Abstract and Applied Analysis

VolumeΒ 2008, Article IDΒ 295817, 14 pages

http://dx.doi.org/10.1155/2008/295817

## On the Continuity Properties of the Attainable Sets of Nonlinear Control Systems with Integral Constraint on Controls

Department of Mathematics, Anadolu University, Eskisehir 26470, , Turkey

Received 11 June 2007; Revised 30 August 2007; Accepted 6 November 2007

Academic Editor: AgacikΒ Zafer

Copyright Β© 2008 Khalik G. Guseinov and Ali S. Nazlipinar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The attainable sets of the nonlinear control systems with integral constraint on the control functions are considered. It is assumed that the behavior of control system is described by differential equation which is nonlinear with respect to phase-state vector and control vector. The admissible control functions are chosen from the closed ball centered at the origin with radius in . Precompactness of the solutions set is specified, and dependence of the attainable sets on the initial conditions and on the parameters of the control system is studied.

#### 1. Introduction

Control problems with integral constraints on control arise in various problems of mathematical modeling. For example, the motion of flying apparatus with variable mass is described in the form of controllable system, where the control function has integral constraints (see, e.g., [1β3]). One of the important constructions of the control systems theory is the attainable set notion. Attainable set is the set of all points to which the system can be steered at the instant of given time. Attainable sets of control systems are very useful tools in the study of various problems of optimization, dynamical systems and differential game theory.

In [4β10], topological properties and numerical construction methods of the attainable sets of linear control systems with integral constraint on control functions are investigated. The attainable sets of affine control systems, that is, the attainable sets of control systems which are nonlinear with respect to the phase-state vector, but are linear with respect to the control vector have been considered in [11β14]. The properties of the attainable sets of the nonlinear control systems have been studied in [15β18].

Approximation method for the construction of attainable sets of affine control systems with integral constraints on the control is given in [11, 13]. In [14], using the topological properties of attainable sets of affine control systems, the continuity properties of minimum time and minimum energy functions are discussed.

The dependence of the attainable set on is studied in [8, 12, 15]. In [15], it is proved that attainable set of affine control system depends on continuously. In [15], the same property is shown for nonlinear control systems.

In [17], if the control resource is sufficiently small, then under some suitable assumptions on the right-hand side of the system, it is proved that the attainable set of the nonlinear control system with integral constraints on control is convex.

The value function of nonlinear optimal control problem with generalized integral constraints on control and phase-state vectors is investigated in [16, 18].

In this article, we consider the attainable sets of the control systems the behavior of which is described by nonlinear differential equations. It is assumed that the admissible control functions are chosen from the closed ball centered at the origin with radius in

In Section 2, it is illustrated that, in general, the attainable set is not closed (Example 2.5) and it is shown that the set of solutions generated by all possible admissible control functions is precompact in the space of continuous functions (Corollary 2.4). In Section 3, the diameter of the attainable set is evaluated (Proposition 3.1) and it is proved that the attainable set is HΓΆlder continuous with respect to time variable (Proposition 3.3). In Section 4, it is shown that the attainable set of the control system is continuous with respect to initial condition (Proposition 4.1). In Section 5, it is proved that the attainable set is Lipschitz continuous with respect to a parameter of the system which define the resource of the control effort (Proposition 5.1).

Consider the control system the behavior of which is described by the differential equation where is the phase-state vector of the system, is the control vector, is the time, and is a compact set.

For and , we set where and denotes the Euclidian norm.

A function is said to be an admissible control function. It is obvious that the set of all admissible control functions is the closed ball centered at the origin with the radius in

It is assumed that the right-hand side of the system (1.1) satisfies the following conditions.

(a)The function is continuous.(b)For any bounded set , there exist constants , , and such that for any , , , and .(c) There exists a constant such that for every .

If the right-hand side of the system (1.1) is affine, that is, if and the functions satisfy the assumptions given in [11β14], then, under these assumptions, the conditions (a), (b), and (c) are also fulfilled.

