1 Introduction

Different types of integral equations arise in various problems of theory and applications in mechanics, physics, biology, economics, medicine etc. (see, e.g. [1–12] and references therein). Many processes have exterior influences called control efforts or the system’s disturbances. Therefore mathematical models of such processes include an additional parameter which is called the control or disturbance vector depending on the character of the exterior influences.

In the present paper, the control system described by a Urysohn type integral equation is studied. Note that the solution of the boundary value problem for an ordinary differential equation can be reduced to the solution of the suitable Urysohn type integral equation. Control systems described by a Urysohn type integral equation are considered in [13–15], where it is assumed that the control functions satisfy the geometric constraint, which means that the control resource is not exhausted by consumption. But some kinds of control efforts are exhausted by consumption such as energy, fuel, finance, and food. In this case the integral constraint on the control functions is inevitable (see, e.g. [16–23] and references therein). For example, the mathematical model of the flying object with rapidly changing mass is described by a control system with integral constrained control functions (see, e.g. [17, 19, 23]). The various topological properties of the set of trajectories of the control systems described via an integral equation with integral constraint on the control functions are considered in [24–26].

Compactness of the set of trajectories of the control system described by a Urysohn type integral equation is investigated in this paper. It is assumed that the control functions are chosen from the space \(L_{2} ( [t_{0},\theta ];\mathbb{R}^{m} )\) and satisfy a quadratic integral constraint. Let us mention that compactness of the set of trajectories guaranties existence of the optimal trajectories in the optimal control problems with continuous payoff functionals.

The paper is organized as follows: In Section 2, the set of admissible control functions is defined and the boundedness, closedness, convexity, and weak compactness of the set of admissible control functions is shown (Proposition 2.2 and Proposition 2.3). In Section 3 the system and the basic conditions which satisfies the system is introduced (Conditions (3A), (3B), and (3C)). Existence and uniqueness of the system’s trajectory generated by a given admissible control function are proved (Proposition 3.1). In Section 4 it is shown that the set of trajectories generated by all admissible control functions is a precompact subset of the space of continuous functions (Theorem 4.1). The closedness of the set of trajectories is proved in Section 5 (Proposition 5.1), and hence the compactness of the set of trajectories is obtained (Theorem 5.1).

2 The set of admissible control functions

We begin with the study of the set of admissible control functions. Let \(Q(\cdot):[t_{0},\theta]\rightarrow\mathbb{R}^{m\times m}\) be a continuous matrix function and \(Q(s)\) be a positive definite \(m\times m\) matrix for every \(s\in [t_{0},\theta ]\). The Lebesgue measurable function \(u(\cdot)\in L_{2} ( [t_{0},\theta ];\mathbb{R}^{m} )\) satisfying the inequality \(\int_{t_{0}}^{\theta}\langle Q(s)u(s), u(s)\rangle \,ds \leq1\) is said to be an admissible control function, where \(L_{2} ( [t_{0},\theta ];\mathbb{R}^{m} )\) is the space of Lebesgue measurable function \(u(\cdot): [t_{0},\theta ] \rightarrow\mathbb{R}^{m}\) such that \(\|u(\cdot)\|_{2}<+\infty\). Here \(\Vert u(\cdot)\Vert _{2} = (\int_{t_{0}}^{\theta } \Vert u(t)\Vert ^{2}\,dt )^{\frac{1}{2}}\), \(\Vert \cdot \Vert \) stands for the Euclidean norm, \(\langle\cdot, \cdot\rangle\) denotes the scalar product. The set of all admissible control functions is denoted by the symbol U. Thus

$$ U= \biggl\{ u(\cdot)\in L_{2} \bigl( [t_{0},\theta ]; \mathbb{R}^{m} \bigr): \int_{t_{0}}^{\theta}\bigl\langle Q(s)u(s), u(s)\bigr\rangle \,ds\leq1 \biggr\} . $$

Now let us give an auxiliary proposition which is used in the following arguments.

Proposition 2.1

Let \(Q(\cdot):[t_{0},\theta ]\rightarrow\mathbb{R}^{m\times m}\) be a continuous matrix function and \(Q(s)\) be a positive definite \(m\times m\) matrix for every \(s\in [t_{0},\theta ]\). Then there exist \(c_{1}>0\), \(c_{2}>0\) such that for each \(u(\cdot)\in L_{2} ( [t_{0},\theta ];\mathbb{R}^{m} )\) the inequality

$$ c_{1}^{2} \bigl(\bigl\| u(\cdot)\bigr\| _{2} \bigr)^{2}\leq \int_{t_{0}}^{\theta}\bigl\langle Q(s) u(s), u(s)\bigr\rangle \,ds\leq c_{2}^{2} \bigl(\bigl\| u(\cdot)\bigr\| _{2} \bigr)^{2} $$

holds.

Proof

Let \(S_{m}=\{u\in\mathbb{R}^{m} : \|u\|=1\}\). For given \(s\in [t_{0},\theta ]\) we set

$$\begin{aligned}& \gamma_{1}(s)=\min \bigl\{ \bigl\langle Q(s) u, u\bigr\rangle : u\in S_{m} \bigr\} ,\qquad \gamma_{2}(s)=\max \bigl\{ \bigl\langle Q(s) u, u\bigr\rangle : u\in S_{m} \bigr\} . \end{aligned}$$

Since the matrix function \(Q(\cdot):[t_{0},\theta]\rightarrow\mathbb {R}^{m\times m}\) is continuous and \(Q(s)\) is a positive definite \(m\times m\) matrix for every \(s\in [t_{0},\theta ]\), the functions \(\gamma_{1}(\cdot):[t_{0},\theta]\rightarrow\mathbb{R}\) and \(\gamma_{2}(\cdot):[t_{0},\theta]\rightarrow\mathbb{R}\) are continuous, and \(\gamma_{1}(s)>0\), \(\gamma_{2}(s)>0\) for every \(s\in [t_{0},\theta ]\).

Now we denote

$$\begin{aligned}& c_{1}^{2}=\min \bigl\{ \gamma_{1}(s) : s \in[t_{0},\theta] \bigr\} ,\\& c_{2}^{2}=\max \bigl\{ \gamma_{2}(s) : s \in[t_{0},\theta] \bigr\} . \end{aligned}$$

It is obvious that \(c_{1}>0\) and \(c_{2}>0\). Thus for each \(u\in S_{m}\) and \(s\in[t_{0},\theta]\), we have

$$ c_{1}^{2}\leq\bigl\langle Q(s) u, u\bigr\rangle \leq c_{2}^{2} . $$
(2.1)

Let us choose an arbitrary \(u(\cdot)\in L_{2} ( [t_{0},\theta ];\mathbb{R}^{m} )\). Then it follows from (2.1) that

$$ c_{1}^{2}\leq\biggl\langle Q(s) \frac{u(s)}{\Vert u(s)\Vert }, \frac {u(s)}{\Vert u(s)\Vert }\biggr\rangle \leq c_{2}^{2} $$

for every \(s\in[t_{0},\theta]\), where \(u(s)\neq0\). The last inequality implies the validity of the proposition. □

From Proposition 2.1 follows the validity of the following corollary.

Corollary 2.1

For every \(u(\cdot)\in U\) the inequality

$$ \bigl\Vert u(\cdot)\bigr\Vert _{2} \leq\frac{1}{c_{1}} $$

is satisfied, where the number \(c_{1}\) is defined in Proposition 2.1.

