A comprehensive study of a feedback control problem with a state-dependent implicit pantograph equation of Chandrasekhar type

1.   Introduction

The Chandrasekhar integral equation arises in the context of radiative transfer theory; this type of equation is essential when examining the propagation of radiation through a medium that exhibits scattering ^[14]. The study of Chandrasekhar's integral equation is crucial in a variety of areas and encountered in several applications, including mathematical and computational methods, astrological applications, radiative transfer theory, and stellar atmospheres (see ^[5,11,12,48]). Several researchers have focused their attention on this type of equation as seen in ^{[13,15,26,28,29]} and the references therein.

Feedback control system is a fundamental concept in engineering and automation that controls a system's behavior by modifying inputs based on output ^[1]. A controller is a biological system that regulates the operation of other biological processes. Integrative feedback control is essential for regulation, sensory adaptation, and long-term effects. Control variables are vital in dealing with unanticipated occurrences that disrupt real-world ecosystems, potentially altering biological features. These difficulties are converted into mathematical models ^[17,39,49]. Several authors investigated feedback problems. In ^[18], the authors established a necessary condition for a positive periodic solution of a feedback control model for chemostats. A positive periodic solution with feedback control involving a nonlinear neutral delay population problem has been analyzed in ^[36]. The author of ^[37] studied the asymptotic stability and solvability of nonlinear functional-integral equations with feedback control.

Stability analysis is an extensive and varied field with deep theoretical roots and numerous applications in engineering, economics, biology, physics, and other disciplines. It is a prevalent topic in the mathematical sciences ^[6]. An equation or problem can be used to simulate a physical process if a minor change in it causes a corresponding small change in the outcome. Alsina and Ger ^[4] initially examined the Hyers-Ulam stability of differential equations. Several papers have been devoted to studying the Hyers-Ulam stability of differential and integral equations, for example, (see ^{[2,3,42,43,44]}). Another concept in stability theory is continuous dependency ^[40], which analyzes the behavior of mathematical solutions under different conditions. Hyers-Ulam stability assesses the problem's resilience to disruptions, while continuous dependency examines how even minor parameter changes affect the problem's unique solution. The study of the continuous dependence of the solution has been addressed through many research works ^[22,27].

Pantograph differential equations are an effective tool for modeling systems with feedback interactions. From signal processing to finance, control systems, and neural networks, these equations aid in the capture of complicated dynamics in which the present state-depends on past states at scaled time intervals. Solving pantograph differential equations analytically is difficult because of their nonlocal nature; often, numerical methods such as finite difference schemes are used to obtain approximate solutions. Much research has been conducted on pantograph equations due to their significance in various research areas. For instance, in ^[38], Patade et al. investigated the analytical solution of the pantograph equation with two delays; they analyzed the existence, uniqueness, and stability of the solution. In ^[23], the authors studied the solvability and the Hyers-Ulam stability of non-local fractional orders of the pantograph equation with a feedback control.

State-dependent (self-reference) differential and integral equations are a special new type of functional differential equations in which the deviation of the argument depends on the time and the state together. In most differential and integral equations with deviating arguments that are found in literature, the deviation of the argument involves only the time, although another important case in theory and practice involves deviating arguments that depend on both the state variable $x$ and the time $t$ (see for instance ^[9,10,20,34]). This kind of delay is widely utilized in nonlinear analysis and has a wide range of applications, including mechanical models ^[31], population models ^[7], infectious disease transmission ^[41], the two-body problem of classical electrodynamics ^[19], the dynamics of economical systems ^[8], and it has many applications in hereditary phenomena ^[35,45,46]. One of the first researches in self-reference differential equations was presented by Eder ^[20], he classifies the solutions to the differential equation $x^{\prime}(t) = x\big(x(t)\big), \; t\in A\subset R$, and demonstrated the existence and uniqueness of the solution with the condition $x(t_{0}) = x_{0}$. Fe'ckan ^[25] introduced a generalization of Eder's results by studying the functional differential equation of the form $x^{\prime}(t) = f\big(x(x(t))\big), \; t\in A\subset R$, where $f \in C^{1}(R).$ Buicá ^[10], examined the uniqueness of solution and data dependence of the problem $x'(t) = f(t, x(x(t))), \; t\in [a, b], \; x(t_0) = x_0,$ where $t_0, x_0 \in [a, b]$ and $f\in C([a, b], [a, b])$. Lin and Lu ^[34] studied the qualitative behavior of a state-dependent functional differential equation. Yang et al. ^[47] examined an ordinary differential equation with a state-dependent delay. EL-Sayed et al. ^[24] investigated a nonlocal boundary value problem of a state-dependent differential equation. In ^[22], the authors analyzed the state-dependent Chandrasekhar integral equation