Let . The absolutely continuous function , which satisfies the equation a.e. in , and the initial condition is said to be a solution of the system (1.1) with initial condition generated by the admissible control function By the symbol we denote the solution of the system (1.1) with initial condition which is generated by the admissible control function Note that the conditions (a)β(c) guarantee the existence, uniqueness, and extendability of the solutions up to the instant of time for every given and

Let us define

the sets where .

The set is called the attainable set of the system (1.1) at the instant of time . It is obvious that the set consists of all to which the system (1.1) can be steered at the instant of time

The Hausdorff distance between the sets and is denoted by and is defined as where .

By , we denote the space of continuous functions with norm

Also, denotes the Hausdorff distance between the sets and

#### 2. Precompactness of the Set of Solutions

The following proposition asserts that the set of solutions and the attainable sets of the control system (1.1) with constraint (1.2) are bounded.

Proposition 2.1. *Let , , . Then for any , the inequality
**
holds, where
**and is the constant
given in condition (c).*

The proof of the proposition follows from condition (c) and Gronwall's inequality.

For given , we set

We get from Proposition 2.1 that for every , and compact such that So, we have the validity of the following corollary.

Corollary 2.2. *The
set
is uniformly
bounded, and consequently for every , where is defined by
(2.2). *

Here and henceforth, we will have in mind the cylinder as the set in condition (b). We set also where is defined by (2.2), is defined by (2.4).

Proposition 2.3. *The
set
is
equicontinuous. *

*Proof. *Let be an
arbitrarily given number. Now, let us choose an arbitrary and Without loss of
generality, we assume that . Then from condition (c), we have

According to Proposition 2.1, , where is defined by
(2.2). Then we get from (2.4), (2.6), (2.7), and HΓΆlder's inequality that

Thus for given , setting , we obtain for Since is arbitrarily
chosen, the equicontinuity of the set follows.

From Corollary 2.2 and Proposition 2.3, we get the validity of the following corollary.

Corollary 2.4. * The set
is a precompact
subset of the space . *

Note that if the right-hand side of the system (1.1) is affine with respect to the control vector then the weak compactness of the set of admissible control functions guaranties the closeness of the attainable sets; but the attainable sets of the control system (1.1) with constraint (1.2), in general, are not closed. In [19, 20], the example is given which illustrates that the attainable set of nonlinear control system with geometric constraint on control is not closed. We use that example to show that the attainable set of nonlinear control system with integral constraint on control is not also closed.

*Example 2.5. *Let
us consider the control system
where is the
phase-state vector of the system, is the control
vector, It is assumed
that and the control
function of the system
(2.9) satisfies the integral constraint
that is, Let us denote

Thus is the set of
solutions, is the
attainable set of the control system (2.9) at the instant of time generated by
control functions

Now, let us prove that the solution set is bounded. Let be an
arbitrarily chosen solution of the system (2.9) with integral constraint (2.10).
Then there exists such that
for any . From (2.10), (2.13), and HΓΆlder's inequality, the
inequality
holds for all Then we get
from (2.10), (2.12), and (2.14) that
for all .

However, (2.14) and (2.15) imply that
Since is arbitrarily
chosen, we get that the set is bounded.

Now we prove that . Let us assume
the contrary, that is, let . Then there
exists such that

Since , then there exists such that
for all

From (2.17), (2.18), and (2.19), it follows that

Since , then it follows from (2.20) that for almost all Then we have
from (2.18) and (2.19) that for every which
contradicts (2.17). Thus

Let us show that .

Let be a uniform
partition of the closed interval , where . Now we define a sequence of functions setting
where .

It is obvious that for all . Let be the solution
of the system (2.9) generated by the admissible control function . Then it follows from (2.9) that
for every

We get from (2.22) and (2.24) that
for every where .

Then, from (2.25) we have
for every , and consequently
for every

According to (2.22), for all Then, from
(2.23) and (2.27) we obtain that
for almost all , and consequently
for every , where .

We conclude from the last inequality that
for every .

It follows from (2.26) and (2.30) that

Since for every , from (2.31) we obtain that

However, (2.21) and (2.32) imply that is not a closed
set.

#### 3. Diameter of the Attainable Set and Continuity with Respect to

In this section we will give an upper estimation for the diameter of the attainable set and will show that the set-valued map is HΓΆlder continuous with respect to .