Let \(u(\cdot)\in U\). Then from Hölder’s inequality and Corollary 2.1 it follows that the inequality

$$ \int_{t_{0}}^{\theta}\bigl\Vert u(s)\bigr\Vert \,ds \leq \frac{1}{c_{1}} \sqrt {\theta-t_{0}} $$
(2.2)

is verified.

Proposition 2.2

The set of admissible control functions U is a bounded, closed, and convex subset of the space \(L_{2} ([t_{0},\theta] ; \mathbb{R}^{m} )\).

Proof

The boundedness of the set of admissible control functions U follows from Corollary 2.1.

Let us show closedness of the set U. Assume that \(u_{k}(\cdot)\in U\) for \(k=1,2,\ldots\) and \(\Vert u_{k}(\cdot )-u_{*}(\cdot)\Vert _{2} \rightarrow0\) as \(k\rightarrow\infty\). We will show that \(u_{*}(\cdot) \in U\), i.e., \(\int _{t_{0}}^{\theta}\langle Q(s) u_{*}(s), u_{*}(s)\rangle \,ds\leq1\).

It is not difficult to verify that

$$\begin{aligned} & \biggl\vert \int_{t_{0}}^{\theta}\bigl\langle Q(s) u_{k}(s),u_{k}(s) \bigr\rangle \,ds- \int_{t_{0}}^{\theta}\bigl\langle Q(s) u_{*}(s), u_{*}(s)\bigr\rangle \,ds \biggr\vert \\ &\quad \leq \int_{t_{0}}^{\theta}\bigl\| Q(s)\bigr\| \bigl\| u_{k}(s)\bigr\| \bigl\| u_{k}(s)-u_{*}(s)\bigr\| \,ds \\ &\qquad{}+ \int_{t_{0}}^{\theta}\bigl\| Q(s)\bigr\| \bigl\| u_{*}(s)\bigr\| \bigl\| u_{k}(s)-u_{*}(s)\bigr\| \,ds \end{aligned}$$
(2.3)

for every \(k=1,2,\ldots\) . Since the function \(Q(\cdot):[t_{0},\theta]\rightarrow\mathbb {R}^{m\times m}\) is continuous, there exists \(a_{*} >0\) such that \(\| Q(s)\|\leq a_{*}\) for every \(s\in[t_{0},\theta]\). Then (2.3) and Hölder’s inequality imply that

$$\begin{aligned} & \biggl|\int_{t_{0}}^{\theta}\bigl\langle Q(s) u_{k}(s), u_{k}(s)\bigr\rangle \,ds - \int _{t_{0}}^{\theta}\bigl\langle Q(s) u_{*}(s), u_{*}(s)\bigr\rangle \,ds \biggr| \\ &\quad \leq a_{*}\bigl\| u_{k}(\cdot)\bigr\| _{2} \cdot\bigl\| u_{k}( \cdot)-u_{*}(\cdot)\bigr\| _{2}+a_{*}\bigl\| u_{*}(\cdot)\bigr\| _{2} \cdot \bigl\| u_{k}(\cdot)-u_{*}(\cdot)\bigr\| _{2} \end{aligned}$$

for every \(k=1,2,\ldots\) . Since \(\Vert u_{k}(\cdot)-u_{*}(\cdot)\Vert _{2} \rightarrow0\) as \(k\rightarrow\infty\), there exists \(a_{1}>0\) such that \(\|u_{*}(\cdot)\|_{2}\leq a_{1}\), \(\|u_{k}(\cdot)\|_{2}\leq a_{1}\) for every \(k=1,2,\ldots\) . Thus the last inequality yields

$$\begin{aligned} \biggl\vert \int_{t_{0}}^{\theta}\bigl\langle Q(s) u_{k}(s), u_{k}(s)\bigr\rangle \,ds- \int_{t_{0}}^{\theta}\bigl\langle Q(s) u_{*}(s), u_{*}(s)\bigr\rangle \,ds \biggr\vert \leq2a_{*}a_{1}\bigl\| u_{k}(\cdot)-u_{*}( \cdot)\bigr\| _{2} \end{aligned}$$
(2.4)

for every \(k=1,2,\ldots\) . The inclusions \(u_{k}(\cdot)\in U\), \(k=1,2,\ldots\) , imply that

$$\begin{aligned} \int_{t_{0}}^{\theta }\bigl\langle Q(s) u_{k}(s),u_{k}(s) \bigr\rangle \,ds\leq1 \end{aligned}$$
(2.5)

for every \(k=1,2,\ldots\) . From (2.4) and (2.5) we obtain

$$\begin{aligned} \int_{t_{0}}^{\theta}\bigl\langle Q(s) u_{*}(s), u_{*}(s)\bigr\rangle \,ds \leq& \int_{t_{0}}^{\theta}\bigl\langle Q(s) u_{k}(s), u_{k}(s)\bigr\rangle \,ds + 2a_{*}a_{1}\bigl\| u_{k}( \cdot)-u_{*}(\cdot)\bigr\| _{2} \\ \leq&1+2a_{*}a_{1}\bigl\| u_{k}(\cdot )-u_{*}(\cdot) \bigr\| _{2} \end{aligned}$$

for every \(k=1,2,\ldots\) and hence

$$\begin{aligned} \int_{t_{0}}^{\theta}\bigl\langle Q(s) u_{*}(s), u_{*}(s)\bigr\rangle \,ds \leq 1. \end{aligned}$$

Thus \(u_{*}(\cdot) \in U\).

Now, let us show the convexity of the set U.

Since the matrix \(Q(s)\) is positive definite for every \(s\in[t_{0},\theta ]\), then it is possible to specify that the function \(u\rightarrow \langle Q(s)u,u\rangle\), \(u\in\mathbb{R}^{m}\), is convex for every \(s\in [t_{0},\theta]\) (see [27]).

Let \(u_{1}(\cdot)\in U\), \(u_{2}(\cdot)\in U\), and \(\alpha\in[0,1]\). Then from the convexity of the function \(u\rightarrow\langle Q(s)u,u\rangle \), \(u\in\mathbb{R}^{m}\), for every \(s\in[t_{0},\theta]\) it follows that

$$\begin{aligned} &\bigl\langle Q(s) \bigl(\alpha u_{1}(s)+ (1-\alpha )u_{2}(s) \bigr), \bigl(\alpha u_{1}(s)+ (1-\alpha )u_{2}(s) \bigr) \bigr\rangle \\ &\quad\leq\alpha\bigl\langle Q(s) u_{1}(s),u_{1}(s) \bigr\rangle + (1-\alpha )\bigl\langle Q(s) u_{2}(s),u_{2}(s)\bigr\rangle \end{aligned}$$

for every \(s\in[t_{0},\theta]\), and consequently

$$\begin{aligned} &\int_{t_{0}}^{\theta}\bigl\langle Q(s) \bigl(\alpha u_{1}(s)+ (1-\alpha ) u_{2}(s) \bigr), \bigl(\alpha u_{1}(s)+ (1-\alpha ) u_{2}(s) \bigr) \bigr\rangle \,ds \\ &\quad \leq \alpha \int_{t_{0}}^{\theta}\bigl\langle Q(s) u_{1}(s),u_{1}(s) \bigr\rangle \,ds+ (1-\alpha ) \int_{t_{0}}^{\theta}\bigl\langle Q(s) u_{2}(s),u_{2}(s) \bigr\rangle \,ds \\ &\quad \leq \alpha+ (1-\alpha ) =1. \end{aligned}$$

This means that \(\alpha u_{1}(\cdot)+ (1-\alpha )u_{2}(\cdot)\in U\) and the proof is completed. □

Proposition 2.3

The set of admissible control functions U is a weakly compact subset of the space \(L_{2} ([t_{0},\theta], \mathbb{R}^{m} )\).