they proved the existence of the solution for this equation and analyzed its continuous dependency on function a. In ^[30], Humphries et al. examined the state-dependent delay system

where the delay $a + \eta x(t - b)$ depends linearly on the state $x$ with strength $\eta$. Lauran in ^[33], analyzed iterative and non iterative first order differential equations of the form

and

respectively, with the initial condition $x(t_{0}) = x_{0}.$ A generalization of the result in ^[33] was introduced by Hashem et al. ^[27], they studied the system of the state dependent functional equations

the authors proved the existence of a unique solution of the system and discussed the continuous dependence of the solution. In ^[21], the authors examined the existence of a unique solution of a feedback control problem with an implicit state-dependent pantograph equation; in addition, they investigated the Hyres-Ulam stability of the problem and the continuous dependence of the solution.

Inspired by modern literature, we consider the state-dependent implicit pantograph equation of the Chandrasekhar type

with the feedback control

Where $\lambda_i\in(0, 1)$ and $x_0, \; y_0 \in R$ are the initial data.

Our goal in this work is to investigate the existence and the uniqueness of the solution $(x, y)\in X$ of the problems (1.1) and (1.2). We prove the continuous dependence of the solution on the initial data $x_0, y_0$ and the parameters $\lambda_i$. Furthermore, we establish the Hyres-Ulam stability of the problem. Next, as a particular case of our work, we discuss an issue of the same type without feedback control.

We outline the main contributions of this paper as follows:

● We examine the feedback control problems (1.1) and (1.2) of the state-dependent pantograph equation of the Chandrasekhar type; we study the qualitative properties of the solution of (1.1) under the feedback control (1.2).

● We explore the problem (5.1) of the state-dependent pantograph equation of the Chandrasekhar type; we refer to the qualitative properties of the solution of (5.1) in the absence of the feedback control.

This study enhances the qualitative analysis of a state-dependent pantograph differential equation with feedback control. The article is structured as follows: Section 2 presents the appropriate assumptions and proves the existence results for the solution of the implicit state-dependent problem (1.1) with the feedback control (1.2) using Schauder fixed point theorem. The suitable assumptions and proofs for the uniqueness of the solution will be provided in Section 3. Section 4 investigates the stability analysis of the problem due to the Hyers-Ulam stability. Additionally, we test the possibility of the solution resisting disturbances through the study of the continuous dependency on $x_0, y_0$ and the parameters $\lambda_i$. Furthermore, in Section 5, we examine a special case of our problem without the control variable; we present some results supporting the problem's existence and stability. In Section 6, we give some instances that illustrate the findings. Finally, Section 7 provides a conclusion.

Let $C(I)$ be the class of all continuous functions on $I = [0, 1]$ with the standard norm $\|u\|_C = \sup_{t\in I}|u(t)|,$ and $\; X = C(I)\times C(I)$ be the Banach space with the norm $\|(u, v)\|_X = \max\{\|u\|_C, \|v\|_C\}$. Let $L^1(I)$ be the space of real functions defined and Lebesgue integrable on the interval I, which is equipped with the standard norm.

2.   Existence results

Consider the problems (1.1) and (1.2) under the assumptions:

$(1)$ $g_i:I\times R \rightarrow R$ satisfies Carath$\grave{e}$odory condition ^[16], i.e., it is measurable in $t \in I$ $\forall\; x \in R$ and continuous in $x \in R,$ $\forall\; t \in I$, and there exist functions $a_i:I \rightarrow R^+ = [0, \infty), \; a_i \in L^1(I)$ and $a_i(t)\leq K_it^n, \; K_i\in(0, 1)$ where $n\in N$, $\forall\; t \in I$, such that

$(2)$ $b_i:I \rightarrow R$ are continuous functions on I.

