The second one relates to the control of such models, aiming to reach a desired state of the process. For example, if the state variable represents a density, one might be interested in controlling its cumulative value or average. This corresponds precisely to our control problem in Section 2.

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Mathematical models expressed through equations with singularities include the Briot-Bouquet equation, which has applications in complex analysis, specifically in the theory of univalent functions; equations arising in Michaelis-Menten kinetics, modeling oxygen diffusion in cells; the Thomas-Fermi equation in atomic physics; and the Emden-Fowler equation, in the study of phenomena in non-Newtonian fluid mechanics [1, 2, 3].

Inspired by the investigation in [4, 5, 6], this paper will explore the following problem

Here, $\ f:[0,\infty)\times\mathbb{R}\times(0,\infty)\rightarrow\bar{\mathbb{R}}(:=\mathbb{R\cup\{\pm\infty\}})$ is a function that possesses a singularity in the first variable, influenced by the third one, that is, for each $\lambda>0$, there exists $\theta(\lambda)>0$ such that

Throughout this paper, we use $u_{\lambda}$ to denote the unique solution to problem (1.1) for a given $\lambda>0$. Since the singularity point $\theta(\lambda)$ varies with $\lambda$, the differential equation (1.1) is said to be with moving singularity.

For each $\lambda>0$, we consider the functional $\psi_{\lambda}\colon C[0,\theta(\lambda))\rightarrow\bar{\mathbb{R}}$,

where the integration over the noncompact interval $\left[0,\,\theta(\lambda)\right)$ is understood in the usual sense (see, e.g., [7]),

Our goal in this paper is the following control problem:

The particularity of the controllability condition resides in the fact that it depends on the singular point which in turn depends on the control variable. To establish sufficient conditions for the existence of a solution to this control problem, we first guarantee that the mapping $\varphi\colon\left(0,\infty\right)\rightarrow\mathbb{R}$,

is well-defined and continuous on $(0,\infty)$. Then, we are able to use a lower and upper solution argument to guarantee the existence of a $\lambda^{\ast}$ with the desired property (1.3). Moreover, by bisection algorithm, we have a method of approximation of the value $\lambda^{\ast}.$

Control problems on a fixed interval that require to find a parameter in order to achieve a specific controllability condition are well-documented in the literature (see, e.g., [8, 9, 10]). The novelty of this paper lies in determining such a parameter where each solution is defined on a different interval. This approach requires a more refined analysis and leads to more complex problems.

The main assumptions we use in our analysis are

• (h1)

  For each $\lambda>0$ and $0<\varepsilon<\theta(\lambda)$, there exists a constant $L=L(\lambda,\varepsilon)$ such that for all $x,\overline{x}\in\mathbb{R}$ and for all $t\in\left[0,\,\theta(\lambda)-\varepsilon\right]$, we have

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  $$|f(t,x,\lambda)-f(t,\overline{x},\lambda)|\leq L(\lambda,\varepsilon)\,|x-\overline{x}|.$$

• (h2)

  The mappings

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  $$\lambda\mapsto\theta(\lambda),\quad\lambda\mapsto u_{0}(\lambda),\quad\lambda\mapsto L(\lambda,\varepsilon)\,\,(\text{for each }\varepsilon>0),$$

  are continuous. Additionally, the map $(t,x,\lambda)\mapsto f(t,x,\lambda)$ is continuous for $t\in[0,\,\theta(\lambda))$, $x\in\mathbb{R}$, and $\lambda\in(0,\infty)$.

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In the next lemma, we show that assumptions (h1), (h2) are sufficient to guarantee the existence of a unique solution of problem (1.1).

The paper is structured as follows: Section 2 presents the original results on the controllability of equations with moving singularities. Before establishing the main result (Theorem 2.6), we derive several auxiliary results concerning the continuous dependence of solutions on the control parameter and the continuity of the control functional with respect to the control variable. In Section 3, we provide a theoretical algorithm for obtaining the solution of the control problem using the method of lower and upper solutions. We also provide an example that illustrates the applicability of the obtained results. Finally, in Section 4 we suggest a possible extension of our approach to fractional differential equations with moving singularities.

In the following, we present some well-known results from the literature that will be used throughout this paper. The first result is the Arzelà-Ascoli theorem (see, e.g., [13, 14]).

The next lemma provides an alternative condition to ensure the convergence of a sequence based on the behavior of its subsequences (see, e.g., [15, Lemma 1.1]).

Finally, we have the well known Grönwall’s lemma (see, [16, 17]).

Starting from the lower and upper solutions $\underline{\lambda},$ $\overline{\lambda},$ one can approximate $\lambda^{\ast}$ by using the following algorithm. For a similar use of this method, we refer the reader to [19].

We note that this step-by-step algorithm iteratively approximates the control solution. At each step, based on the obtained feedback, either the subsolution or the supersolution is improved.

1. (h3’)

   There exists a constant $a>p$ such that, for all $\lambda>0$, one has

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   $$\left|f(t,x,\lambda)\right|\leq\frac{|x|}{a(\theta(\lambda)-t)}+C_{\lambda},$$

   for all $t\in[0,\theta(\lambda))$ and all $x\in\mathbb{R}$, where $C_{\lambda}\geq 0$ and the map $\lambda\mapsto C_{\lambda}$ is continuous.

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The above results can be generalized to fractional differential equations with moving singularities. Such problems more accurately describe various physical, biological, or medical processes (see, e.g., [20, 21, 22, 23]). Therefore, our results related to problem (1.1) can be extended to the following problem:

where ${}^{c}D^{\alpha}$ is the Caputo fractional derivative and $0<\alpha<1$. As shown in the literature, the above problem is equivalent to the Voltera integral equation

Note that, in our case, the control problem remain unchanged, i.e., find $\lambda$ such that

Since our entire analysis is grounded in the integral form of the Cauchy problem, we can easily extend the proof steps to address this more general case. The flexibility of the integral formulation allows for the adaptation of our methods without significant modifications. By imposing conditions similar to those outlined in (h1) and (h2), we can rigorously establish the controllability of the problem.

The analyzed control problem in this paper is atypical in several aspects: (a) it refers to equations with singularity; (b) the singularity itself depends on the control variable; (c) the controllability condition involves the moving singularity. All these aspects make the analysis much more complex and adapted to the specifics of the problem. The working techniques can also be taken into account for the investigation of other types of singular equations and controllability conditions including singular partial differential equations (see, e.g., [6]). We believe and anticipate that the ideas and techniques used in this article will have the high degree of suitability for the specifics of each individual problem in future research.