Proof

Let \(u_{k}(\cdot) \in U\) for every \(k=1,2,\ldots\) . Let us show that there exists a subsequence \(\{u_{k_{i}}(\cdot) \}_{i=1}^{\infty}\) of the sequence \(\{u_{k}(\cdot) \}_{k=1}^{\infty}\) and \(u_{*}(\cdot )\in U\) such that \(u_{k_{i}}(\cdot)\stackrel{\mathrm{weak}}{\longrightarrow} u_{*}(\cdot)\) as \(i\rightarrow\infty\).

Since \(u_{k}(\cdot)\in U\) for every \(k=1,2,\ldots\) , by virtue of the Corollary 2.1 we see that the sequence \(\{u_{k}(\cdot) \}_{k=1}^{\infty}\) is bounded in the space \(L_{2} ([t_{0},\theta], \mathbb{R}^{m} )\), and hence according to [28] it has a weakly convergent subsequence \(\{u_{k_{i}}(\cdot) \}_{i=1}^{\infty}\). Let \(u_{k_{i}}(\cdot )\stackrel{\mathrm{weak}}{\longrightarrow} u_{*}(\cdot)\) as \(i\rightarrow\infty\).

By Mazur’s theorem (see, e.g. [29]), for each \(j>0\), there exist \(\alpha_{1}^{j}\geq0, \alpha_{2}^{j}\geq0, \ldots,\alpha_{j}^{j} \geq0\) such that \(\sum_{i=1}^{j}\alpha_{i}^{j}=1\) and

$$ \Biggl\Vert \sum_{i=1}^{j} \alpha_{i}^{j} u_{k_{i}}(\cdot)-u_{*}(\cdot)\Biggr\Vert _{2}< \frac{1}{j} . $$
(2.6)

Let us denote \(z_{j}(\cdot)=\sum_{i=1}^{j}\alpha_{i}^{j} u_{k_{i}}(\cdot)\). Since \(\alpha_{1}^{j}\geq0, \alpha_{2}^{j}\geq0, \ldots ,\alpha_{j}^{j} \geq0\), \(\sum_{i=1}^{j}\alpha_{i}^{j}=1\), \(u_{k_{i}}(\cdot)\in U\) for every \(i=1,2,\ldots\) and \(U\subset L_{2} ([t_{0},\theta], \mathbb{R}^{m} )\) is a convex set (according to the Proposition 2.2), we have \(z_{j}(\cdot)\in U\) for every \(j=1,2,\ldots\) . Thus, from (2.6) we conclude that for a given \(j>0\) there exists \(z_{j}(\cdot)\in U\) such that the inequality

$$ \bigl\Vert z_{j}(\cdot)-u_{*}(\cdot)\bigr\Vert _{2}< \frac{1}{j} $$
(2.7)

holds. This means that \(u_{*}(\cdot)\in cl (U)\), where cl denotes the closure of a set. Via Proposition 2.2, U is a closed set. Then we obtain \(u_{*}(\cdot)\in U\). □

3 The system and the set of trajectories

We consider a control system the behavior of which is described by a Urysohn type integral equation

$$ x(t)=f \bigl(t,x(t) \bigr)+\lambda \int_{t_{0}}^{\theta} \bigl[K_{1} \bigl(t,s,x(s) \bigr)+K_{2} \bigl(t,s,x(s) \bigr)u(s) \bigr]\,ds, $$
(3.1)

where \(t\in [t_{0},\theta ]\), \(s\in [t_{0},\theta ]\), \(x(s)\in\mathbb{R}^{n}\) is the state vector, \(u(s)\in\mathbb{R}^{m}\) is the control vector and \(\lambda\geq0\).

We assume that the functions and the number \(\lambda\geq0\) given in system (3.1) satisfy the following conditions:

  1. (3A)

    the functions \(f(\cdot) : [t_{0},\theta ]\times \mathbb{R}^{n}\rightarrow\mathbb{R}^{n}\), \(K_{1}(\cdot) : [t_{0},\theta ]\times [t_{0},\theta ]\times\mathbb{R}^{n}\rightarrow\mathbb {R}^{n}\), and \(K_{2}(\cdot): [t_{0},\theta ]\times [t_{0},\theta ]\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{n\times m}\) are continuous;

  2. (3B)

    there exist \(L_{0}\in[0,1)\), \(L_{1}\geq0\), and \(L_{2}\geq0\) such that

    $$\begin{aligned}& \bigl\Vert f(t,x_{1})-f(t,x_{2})\bigr\Vert \leq L_{0} \Vert x_{1}-x_{2} \Vert , \\& \bigl\Vert K_{1}(t,s,x_{1})-K_{1}(t,s,x_{2}) \bigr\Vert \leq L_{1} \Vert x_{1}-x_{2} \Vert , \\& \bigl\Vert K_{2}(t,s,x_{1})-K_{2}(t,s,x_{2}) \bigr\Vert \leq L_{2} \Vert x_{1}-x_{2} \Vert \end{aligned}$$

    for every \((t,x_{1}) \in [t_{0},\theta ]\times\mathbb{R}^{n}\), \((t,x_{2}) \in [t_{0},\theta ]\times\mathbb{R}^{n}\), \((t,s,x_{1}) \in [t_{0},\theta ]\times [t_{0},\theta ]\times\mathbb {R}^{n}\), \((t,s,x_{2}) \in [t_{0},\theta ]\times [t_{0},\theta ]\times\mathbb{R}^{n}\);

  3. (3C)

    the inequality \(\lambda L_{1} ( \theta-t_{0} )+\lambda L_{2}\sqrt{\theta-t_{0}}\frac{1}{c_{1}}< 1-L_{0}\) is satisfied, where \(c_{1}\) is defined in Proposition 2.1.

We denote

$$ L(\lambda)= L_{0}+\lambda L_{1} ( \theta-t_{0} )+\lambda L_{2}\sqrt {\theta-t_{0}} \frac{1}{c_{1}}. $$
(3.2)

By virtue of condition (3C) we have \(L(\lambda) < 1\).

Now, let us define the trajectory of the system (3.1) generated by a given admissible control function. Let \(u_{*}(\cdot)\in U\). A continuous function \(x_{*}(\cdot): [t_{0},\theta ]\rightarrow \mathbb{R}^{n}\) satisfying the integral equation

$$ x_{*}(t)=f \bigl(t,x_{*}(t) \bigr)+\lambda \int_{t_{0}}^{\theta} \bigl[K_{1} \bigl(t,s,x_{*}(s) \bigr)+K_{2} \bigl(t,s,x_{*}(s) \bigr)u_{*}(s) \bigr]\,ds $$

for each \(t\in [t_{0},\theta ]\) is called a trajectory of the system (3.1), generated by the admissible control function \(u_{*}(\cdot)\in U\).