$(3)$ There exists a real positive root $r$ of the algebraic equation

where $K\lambda r < 1,$ $K = \max\{K_i\}$, $\lambda = \max\{\lambda_i\}$, $B = \max\{\|b_i\|\}$, $A = \max\{x_0, y_0\}.$

2.1.   Formulation of problem

Let $\frac{dx}{dt} = u(t), \; \; \frac{dy}{dt} = v(t)$, then

Then the problems (1.1) and (1.2) will be given by

Where

and

Define the operator F associated with (2.3) and (2.4) by

where

Theorem 2.1. Let the assumptions (1)–(3) be satisfied. Then the problems (2.3) and (2.4) has at least one solution.

Proof. Define set $Q_r \subset X$ as

Obviously, $Q_r$ is a closed convex bounded set. Now for $(u, v)\in Q_{r}, \; t \in I$, we have

Similarly,

Then

Then the class of functions $\{F(u, v)\}$ is uniformly bounded on $I$. Let $(u, v)\in Q_r$ and $t_1, \; t_2\in I$ with $t_1 < t_2$ such that $|t_2-t_1| < \delta$, then

Then the class of functions $\{F_1 u\}$ is equi-continuous.

Similarly,

Then the class of functions $\{F_2 u\}$ is equi-continuous. We deduce that $F:Q_r \rightarrow Q_r$ and the class functions $\{F(u, v)\}$ is equi-continuous. By Arzela-Theorem ^[32], $\{F(u, v)\}$ is compact, then $F$ is compact.

Now, let $\{u_n\}, \; \{v_n\} \subset Q_r$ such that $u_n(t) \rightarrow u(t)$, $v_n(t) \rightarrow v(t),$ where $n \rightarrow \infty,$ then

and

Then, from assumption $(1)$, we have

and

Applying Lebesgue dominated convergence theorem ^[32], we have

Similarly,

Now,

Then $F(u, v)$ is continuous. Now all conditions of Schauder's fixed point theorem ^[32] are satisfied, then the operator F has at least one fixed point $(u, v)\in Q_r$, Consequently, there exist at least one solution of the problems (2.3) and (2.4). □

Corollary 2.1. Let the assumptions of Theorem 2.1 be satisfied, by using Eqs (2.1) and (2.2), we deduce that the problems (1.1) and (1.2) has at least one solution $x \in C(I)$.

3.   The uniqueness of the solution

Consider the following additional assumptions:

($1^*$) $g_i:I\times R \rightarrow R$ satisfies the Lipschitz condition with positive Lipschitz constants $M_i$ such that

Theorem 3.1. Let the assumptions (1)–(3) and $(1^*)$ be satisfied, If $0 < \frac{\lambda rM}{(1-\lambda rK)} < 1,$ then the solution of the problems (1.1) and (1.2) is unique.

Proof. let $(u_1, v_1), (u_2, v_2)$ be two solutions of (2.3) and (2.4), then

and

then

Similarly,

Then

Then

Since $0 < \frac{\lambda rM}{1-\lambda rK} < 1$, then the solution of the problems (2.3) and (2.4) is unique. □

Corollary 3.1. Let the assumptions of Theorem 3.1 be satisfied; according to the Eqs (2.1) and (2.2), the solution $(x, y)\in X$ of (1.1) and (1.2) is unique.

4.   Stability analysis of the problem

4.1.   Hyres-Ulam stability

Definition 4.1. Let the solution $(x, y) \in X$ of (1.1) and (1.2) be exists, then the problems (1.1) and (1.2) is Hyers-Ulam stable if $\; \forall\epsilon > 0,$ there exists $\delta(\epsilon) > 0\;$ such that for any $\delta$-approximate solution $(x_s, y_s)\in X$ of (1.1) and (1.2) satisfies

implies

Theorem 4.1. If the assumptions of Theorem 3.1 are met, then the problems (1.1) and (1.2) is Hyers-Ulam stable.