We denote the diameter of a set by and define it as

The following proposition characterizes the diameter of the attainable set

Proposition 3.1. *For
every , the inequality
**
holds for any , where
*

*Proof. *Let and be arbitrarily
chosen. Then there exist , , , such that

Since , then from (3.6), and the condition (b), we
get

Since , then the
HΓΆlder and Minkowski inequalities imply that
where is defined by
(3.5). Since is arbitrarily
chosen, we obtain from (3.7), (3.8), and Gronwall's inequality that
where is defined by
(3.4).

Note that an estimation for diameter of the attainable set can be obtained from Propo-sition 2.1; but the estimation given by Proposition 3.1 is more precise.

Corollary 3.2. * as *

The following proposition asserts that the attainable set is HΓΆlder continuous with respect to

Proposition 3.3. *Let Then
**
where is defined by
(2.6).*

*Proof. *Without loss of generality, let us
assume that Let be arbitrarily
chosen. Then there exist , and such that
Let

It is obvious that From
Proposition 2.1, relations (2.4), (2.6), (3.11), (3.12), and the condition (c),
we have
where is the constant
given in condition (c).

Since is arbitrarily
chosen, then (3.13) implies that

Analogously, it is possible to show that

In fact, (3.14) and (3.15)
yield the proof.

From Proposition 3.3, we obtain the following corollary.

Corollary 3.4. *The
set-valued map , is -HΓΆlder
continuous. *

#### 4. Dependence of the Attainable Sets on Parameters and

The following proposition characterizes the continuity of the set-valued map in the Hausdorff metric.

Let us denote where is defined by (2.4), and are the constants given in condition (b).

Proposition 4.1. *Let and
be compact
sets. Then the inequality
**
holds for all , where is defined by
(2.6), is defined by
(4.1).*

*Proof. *Let us choose arbitrary and , where . Then there
exist and such that
holds.
According to the definition of Hausdorff distance, there exists such that . Let be a solution
of the control system (1.1), generated by the admissible control function with initial
condition Then
and

From (4.3), (4.4), and conditions (b) and (c), we have

Proposition 2.1 implies that
where is defined by
(2.6). Since is arbitrarily
chosen, from (4.5), (4.6), and Gronwall's inequality, we get

Hence, we obtain from (4.7) that

Similarly, one can prove that

Finally, (4.8) and (4.9) complete the proof.

From Proposition 4.1, the validity of the following corollaries follow.

Corollary 4.2. *The
inequality
**
holds for all , where is defined by
(4.1).*

Corollary 4.3. *The
inequality
**
holds.*

Corollary 4.4. *Let and
be compact sets
for all . Assume that and as . Then for all *

#### 5. Dependence of the Attainable Sets on

In this section we specify dependence of the set on the constraint parameter Let where is defined by (4.1).

The following proposition characterizes the relation between the solutions sets and .

Proposition 5.1. *The
inequality
**
is satisfied, where is defined by
(5.1).*

*Proof. *Let be an
arbitrarily chosen solution. Then there exist and such that
for every

We define a new control function , setting

It is not difficult to verify that . Let be a solution
of the control system (1.1), generated by from the
initial point . Then and
for every . From (5.3), (5.4), (5.5), and condition (b), we get
for every where is defined by
(2.4).

The Gronwall inequality, (5.1), and (5.6)
yield that
for all .

Thus from (5.7) we get that for any fixed there exists such that
and consequently
where is the closed
unit ball centered at the origin in the space .

Analogously, it is possible to prove that
Hence, from
(5.9) and (5.10), we obtain the proof of the proposition.

From Proposition 5.1, it follows that the following corollaries are satisfied.

Corollary 5.2. *The
inequality
**
is satisfied for any where is defined by
(5.1).*

Corollary 5.3. *Let as . Then
**
for every *

#### Acknowledgments

This research was supported by the Scientific and Technological Research Council of Turkey (TUBITAK) Project no. 106T012. The authors thank the referees for the careful reading of the manuscript and the helpful suggestions.