The trajectory of the system (3.1) generated by the control function \(u(\cdot)\in U\) is denoted by \(x (\cdot ;u(\cdot) )\) and the set

$$ \mathbf{X}=\bigl\{ x\bigl(\cdot ;u(\cdot)\bigr): u(\cdot)\in U\bigr\} $$

is called the set of trajectories of the system (3.1). It is obvious that \(\mathbf{X} \subset C ( [t_{0},\theta ];\mathbb {R}^{n} )\), where \(C ( [t_{0},\theta ];\mathbb{R}^{n} )\) is the space of continuous functions \(x(\cdot) : [t_{0},\theta ]\rightarrow\mathbb{R}^{n}\) with norm

$$ \bigl\| x(\cdot)\bigr\| _{C} = \max\bigl\{ \bigl\| x(t)\bigr\| : t\in[t_{0},\theta] \bigr\} . $$

For \(t\in[t_{0},\theta]\) we denote

$$ \mathbf{X}(t)=\bigl\{ x(t)\in\mathbb{R}^{n}: x(\cdot)\in \mathbf{X} \bigr\} . $$
(3.3)

The set \(\mathbf{X}(t)\) consists of points to which arrive the trajectories of the system at the instant of t.

Proposition 3.1

Every \(u(\cdot)\in U\) generates a unique trajectory of the system (3.1).

Proof

Let \(u_{*}(\cdot)\in U\) be a fixed admissible control function. Define an operator \(F (x (\cdot ) )\) by setting

$$ F\bigl(x(\cdot)\bigr)|(t)=f \bigl(t,x(t) \bigr)+\lambda \int_{t_{0}}^{\theta} \bigl[K_{1} \bigl(t,s,x(s) \bigr)+K_{2} \bigl(t,s,x(s) \bigr)u_{*}(s) \bigr]\,ds, \quad t \in[t_{0},\theta], $$

where \(x(\cdot)\in C ( [t_{0},\theta ],\mathbb{R}^{n} ) \).

It is not difficult to prove that, for each fixed \(x(\cdot)\in C ( [t_{0},\theta ];\mathbb{R}^{n} )\), the function \(t\mapsto F (x(\cdot) )|(t)\), \(t\in [t_{0},\theta ]\), is continuous. So is the operator

$$ F (\cdot ): C \bigl( [t_{0},\theta ];\mathbb{R}^{n} \bigr) \rightarrow C \bigl( [t_{0},\theta ];\mathbb{R}^{n} \bigr). $$

Let us choose arbitrarily \(x_{1}(\cdot)\in C ( [t_{0},\theta ];\mathbb{R}^{n} )\), \(x_{2}(\cdot)\in C ( [t_{0},\theta ];\mathbb{R}^{n} )\) and \(t\in[t_{0},\theta]\). Then from condition (3B) and (2.2) it follows that

$$\begin{aligned} & \bigl\| F \bigl(x_{2}(\cdot) \bigr)|(t)-F \bigl(x_{1}(\cdot) \bigr)|(t) \bigr\| \\ &\quad\leq L_{0}\bigl\| x_{2}(t)-x_{1}(t)\bigr\| +\lambda L_{1} \int_{t_{0}}^{\theta}\bigl\| x_{2}(s)-x_{1}(s) \bigr\| \,ds \\ &\qquad{}+\lambda L_{2} \int_{t_{0}}^{\theta} \bigl\| x_{2}(s)-x_{1}(s) \bigr\| \bigl\| u_{*}(s)\bigr\| \,ds \\ &\quad\leq \biggl[ L_{0}+\lambda L_{1} (\theta-t_{0} )+\lambda L_{2} \int _{t_{0}}^{\theta}\bigl\| u_{*}(s)\bigr\| \,ds \biggr] \bigl\| x_{2}( \cdot)-x_{1}(\cdot)\bigr\| _{C} \\ &\quad\leq \biggl[L_{0}+\lambda L_{1} (\theta-t_{0} )+\lambda L_{2}\sqrt {\theta-t_{0}}\cdot\frac{1}{c_{1}} \biggr]\bigl\| x_{2}(\cdot)-x_{1}(\cdot)\bigr\| _{C} \end{aligned}$$

for every \(t\in[t_{0},\theta]\), and hence

$$\begin{aligned} \bigl\| F \bigl(x_{2}(\cdot) \bigr)-F \bigl(x_{1}( \cdot) \bigr) \bigr\| _{C}\leq L(\lambda) \bigl\| x_{2}( \cdot)-x_{1}(\cdot)\bigr\| _{C} , \end{aligned}$$
(3.4)

where \(L(\lambda)\) is defined by (3.2).

Via Condition (3C) we have \(L(\lambda)<1\). Then (3.4) shows that the operator \(F(\cdot):C ( [t_{0},\theta ]; \mathbb {R}^{n} )\rightarrow C ( [t_{0},\theta ];\mathbb {R}^{n} )\) is contractive. Since \(C ( [t_{0},\theta ],\mathbb{R}^{n} )\) is a complete metric space, by the Banach fixed point theorem, the operator \(F(\cdot)\) has a unique fixed point, that is, there exists a unique \(x_{*}(\cdot)\in C ( [t_{0},\theta ];\mathbb{R}^{n} )\) such that \(F (x_{*}(\cdot) )=x_{*}(\cdot)\), which means that there exists a unique \(x_{*}(\cdot)\in C ( [t_{0},\theta ];\mathbb{R}^{n} )\) such that

$$ x_{*}(t)=f \bigl(t,x_{*}(t) \bigr)+\lambda \int_{a}^{b} \bigl[K_{1} \bigl(t,s,x_{*}(s) \bigr)+K_{2} \bigl(t,s,x_{*}(s) \bigr)u_{*}(s) \bigr]\,ds $$

for every \([t_{0},\theta ]\). □

4 Precompactness of the set of trajectories

First of all we will prove that the set of trajectories X of the system (3.1) is a bounded subset of the space \(C ( [t_{0},\theta ];\mathbb{R}^{n} )\).

Proposition 4.1

There exist \(\beta_{0}\geq0\), \(\beta_{1}\geq0\), \(\beta_{2}\geq0\) such that

  1. (i)

    \(\|f (t,x )\| \leq \beta_{0}+L_{0}\|x\|\),

  2. (ii)

    \(\|K_{1} (t,s,x )\| \leq \beta_{1}+L_{1}\|x\|\),

  3. (iii)

    \(\|K_{2} (t,s,x )\| \leq \beta_{2}+L_{2}\|x\|\),

for every \((t,x )\in [t_{0},\theta ]\times\mathbb {R}^{n}\) and \((t,s,x )\in [t_{0},\theta ]\times [t_{0},\theta ]\times\mathbb{R}^{n}\), where \(L_{0}\), \(L_{1}\), and \(L_{2}\) are defined in Condition (3B).