Proof. Let

then

Similarly,

Let $\frac{dx_s}{dt} = u_s$ and $\frac{dy_s}{dt} = v_s$, then

Hence

Then,

and

Hence

Similarly,

Then

and

then

Now,

Then

Then the problems (1.1) and (1.2) is Hyers-Ulam stable. □

4.2.   Continuous dependence

Definition 4.2. The solution $(u, v)\in Q_r$ of (2.3) and (2.4) depends continuously on $x_0, \; y_0, \; \lambda$ if $\forall\; \epsilon > 0$ there exists$\; \; \delta(\epsilon) > 0$ such that

where

Theorem 4.2. Let the assumptions of Theorem 3.1 be satisfied, then $(u, v)$ depends continuously on the initial data $x_0, \; y_0$ and the parameters $\lambda_i, \; i = 1, 2.$

Proof. Let $\delta(\epsilon) > 0$ be given such that

and let $(u^*, v^*)$ be the solution of (4.1) and (4.2), then

and

Then

Similarly,

Then

and

Then

Thus

Which meas that $\delta = \delta(\epsilon).$ Moreover, we have $\lambda r\; M+\lambda r\; K < 1$ and $A, r, k, \lambda > 0,$ then $\delta$ is positive. This means that the solution of (2.3) and (2.4) depends continuously on the initial data $x_0, \; y_0$ and the parameters $\lambda_i, \; i = 1, 2.$ □

Definition 4.3. The solution $(x, y) \in X$ of (1.1) and (1.2) depends continuously on $u, \; v$ if $\forall\; \epsilon > 0$ there exists $\delta(\epsilon) > 0$ such that

where

Theorem 4.3. Let the assumptions of Theorem 4.2 be satisfied, then the solution $(x, y)\in X$ depends continuously on $u, \; v$.

Proof. Let $(x^*, y^*)$ be the solution of (4.3) and (4.4), then

then

Similarly,

then

Now,

Then

□

This means that the solution $(x, y)\in X$ of (1.1) and (1.2) depends continuously on $u, \; v$.

Corollary 4.1. Let the assumptions of Theorem 4.3 be satisfied, then the solution $(x, y)\in X$ of (1.1) and (1.2) depends continuously on $x_0, y_0$ and the parameter $\lambda_i$.

5.   Special case

In the lack of feedback control; as a special case of our work, we can investigate the following problem of the state-dependent implicit pantograph differential equation of the chandrasekhar type.

where $\lambda\in(0, 1)$, $b : I\rightarrow R$ are continuous function and the function $g : I\times R \rightarrow R$ satisfies Caratheodory condition.

The existence of the unique solution of (5.1) will be studied. We will also prove the continuous dependence of the solution on the initial data $x_0$ and the parameter $\lambda$. Furthermore, we will establish the Hyres-Ulam stability. This problem can be addressed under the following assumptions:

$(i)$ $g:I\times R \rightarrow R$ satisfies Carath$\grave{e}$odory condition^[16]. (i.e., it is measurable in $t \in I$ $\forall x \in R$ and continuous in $x \in R$ $\forall t \in I$) and there exists a function $a:I \rightarrow R^+ = [0, \infty), \; a \in L^1(I)$ and $a(t)\leq Kt^n, \; K\in(0, 1), \; n \in N$ $\forall \; t \in I$, such that

$(ii)$ There exists a real positive root $r$ of the algebraic equation

such that $K \lambda r < 1.$

We can formulate the problem as follows: Put $\frac{dx}{dt} = y(t)$, we get

Using the same techniques, we can derive the following theorems:

Theorem 5.1. Let the assumptions $(i)$–$(ii)$ be satisfied, then integral equation (5.2) has at least one solution $y \in C(I)$. Consequently, the problem (5.1) has at least one solution $x \in C(I)$.

Theorem 5.2. Let the assumptions $(i)$–$(ii)$ of Theorem 5.1 be satisfied, if $g$ satisfies lipschetz condition such that $|g(t, x)-g(t, y)|\leq c|x-y|$ $\forall t\in I$ and $x, \; y \in Q_r$ where $c$ is a positive constant. If $\lambda r(c+K) < 1,$ then the solution of the problem (5.1) is unique.