#### References

- V. V. Beletskii,
*Studies of Motions of Celestial Bodies*, Nauka, Moscow, Russia, 1972. - N. N. Krasovskii,
*Theory of Control of Motion: Linear Systems*, Nauka, Moscow, Russia, 1968. View at Zentralblatt MATH Β· View at MathSciNet - V. I. Ukhobotov,
*One Dimensional Projection Method in Linear Differential Games with Integral Constraints*, Chelyabinsk State University Press, Chelyabinsk, Russia, 2005. - A. G. Chentsov, βAsymptotic attainability with perturbation of integral constraints in an abstract control problemβI,β
*Russian Mathematics*, vol. 39, no. 2, pp. 57β68, 1995. View at Google Scholar Β· View at MathSciNet - A. G. Chentsov,
*Asymptotic Attainability*, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997. View at Zentralblatt MATH - R. Conti,
*Problemi di Controllo e di Controllo Ottimale*, UTET, Torino, Italy, 1974. - F. Gozzi and P. Loreti, βRegularity of the minimum time function and minimum energy problems: the linear case,β
*SIAM Journal on Control and Optimization*, vol. 37, no. 4, pp. 1195β1221, 1999. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - H. W. Lou, βOn the attainable sets of control systems with $p$-integrable controls,β
*Journal of Optimization Theory and Applications*, vol. 123, no. 1, pp. 123β147, 2004. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - M. Motta and C. Sartori, βMinimum time and minimum energy function for linear systems with controls in ${L}^{p}$ for $p\beta \x89\u20af1$,β preprint, 200.
- A. N. Sirotin and A. M. Formalskii, βReachability and controllability of discrete-time systems under control actions bounded in magnitude and norm,β
*Automation and Remote Control*, vol. 64, no. 12, pp. 1844β1857, 2003. View at Publisher Β· View at Google Scholar - Kh. G. GuseΔnov, A. A. Neznakhin, and V. N. Ushakov, βApproximate construction of reachable sets of control systems with integral constraints on the controls,β
*Journal of Applied Mathematics and Mechanics*, vol. 63, no. 4, pp. 557β567, 1999. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - Kh. G. Guseinov, O. Ozer, and E. Akyar, βOn the continuity properties of the attainable sets of control systems with integral constraints on control,β
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 56, no. 3, pp. 433β449, 2004. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - Kh. G. Guseinov, O. Ozer, E. Akyar, and V. N. Ushakov, βThe approximation of reachable sets of control systems with integral constraint on controls,β
*Nonlinear Differential Equations and Applications*, vol. 14, no. 1-2, pp. 57β73, 2007. View at Publisher Β· View at Google Scholar Β· View at MathSciNet - M. Motta and C. Sartori, βMinimum time with bounded energy, minimum energy with bounded time,β
*SIAM Journal on Control and Optimization*, vol. 42, no. 3, pp. 789β809, 2003. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - Kh. G. Guseinov and A. S. Nazlipinar, βOn the continuity property of ${L}_{p}$ balls and an application,β
*Journal of Mathematical Analysis and Applications*, vol. 335, no. 2, pp. 1347β1359, 2007. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - M. Motta and F. Rampazzo, βMultivalued dynamics on a closed domain with absorbing boundary. Applications to optimal control problems with integral constraints,β
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 41, no. 5-6, pp. 631β647, 2000. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - B. T. Polyak, βConvexity of the reachable set of nonlinear systems under ${L}_{2}$ bounded controls,β
*Tech. Rep. 02.2002/2003*, Institut Mittag-Leffler, Djursholm, Sweden, 2003. View at Google Scholar - P. Soravia, βViscosity solutions and optimal control problems with integral constraints,β
*Systems & Control Letters*, vol. 40, no. 5, pp. 325β335, 2000. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - A. F. Filippov, βOn some questions in the theory of optimal regulation: existence of a solution of the problem of optimal regulation in the class of bounded measurable functions,β
*Vestnik Moskovskogo Universiteta, Seriya Matematika, Mekhanika*, vol. 1959, no. 2, pp. 25β32, 1959. View at Google Scholar Β· View at MathSciNet - A. F. Filippov,
*Differential Equations with Discontinuous Righthand Sides*, vol. 18 of*Mathematics and Its Applications*, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1988. View at Zentralblatt MATH Β· View at MathSciNet