Proof

We just show the proof for (iii). According to Condition (3B)

$$ \bigl\| K_{2} (t,s,x )-K_{2} (t,s,0 )\bigr\| \leq L_{2}\|x\| $$

for every \((t,s,x ) \in [t_{0},\theta ]\times [t_{0},\theta ]\times\mathbb{R}^{n}\) and consequently,

$$ \bigl\| K_{2} (t,s,x )\bigr\| \leq\bigl\| K_{2} (t,s,0 )\bigr\| + L_{2}\|x\| $$
(4.1)

for every \((t,s,x )\in [t_{0},\theta ]\times [t_{0},\theta ]\times\mathbb{R}^{n}\). Since the function \(K_{2}(\cdot)\) is continuous, setting

$$ \beta_{2}=\max \bigl\{ \bigl\| K_{2}(t,s,0)\bigr\| : (t,s )\in [t_{0},\theta ]\times [t_{0},\theta ] \bigr\} $$

we obtain the proof of (iii) from (4.1). The proofs of (i) and (ii) are carried out similarly. □

Denote

$$ q_{*}= \frac{ \beta_{0}+\lambda\beta_{1} (\theta-t_{0} )+\lambda\beta_{2}\sqrt{\theta-t_{0}}\frac{1}{c_{1}}}{1-L(\lambda)} , $$
(4.2)

where \(L(\lambda)\) is defined by (3.2).

Proposition 4.2

For every \(x(\cdot) \in\mathbf{X}\) the inequality

$$ \bigl\Vert x(\cdot)\bigr\Vert _{C} \leq q_{*} $$

is satisfied.

Proof

Let \(x(\cdot)\in\mathbf{X}\) be an arbitrary chosen trajectory, generated by the control function \(u(\cdot)\in U\). Then

$$ x(t)=f \bigl(t,x(t) \bigr)+\lambda \int_{t_{0}}^{\theta} \bigl[K_{1} \bigl(t,s,x(s) \bigr)+ K_{2} \bigl(t,s,x(s) \bigr)u(s) \bigr]\,ds $$

for each \(t\in [t_{0},\theta ]\). Proposition 4.1, (2.2), and (3.2) imply that

$$\begin{aligned} \bigl\| x(t)\bigr\| \leq& \beta_{0}+L_{0}\bigl\| x(t)\bigr\| +\lambda \int_{t_{0}}^{\theta} \bigl[ \bigl(\beta_{1}+L_{1} \bigl\| x(s)\bigr\| \bigr)+ \bigl(\beta_{2}+L_{2}\bigl\| x(s)\bigr\| \bigr)\bigl\| u(s)\bigr\| \bigr]\,ds \\ \leq&\beta_{0}+L_{0}\bigl\| x(\cdot)\bigr\| _{C}+\lambda \beta_{1} (\theta-t_{0} )+\lambda L_{1}\bigl\| x(\cdot) \bigr\| _{C} (\theta-t_{0} ) \\ &{}+ \lambda\beta _{2} \int_{t_{0}}^{\theta}\bigl\Vert u(s)\bigr\Vert \,ds + \lambda L_{2}\bigl\| x(\cdot)\bigr\| _{C} \int_{t_{0}}^{\theta}\bigl\| u(s)\bigr\| \,ds \\ \leq&\beta_{0}+\lambda\beta_{1} (\theta-t_{0} )+\lambda\beta_{2}\sqrt {\theta-t_{0}}\frac{1}{c_{1}} \\ &{}+\bigl\| x(\cdot)\bigr\| _{C} \biggl(L_{0}+\lambda L_{1} (\theta-t_{0} )+\lambda L_{2}\sqrt{\theta-t_{0}} \frac{1}{c_{1}} \biggr) \\ =& \beta_{0}+\lambda\beta _{1} (\theta-t_{0} )+\lambda\beta_{2}\sqrt{\theta-t_{0}}\frac {1}{c_{1}}+L( \lambda)\bigl\| x(\cdot)\bigr\| _{C} \end{aligned}$$

for every \(t\in [t_{0},\theta ]\), and consequently

$$\begin{aligned} \bigl\| x(\cdot)\bigr\| _{C}\leq \beta_{0}+\lambda\beta_{1} (\theta-t_{0} )+\lambda\beta_{2}\sqrt{\theta-t_{0}} \frac{1}{c_{1}}+L(\lambda)\bigl\| x(\cdot)\bigr\| _{C} . \end{aligned}$$

Since \(L(\lambda)<1\), the last inequality and (4.2) complete the proof. □

Proposition 4.2 shows that the set of trajectories X of the system (3.1) is bounded.

Now let us prove that the set of trajectories X of the system (3.1) is a family of equicontinuous functions. Denote

$$\begin{aligned} &B_{n}(q_{*})= \bigl\{ x\in\mathbb{R}^{n} : \|x\|\leq q_{*} \bigr\} , \\ &D_{1}= [t_{0},\theta ]\times B_{n}(q_{*}), \qquad D_{2}= [t_{0},\theta ]\times [t_{0},\theta ] \times B_{n}(q_{*}), \\ &\omega_{0} (\Delta )=\max \bigl\{ \bigl\Vert f ( t_{2},x )-f(t_{1},x) \bigr\Vert : ( t_{1},x ) \in D_{1}, ( t_{2},x )\in D_{1}, \vert t_{2}-t_{1} \vert \leq\Delta \bigr\} , \end{aligned}$$
(4.3)
$$\begin{aligned} & \begin{aligned}[b]\omega_{1} (\Delta )={}&\max \bigl\{ \bigl\Vert K_{1} ( t_{2},s,x )-K_{1}(t_{1},s,x) \bigr\Vert : \\ &{}( t_{1},s,x )\in D_{2}, ( t_{2},s,x )\in D_{2}, \vert t_{2}-t_{1} \vert \leq\Delta \bigr\} , \end{aligned} \end{aligned}$$
(4.4)
$$\begin{aligned} & \begin{aligned}[b] \omega_{2} (\Delta )={}&\max \bigl\{ \bigl\Vert K_{2} ( t_{2},s,x )-K_{2}(t_{1},s,x) \bigr\Vert : ( t_{1},s,x )\in D_{2},\\ &{}( t_{2},s,x )\in D_{2}, \vert t_{2}-t_{1} \vert \leq\Delta \bigr\} , \end{aligned} \end{aligned}$$
(4.5)
$$\begin{aligned} &\varphi(\Delta) = \frac{1}{1-L_{0}} \biggl[ \omega_{0} ( \Delta ) +\lambda ( \theta-t_{0} ) \omega_{1} ( \Delta )+ \frac{\lambda\sqrt{\theta-t_{0}}}{c_{1}} \omega_{2} ( \Delta ) \biggr]. \end{aligned}$$
(4.6)

It is obvious that \(\varphi(\cdot):[0,\infty)\rightarrow[0,\infty)\) is not decreasing and \(\varphi(\Delta) \rightarrow0^{+}\) as \(\Delta \rightarrow0^{+}\).

Proposition 4.3

For every \(t_{1}\in [t_{0},\theta ]\), \(t_{2}\in [t_{0},\theta ]\), and \(x(\cdot)\in\mathbf{X}\) the inequality

$$ \bigl\Vert x ( t_{2} )-x ( t_{1} ) \bigr\Vert \leq \varphi \bigl( \vert t_{2}-t_{1} \vert \bigr) $$

holds, where \(\varphi(\cdot)\) is defined by (4.6).

