Abstract
The control systems described by the Urysohntype integral equations and integral constraints on the control functions are considered. The functions from the closed ball of the space
1 Introduction
In general, depending on the character of the control efforts, control systems can be classified as follows: control systems with geometric constraints on the control functions; control systems with integral constraints on the control functions; and control systems with mixed constraints on the control functions, which include both the geometric and integral constraints on the control functions. Geometric constraint characterizes the control efforts that are not exhausted by consumption, and these systems are a wellstudied chapter of the control systems theory (see, e.g., [1,2,3]). They are also investigated in the framework of differential and integral inclusion theory (see, e.g., [4,5, 6,7] and references therein). Integral constraint on the control functions appears in the case if the control resource is exhausted by consumption such as energy, fuel, finance, and food. For example, the motion of a flying object with rapidly changing mass is described by a control system with integral constraints on the control functions (see, e.g., [8,9,10]). Integrally boundedness of the function does not imply geometric boundedness, and therefore, additional difficulties are encountered in the investigation of these systems.
In theory and applications, the control systems described by various types of evolution equations are studied. One of the more interesting systems is the control system described by integral equations. The solution concepts for different types of initial and boundary value problems can be reduced to the appropriate solution notions for Volterra and UrysohnHammersteintype integral equations. It should be noted that integral models have some advantages over differential ones. For example, the trajectories for such systems can be defined as continuous, even as integrable functions. The integrable solution notion is an adequate way to describe the behavior of some physical processes (see, e.g., [11,12]). Therefore, the investigation of control systems described by the Urysohntype integral equations is important for the theory and applications of control systems theory. Note that the controllability properties and existence of the optimal trajectories of the control system described by the Urysohntype integral equation are investigated in [13,14,15, 16,17] (see also references therein). The articles [18,19] consider the approximate construction of the set of trajectories of the control systems described by Urysohntype integral equations and integral constraints on the control functions. In [13,14,19], it is assumed that the trajectory of the system is a continuous function and satisfies the system’s equation everywhere. In articles [15,16,17], the integrable functions are chosen as the system’s trajectory. In this article, the properties of the set of pintegrable trajectories of the control system described by the Urysohntype integral equation with the integral constraint on the control functions are studied where the system’s equation is nonlinear with respect to state and control vectors. The boundedness and pathconnectedness of the set of trajectories are established, and it is illustrated that, in general, the set of trajectories is not a closed subset of the space of pintegrable functions. It is proved that the consumption of the remaining control resource on a domain with sufficiently small measure causes a nonessential change in the trajectory of the system. It is shown that the set of trajectories generated by a full consumption of the control resource is dense in the set of trajectories generated by all admissible control functions.
The article is organized as follows. In Section 2, the basic conditions, which are used in the following arguments, are given. In Section 3, the existence and uniqueness of pintegrable trajectories generated by a given admissible control function (Proposition 1), the boundedness of the set of trajectories (Proposition 2), the dependence of the trajectories on the generating admissible control functions (Proposition 3), and the pathconnectedness of the set of trajectories (Theorem 1) are presented. In Section 4, it is shown that the set of trajectories, in general, is not a closed subset of the space
2 The system
Consider a control system described by the Urysohntype integral equation as follows:
where
For given
where
Let
A. The function
is verified for every
B. The function
and for a.a.
is held for each
C. The inequality
Let
3 Basic properties of the set of trajectories
The following propositions characterize the basic properties of the set of trajectories. Denote
Proposition 1
Every admissible control function
Proof
For given
First, let us show that
Let
for a.a.
for a.a.
From the last inequality, we conclude that
and consequently,
for a.a.
According to the condition C, we have
for a.a.
The following proposition characterizes boundedness of the set of trajectories in the space
where
Proposition 2
For every
Proof
Choose an arbitrary
for a.a.
for a.a.
where
Let us set
In the following, we give an evaluation between the trajectories of system (1) generated by different admissible control functions.
Proposition 3
Let
is held where
Proof
From conditions A and B and Hölder’s inequality, we obtain that
for a.a.
where
From Proposition 3, it follows that every solution depends on its generating control function Lipschitz continuously.
Now, let us give a definition of pathconnectedness.
Definition 1
Let
Let
we conclude that the set
Theorem 1
The set of trajectories
4 Closedness of the set of trajectories
In this section, it is shown that the set of trajectories
Proposition 4
Let
Example 1
Consider the control system described by the following system of integral equations:
where
The set of trajectories of system (9) generated by all admissible control functions
First of all, let us show that
for a.a.
for a.a.
Inequality (14) means that the set of trajectories
Choose a sequence of uniform partitions
where
for a.a.
for a.a.
for a.a.
for every
for a.a.
for a.a.
Note that along with compactness, the closedness of the set of trajectories
Consider the following optimal control problem. Let the dynamics of the control system be described by system of integral equations (9), where
on the set of trajectories
for every
Let
Since
5 Robustness of the trajectories
In this section, in addition to the conditions A–C it will be assumed that the function
Theorem 2
Let
be such that
then
Proof
From conditions A and B, the inclusion of
where
where
By virtue of the condition C, (25), and (26), we obtain that
From Theorem 2, it follows that consuming a big quantity of the control resource on the domain with a sufficiently small measure is not an effective way to change the system’s trajectory. In addition, Theorem 2 implies that if we have a superfluous control resource and we want to get rid of it, then consuming the all remaining control resource on the domain with sufficiently small measure, we will obtain a minor deviation of the system’s trajectory.
Denote
and let
Theorem 3
The equality
Proof
Let us choose an arbitrary
and
where
where
Taking into consideration that
Since
Theorem 3 implies the validity of the following corollary.
Corollary 1
The equality
6 Conclusion
This article investigates the boundedness, pathconnectedness, and closedness properties of the set of pintegrable trajectories of the control system described by the Urysohntype integral equation. The control functions have an integral constraint, i.e., the control resource is exhausted by consumption, which often arises in different problems of theory and applications. The specified pathconnectedness property does not permit splitting of the set of trajectories, and not closedness of the set of trajectories in the considered example allows the existence of the sliding modes for approximate optimal trajectories. The robustness of the trajectory with respect to the fast consumption of the remaining control resource is shown. This means that in domains with sufficiently small measures, it is efficient to consume energy like control resources in small portions. If there is an excess control resource and it is required to get rid of it, then by consuming all the remaining control resources in the domain with a sufficiently small measure, it is possible to obtain a minor variation of the system’s trajectory.

Conflict of interest: The author states no conflict of interest.
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