Theorem 5.3. Let the assumptions of Theorem (5.2) be satisfied, then the problem (5.1) is Hyers-Ulam stable.

Theorem 5.4. Let the assumptions of Theorem 5.2 be satisfied, then $y\in C(I)$ depends continuously on the initial data $x_0$ and the parameter $\lambda$.

Theorem 5.5. Let the assumptions of Theorem 5.2 be satisfied, then the solution $x\in C(I)$ depends continuously on $y$.

Corollary 5.1. Let the assumptions of Theorems 5.4 and 5.5 be satisfied, then the solution $x\in C(I)$ depends continuously on the initial data $x_0$ and the parameter $\lambda$.

6.   Examples

Example 1. Consider the problem

Let $\frac{dx}{dt} = u, \; \; \frac{dy}{dt} = v$ then

Set

and

then

and

Where $K = \max\{\frac{1}{5}, \frac{1}{7}\} = \frac{1}{5},$ $\lambda = \max\{\frac{1}{2}, \; \frac{1}{3}\} = \frac{1}{2}$, $A = \max\{\frac{1}{7}, \; \frac{1}{10}\} = \frac{1}{7}$, $B = \max\{0.6667, 0.043\} = 0.6667, \; M = \max\{\frac{1}{5}, \frac{1}{7}\} = \frac{1}{5}$. Then we get $r_1 = 0.8027, \; r_2 = 9.1973,$ such that $\lambda r_1 K = 0.0803 < 1$ and $\lambda r_2 K = 0.9197 < 1.$ It is clear that all assumptions of Corollary 1 are satisfied. Hence there exist at least one solution $(x, y) \in X$ of (6.1) and (6.2). Moreover, we have only $r_1$ satisfies $\lambda r_1(M+K) = 0.1605 < 1.$ Thus all assumptions of Corollary 2 are satisfied, then the solution of problems (6.1) and (6.2) is unique.

Example 2. Consider the problem

Let $\frac{dx}{dt} = y$ then

Set

and

thus

and

Where $K = \frac{1}{4},$ $\lambda = \frac{1}{3}$, $x_0 = \frac{1}{2}$, $\|b\| = \frac{1}{6}, \; c = \frac{1}{4}$. Then we get $r_1 = 0.343, \; r_2 = 11.657$, such that $\lambda r_1 K = 0.0286 < 1$ and $\lambda r_2 K = 0.9714 < 1.$

It is clear that all assumptions of Theorem 5.1 are satisfied. Hence there exist at least one solution $x \in C[0, 1]$ of (6.3). Moreover, we have only $r_1$ satisfies $\lambda r_1(c+K) = 0.0572 < 1.$

Thus all assumptions of Theorem 5.2 are satisfied, then the solution of Problem 6.3 is unique.

7.   Conclusions

Differential equations with control variables are frequently encountered within several domains such as control theory, optimization, dynamic systems, etc...The existence and uniqueness of solutions of this type of equations is crucial. Sometimes the equations may not have a unique solution, or solutions may need particular conditions to exist. Determining whether a solution is stable or not is often challenging. Stability analysis is critical for control systems to guarantee that small perturbations do not cause significant variations in system behavior. Research on feedback control problems with state-dependent delays has numerous applications in biology, ecology, physics, engineering, and other fields. This type of delay imparts memory effects into the system, which implies intricate and rich dynamics. In this paper, we investigate the existence of at least one continuous solution to a feedback control problem including implicit pantograph equations of the Chandrasekhar type. In addition, we analyzed the uniqueness of the solution in light of appropriate assumptions. Also, we investigated the problem's Hyers-Ulam stability and the continuous dependence of the solution on the original data and parameter. Furthermore, we introduce a brief study for the problem in the absence of feedback control. Finally, we presented few examples in both cases when the problem contains a control variable and in the absence of a control variable.

Author contributions

The authors contributed equally to this paper.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors thank the referees for their useful suggestions and comments, which helped strengthen our paper.

Conflict of interest

The authors declare no conflicts of interest.