Proof

Let \(x(\cdot)\in\mathbf{X}\) be an arbitrarily chosen trajectory generated by the admissible control function \(u(\cdot)\in U\). Then

$$ x(t)=f \bigl(t,x(t) \bigr)+\lambda \int_{t_{0}}^{\theta} \bigl[K_{1} \bigl(t,s,x(s) \bigr)+K_{2} \bigl(t,s,x(s) \bigr)u(s) \bigr]\,ds $$

for every \(t\in[t_{0},\theta]\), and hence

$$\begin{aligned} \bigl\Vert x ( t_{2} )-x ( t_{1} ) \bigr\Vert \leq& \bigl\Vert f \bigl( t_{2},x ( t_{2} ) \bigr)- f \bigl( t_{1},x ( t_{1} ) \bigr) \bigr\Vert +\lambda \int_{t_{0}}^{\theta }\bigl\Vert K_{1} \bigl( t_{2},s,x ( s ) \bigr)-K_{1} \bigl( t_{1},s,x ( s ) \bigr) \bigr\Vert \,ds \\ &{}+\lambda \int_{a}^{b}\bigl\Vert K_{2} \bigl( t_{2},s,x ( s ) \bigr)-K_{2} \bigl( t_{1},s,x ( s ) \bigr) \bigr\Vert \bigl\Vert u ( s ) \bigr\Vert \,ds. \end{aligned}$$
(4.7)

Since \(x(\cdot)\in\mathbf{X}\), according to Proposition 4.2 we have \(\Vert x(s)\Vert \leq q_{*}\) for every \(s\in[t_{0},\theta]\). From Condition (3B), (4.3), (4.4), and (4.5) it follows that

$$\begin{aligned} &\bigl\Vert f \bigl( t_{2},x ( t_{2} ) \bigr)- f \bigl( t_{1},x ( t_{1} ) \bigr) \bigr\Vert \\ &\quad\leq \bigl\Vert f \bigl( t_{2},x ( t_{2} ) \bigr)- f \bigl( t_{2},x ( t_{1} ) \bigr) \bigr\Vert +\bigl\Vert f \bigl( t_{2},x ( t_{1} ) \bigr)- f \bigl( t_{1},x ( t_{1} ) \bigr) \bigr\Vert \\ &\quad\leq L_{0}\bigl\Vert x ( t_{2} )-x ( t_{1} ) \bigr\Vert +\omega_{0} \bigl( \vert t_{2}-t_{1} \vert \bigr), \end{aligned}$$
(4.8)
$$\begin{aligned} &\bigl\Vert K_{1} \bigl( t_{2},s,x ( s ) \bigr)-K_{1} \bigl( t_{1},s,x ( s ) \bigr) \bigr\Vert \leq \omega_{1} \bigl( \vert t_{2}-t_{1} \vert \bigr), \end{aligned}$$
(4.9)
$$\begin{aligned} &\bigl\Vert K_{2} \bigl( t_{2},s,x ( s ) \bigr)-K_{2} \bigl( t_{1},s,x ( s ) \bigr) \bigr\Vert \leq \omega_{2} \bigl(\vert t_{2}-t_{1} \vert \bigr) \end{aligned}$$
(4.10)

for any \(s\in [t_{0},\theta ]\). From (2.2), (4.7), (4.8), (4.9), and (4.10) we obtain

$$\begin{aligned} \bigl\Vert x ( t_{2} )-x ( t_{1} ) \bigr\Vert \leq& L_{0}\bigl\Vert x ( t_{2} )-x ( t_{1} ) \bigr\Vert +\omega_{0} \bigl( \vert t_{2}-t_{1} \vert \bigr)+\lambda ( \theta-t_{0} )\omega _{1} \bigl( \vert t_{2}-t_{1} \vert \bigr) \\ &{}+\lambda\omega_{2} \bigl( \vert t_{2}-t_{1} \vert \bigr) \int _{t_{0}}^{\theta}\bigl\Vert u ( s ) \bigr\Vert \,ds \\ \leq& L_{0}\bigl\Vert x ( t_{2} )-x ( t_{1} ) \bigr\Vert + \omega_{0} \bigl( \vert t_{2}-t_{1} \vert \bigr)+\lambda ( \theta-t_{0} )\omega_{1} \bigl( \vert t_{2}-t_{1} \vert \bigr) \\ &{}+\lambda \omega_{2} \bigl( \vert t_{2}-t_{1} \vert \bigr)\cdot\frac{\sqrt{\theta -t_{0}}}{c_{1}} . \end{aligned}$$

Since \(L_{0}\in[0,1)\), the last inequality and (4.6) imply the proof. □

Proposition 4.4

The set of trajectories X of the system (3.1) is a family of equicontinuous functions.

Proof

Let us choose an arbitrary \(\varepsilon> 0\). Since \(\varphi(\Delta )\rightarrow0^{+}\) as \(\Delta\rightarrow0^{+}\), for \(\varepsilon>0\) there exists \(\delta ( \varepsilon )> 0\) such that for each \(\Delta\in (0, \delta ( \varepsilon ) )\) the inequality \(\varphi(\Delta)<\varepsilon\) is satisfied.

Choose an arbitrary \(x(\cdot)\in\mathbf{X}\) and \(t_{1}\in [t_{0},\theta ]\), \(t_{2}\in [t_{0},\theta ]\) such that \(\vert t_{2}-t_{1}\vert <\delta(\varepsilon)\). Then according to Proposition 4.3 we have

$$ \bigl\| x(t_{2})-x(t_{1})\bigr\| \leq\varphi \bigl(\vert t_{2}-t_{1}\vert \bigr) < \varepsilon $$

and the proof is completed. □

Thus from Proposition 4.2, Proposition 4.4, and the Arzela-Ascoli theorem we obtain the precompactness of the set of trajectories.

Theorem 4.1

The set of trajectories X of the system (3.1) is a precompact subset of the space \(C ( [ t_{0},\theta ],\mathbb{R}^{n} )\).

Let \(h(E,D)\) denote the Hausdorff distance between the sets \(E\subset \mathbb{R}^{n}\) and \(D\subset\mathbb{R}^{n}\). From Proposition 4.3 follows the validity of the following corollary.

Corollary 4.1

For every \(t_{1}\in [t_{0},\theta ]\) and \(t_{2}\in [t_{0},\theta ]\) the inequality

$$ h \bigl(\mathbf{X}(t_{1}),\mathbf{X}(t_{2}) \bigr)\leq \varphi \bigl(\vert t_{2}-t_{1}\vert \bigr) $$

is satisfied, and hence the set valued map \(t\rightarrow\mathbf {X}(t)\), \(t\in [t_{0},\theta ]\), is continuous in the Hausdorff metric, where the set \(\mathbf{X}(t)\) is defined by (3.3).

5 Closedness of the set of trajectories

The next theorem specifies closedness of the set of trajectories X of the system (3.1).

Proposition 5.1

The set of trajectories X of the system (3.1) is a closed subset of the space \(C ( [ t_{0},\theta ];\mathbb {R}^{n} )\).

Proof

Suppose that \(x_{k}(\cdot)\in\mathbf{X}\) for every \(k=1,2,\ldots\) and \(\Vert x_{k}(\cdot)-x_{0}(\cdot)\Vert _{C} \rightarrow0\) as \(k\rightarrow \infty\). Let the trajectory \(x_{k}(\cdot)\) be generated by the admissible control function \(u_{k}(\cdot)\in U\), where \(k=1,2,\ldots\) . According to Proposition 2.3, the set of admissible control functions U is weakly compact in the space \(L_{2} ( [ t_{0},\theta ]; \mathbb{R}^{m} )\). Then, without loss of generality, one can assume that \(u_{k}(\cdot)\stackrel{\mathrm{weak}}{\longrightarrow}u_{*}(\cdot)\) as \(k\rightarrow \infty\), where \(u_{*}(\cdot)\in U\). Let \(x_{*}(\cdot):[t_{0},\theta] \rightarrow\mathbb{R}^{n}\) be the trajectory of system (3.1), generated by \(u_{*}(\cdot)\in U\). Then \(x_{*}(\cdot)\in\textbf{X}\) and via condition (3B) we have

$$\begin{aligned} \bigl\| x_{k}(t)-x_{*}(t)\bigr\| \leq& \bigl\Vert f \bigl(t,x_{k}(t) \bigr)-f \bigl(t,x_{*}(t) \bigr)\bigr\Vert +\lambda \int_{t_{0}}^{\theta}\bigl\Vert K_{1} \bigl(t,s,x_{k}(s) \bigr)- K_{1} \bigl(t,s,x_{*}(s) \bigr) \bigr\Vert \,ds \\ &{}+ \lambda\biggl\Vert \int_{t_{0}}^{\theta} \bigl[K_{2} \bigl(t,s,x_{k}(s) \bigr)u_{k}(s)-K_{2} \bigl(t,s,x_{*}(s) \bigr)u_{*}(s) \bigr]\,ds\biggr\Vert \\ \leq& L_{0} \bigl\Vert x_{k}(s)-x_{*}(s)\bigr\Vert + \lambda L_{1} \int_{t_{0}}^{\theta } \bigl\Vert x_{k}(s)-x_{*}(s) \bigr\Vert \,ds \\ &{}+ \lambda\biggl\Vert \int_{t_{0}}^{\theta} \bigl[K_{2} \bigl(t,s,x_{k}(s) \bigr)u_{k}(s)-K_{2} \bigl(t,s,x_{*}(s) \bigr)u_{*}(s) \bigr]\,ds\biggr\Vert \end{aligned}$$

for any \(t\in [t_{0},\theta ]\). Since \(L_{0} \in[0,1)\), the last inequality yields

$$\begin{aligned} \bigl\| x_{k}(t)-x_{*}(t)\bigr\| \leq&\frac{\lambda L_{1}}{1-L_{0}} \int_{t_{0}}^{\theta}\bigl\| x_{k}(s)-x_{*}(s)\bigr\| \,ds \\ &{} +\frac{\lambda}{1-L_{0}} \biggl\Vert \int_{t_{0}}^{\theta} \bigl[K_{2} \bigl(t,s,x_{k}(s) \bigr)u_{k}(s)-K_{2} \bigl(t,s,x_{*}(s) \bigr)u_{*}(s) \bigr]\,ds \biggr\Vert \end{aligned}$$
(5.1)

for every \(t\in [t_{0},\theta ]\).

Condition (3B) implies that

$$\begin{aligned} & \biggl\Vert \int_{t_{0}}^{\theta} \bigl[K_{2} \bigl(t,s,x_{k}(s) \bigr)u_{k}(s)-K_{2} \bigl(t,s,x_{*}(s) \bigr)u_{*}(s) \bigr]\,ds\biggr\Vert \\ &\quad\leq \int_{t_{0}}^{\theta} L_{2}\bigl\Vert x_{k}(s)-x_{*}(s)\bigr\| \bigl\| u_{k}(s)\bigr\Vert \,ds +\biggl\Vert \int_{t_{0}}^{\theta} K_{2} \bigl(t,s,x_{*}(s) \bigr) \bigl(u_{k}(s)-u_{*}(s) \bigr)\,ds\biggr\Vert . \end{aligned}$$
(5.2)

Setting \(\psi (t,s )= K_{2} (t,s,x_{*}(s) )\), from (5.1) and (5.2) we obtain

$$\begin{aligned} \bigl\| x_{k}(t)-x_{*}(t)\bigr\| \leq& \frac{\lambda}{1-L_{0}} \int_{t_{0}}^{\theta} \bigl(L_{1}+L_{2} \bigl\| u_{k}(s)\bigr\| \bigr)\bigl\| x_{k}(s)-x_{*}(s)\bigr\| \,ds \\ &{}+\frac{\lambda}{1-L_{0}}\biggl\Vert \int_{t_{0}}^{\theta} \psi (t,s ) \bigl(u_{k}(s)-u_{*}(s) \bigr)\,ds\biggr\Vert \end{aligned}$$
(5.3)

for every \(t\in [t_{0},\theta ]\).

Since the function \(\psi(\cdot,\cdot): [t_{0},\theta ]\times [t_{0},\theta ]\rightarrow\mathbb{R}^{n\times m}\) is continuous, \(u_{k}(\cdot) \stackrel{\mathrm{weak}}{\longrightarrow} u_{*}(\cdot)\) as \(k\rightarrow\infty\), for each fixed \(t \in [t_{0},\theta ]\) we have

$$ \int_{t_{0}}^{\theta} \psi (t,s ) \bigl(u_{k}(s)-u_{*}(s) \bigr)\,ds\rightarrow0 \quad\mbox{as } k\rightarrow\infty. $$

Thus, for a fixed \(t \in [t_{0},\theta ]\) and for a given \(\varepsilon>0\) there exists \(K_{*} (t,\varepsilon ) > 0\) such that for each \(k>K_{*} (t,\varepsilon )\) the inequality

$$ \biggl\Vert \int_{t_{0}}^{\theta} \psi (t,s ) \bigl(u_{k}(s)-u_{*}(s) \bigr)\,ds\biggr\Vert < \varepsilon $$
(5.4)

is satisfied.

Now let us prove that for a given \(\varepsilon>0\), there exists \(K^{*}(\varepsilon)>0\) such that for each \(k>K^{*}(\varepsilon)\) and \(t\in [t_{0},\theta ]\) the inequality

$$ \biggl\Vert \int_{t_{0}}^{\theta} \psi (t,s ) \bigl(u_{k}(s)-u_{*}(s) \bigr)\,ds\biggr\Vert < \varepsilon $$
(5.5)

holds.

Assume the contrary. Then there exist \(\varepsilon_{*}>0\), \(t_{i}\in [t_{0},\theta ]\), \(k_{i}>0\) such that \(k_{i}\rightarrow+\infty\) as \(i\rightarrow+\infty\) and the inequality

$$ \biggl\Vert \int_{t_{0}}^{\theta} \psi (t_{i},s ) \bigl(u_{k_{i}}(s)-u_{*}(s) \bigr)\,ds\biggr\Vert \geq\varepsilon_{*} $$
(5.6)

is verified for every \(i=1,2,\ldots\) .

Since \(t_{i}\in [t_{0},\theta ]\) for each \(i=1,2,\ldots\) , without loss of generality, assume that \(t_{i}\rightarrow t_{*}\in [t_{0},\theta ]\) as \(i\rightarrow+\infty\).

According to (5.4), for \(\frac{\varepsilon _{*}}{4}>0\) there exists \(K_{1} (t_{*},\varepsilon_{*} )>0\) such that for each \(i>K_{1} (t_{*},\varepsilon_{*} )\) the inequality

$$ \biggl\Vert \int_{t_{0}}^{\theta} \psi (t_{*},s ) \bigl(u_{k_{i}}(s)-u_{*}(s) \bigr)\,ds\biggr\Vert < \frac{\varepsilon_{*}}{4} $$
(5.7)

holds.

The continuity of the function \(\psi(\cdot,\cdot): [t_{0},\theta ]\times [t_{0},\theta ]\rightarrow\mathbb{R}^{n\times m}\) shows that for \(\frac{\varepsilon_{*} c_{1}}{8\sqrt{\theta -t_{0}}}\) there exists \(K_{2} (t_{*},\varepsilon_{*} )\) such that for every \(i>K_{2} (t_{*},\varepsilon_{*} )\) and \(s\in [t_{0},\theta ]\) the inequality

$$ \bigl\Vert \psi (t_{i},s )-\psi (t_{*},s )\bigr\Vert < \frac {\varepsilon_{*} c_{1}}{8\sqrt{\theta-t_{0}}} $$
(5.8)

is satisfied.

Denote \(K_{3} (t_{*},\varepsilon_{*} )=\max \{K_{1} (t_{*},\varepsilon_{*} ),K_{2} (t_{*},\varepsilon_{*} ) \}\). Since \(u_{k_{i}}(\cdot)\in U\), \(u_{*}(\cdot)\in U\), (2.2), (5.7), and (5.8) yield, for every \(i>K_{3} (t_{*},\varepsilon_{*} )\), the inequality

$$\begin{aligned} \biggl\Vert \int_{t_{0}}^{\theta} \psi (t_{i},s ) \bigl(u_{k_{i}}(s)-u_{*}(s) \bigr)\,ds\biggr\Vert \leq&\biggl\Vert \int_{t_{0}}^{\theta} \bigl[\psi ( t_{i},s )-\psi ( t_{*},s ) \bigr] \bigl( u_{k_{i}}(s)-u_{*}(s) \bigr)\,ds\biggr\Vert \\ &{}+\biggl\Vert \int_{t_{0}}^{\theta} \psi ( t_{*},s ) \bigl( u_{k_{i}}(s)-u_{*}(s) \bigr)\,ds \biggr\Vert \\ < & \frac{\varepsilon_{*}}{4}+ \int_{t_{0}}^{\theta} \bigl\Vert \psi ( t_{i},s )-\psi ( t_{*},s ) \bigr\Vert \bigl\Vert u_{k_{i}}(s)-u_{*}(s) \bigr\Vert \,ds \\ \leq& \frac{\varepsilon_{*}}{4}+ \int_{t_{0}}^{\theta} \frac{\varepsilon_{*} c_{1}}{8\sqrt{\theta-t_{0}}} \bigl[ \bigl\Vert u_{k_{i}}(s)\bigr\Vert +\bigl\Vert u_{*}(s) \bigr\Vert \bigr]\,ds \\ \leq& \frac{\varepsilon_{*}}{4}+\frac{\varepsilon_{*} c_{1}}{8\sqrt{\theta -t_{0}}} \cdot\frac{2\sqrt{\theta-t_{0}}}{c_{1}} = \frac{\varepsilon_{*}}{2} . \end{aligned}$$
(5.9)

Thus (5.6) and (5.9) are in contradiction, and hence the validity of (5.5) is proved.

Now, for a given \(\varepsilon>0\), let us choose an arbitrary \(k>K^{*}(\varepsilon)\). Then from (2.2), (3.2), (5.3), and (5.5) it follows that

$$\begin{aligned} \bigl\Vert x_{k}(t)-x_{*}(t) \bigr\Vert \leq&\frac{\lambda}{1-L_{0}} \varepsilon+\frac{\lambda}{1-L_{0}} \int_{t_{0}}^{\theta} \bigl( L_{1}+L_{2} \bigl\Vert u_{k}(s) \bigr\Vert \bigr) \bigl\Vert x_{k}(s) -x_{*}(s)\bigr\Vert \,ds \\ \leq& \frac{\lambda}{1-L_{0}} \varepsilon+\frac{\lambda}{1-L_{0}}\bigl\Vert x_{k}(\cdot)-x_{*}(\cdot) \bigr\Vert _{C} \int_{t_{0}}^{\theta} \bigl( L_{1}+L_{2} \bigl\Vert u_{k}(s) \bigr\Vert \bigr)\,ds \\ =& \frac{\lambda}{1-L_{0}} \varepsilon+\frac{\lambda}{1-L_{0}} \biggl( L_{1} ( \theta-t_{0} )+L_{2} \int_{t_{0}}^{\theta}\bigl\Vert u_{k}(s) \bigr\Vert \,ds \biggr)\bigl\Vert x_{k}(\cdot)-x_{*}(\cdot) \bigr\Vert _{C} \\ \leq& \varepsilon\frac{\lambda}{1-L_{0}} +\frac{\lambda}{1-L_{0}} \biggl( L_{1} ( \theta-t_{0} )+L_{2} \frac{\sqrt{\theta-t_{0}}}{c_{1}} \biggr) \bigl\Vert x_{k}(\cdot)-x_{*}(\cdot) \bigr\Vert _{C} \\ =& \varepsilon\frac{\lambda}{1-L_{0}} +\frac{L(\lambda)-L_{0} }{1-L_{0}} \bigl\Vert x_{k}(\cdot)-x_{*}(\cdot) \bigr\Vert _{C} \end{aligned}$$

for every \(t\in[t_{0},\theta]\), and consequently

$$ \bigl\Vert x_{k}(\cdot)-x_{*}(\cdot) \bigr\Vert _{C}\leq\frac{\lambda}{1-L_{0}} \varepsilon+\frac{L(\lambda)-L_{0}}{1-L_{0}} \bigl\Vert x_{k}(\cdot)-x_{*}(\cdot) \bigr\Vert _{C} $$
(5.10)

for any \(k>K_{*}(\varepsilon)\). Since

$$\begin{aligned} 1-\frac{L(\lambda)-L_{0} }{1-L_{0}} >0, \end{aligned}$$

(5.10) implies that

$$ \bigl\| x_{k}(\cdot)-x_{*}(\cdot)\bigr\| _{C} \leq\frac{\lambda}{1-L(\lambda)} \cdot \varepsilon $$

for any \(k>K^{*}(\varepsilon)\). This means that \(\Vert x_{k}(\cdot)- x_{*}(\cdot)\Vert _{C} \rightarrow0\) as \(k\rightarrow\infty\). The uniqueness of limit gives us that \(x_{0}(\cdot)=x_{*}(\cdot)\). Since \(x_{*}(\cdot)\in\mathbf{X}\), \(x_{0}(\cdot)\in\mathbf{X}\) and the proof is completed. □

Theorem 4.1 and Theorem 5.1 yield the compactness of the set of trajectories.

Theorem 5.1

The set of trajectories X of the system (3.1) is a compact subset of the space \(C ( [t_{0},\theta ];\mathbb {R}^{n} )\).

6 Conclusion

Compactness of the set of trajectories of the control system described by a Urysohn type integral equation is specified where the system is nonlinear with respect to the state vector and is affine with respect to the control vector. The admissible control functions are chosen from the space \(L_{2} ([t_{0},\theta]; \mathbb{R}^{m} )\) which satisfy an additional quadratic integral constraint. This means that the control resource of the system is limited and it is exhausted by consumption. It is proved that the set of trajectories is a compact subset of the space \(C ( [t_{0},\theta ];\mathbb{R}^{n} )\). This allows one to predict the existence of the optimal trajectory in the optimal control problem for the system described by a Urysohn type integral equation with quadratic integral constraint on the controls and with continuous payoff functional.