D O I: 1 0 .1 5 0 1 / C o m m u a 1 _ 0 0 0 0 0 0 0 7 7 6 IS S N 1 3 0 3 –5 9 9 1
ON THE EXISTENCE OF "-OPTIMAL TRAJECTORIES OF THE
CONTROL SYSTEMS WITH CONSTRAINED CONTROL RESOURCES
ANAR HUSEYIN
Abstract. The control system described by a Urysohn type integral equation is considered. It is assumed that the admissible control functions are chosen from the closed ball of the space Lp; p > 1; with radius r and centered at
the origin. Precompactness of the set of trajectories of the control system in the space of continuous functions is shown. This allows to prove that optimal control problem with lower semicontinuous payo¤ functional has an "-optimal trajectory for every " > 0.
1. Introduction
Integral equations arise in many problems of contemporary physics and mechan-ics (see, e.g. [1], [3], [4], [11], [13], [15], [18] and references therein). Pointing out the importance of the integral equations, W. Heisenberg in his well known "Physics and Philosophy" writes: "The …nal equation of motion for matter will probably be some quantized nonlinear wave equation... This wave equation will probably be equivalent to rather complicated sets of integral equations..." (see, [6], page 68). Often the processes which are described by the integral equations have exterior in‡uences called control e¤orts or uncertainties of the systems, depending on the characters of these in‡uences. In this paper it will be assumed that exterior in‡u-ences are control e¤orts and control functions characterizing the control e¤orts have an integral constraint. Integral constraint on the control functions is inevitable if the control resource is exhausted by consumption, such as energy, fuel, food and …nance (see, e.g. [5], [14], [16], [17]).
In papers [8], [9] various topological properties of the sets of trajectories of the control systems described by the nonlinear Volterra type integral equations with integral constraint on the control functions are studied. In [7] the approximation of
Received by the editors: Feb 04, 2016, Accepted: June 26, 2016. 2010 Mathematics Subject Classi…cation. 45G15, 93B03, 93C23.
Key words and phrases. Urysohn integral equation, control system, integral constraint, set of trajectories, "-optimal trajectory.
c 2 0 1 7 A n ka ra U n ive rsity C o m m u n ic a tio n s d e la Fa c u lté d e s S c ie n c e s d e l’U n ive rs ité d ’A n ka ra . S é rie s A 1 . M a th e m a t ic s a n d S t a tis tic s .
the sets of trajectories of the aforementioned systems is discussed. A similar prob-lem for the systems described by the ordinary di¤erential equations is considered in [5]. Existence of optimal controls and controllability of the systems described by the Urysohn type integral equations are discussed in [2], [12] where it is assumed that control functions have a geometric constraint.
In the presented paper existence of "-optimal trajectories of the control systems described by the Urysohn type integral equations is investigated. The closed ball of the space Lp; p > 1; with radius r and centered at the origin is chosen as the
set of admissible control functions which means that admissible control functions have an integral constraint. Precompactness of the set of trajectories generated by all admissible control functions is established. Using this result it is proved that optimal minimization control problem with lower semicontinuous payo¤ functional has an "-optimal trajectory for every " > 0:
The paper is organized as follows: In Section 2 the conditions are formulated which satisfy the system equation (Conditions A, B and C). In Section 3 it is shown that under accepted conditions, every admissible control function generates unique trajectory of the system (Theorem 3.1). In Section 4 it is proved that the set of trajectories of the system is bounded (Theorem 4.1). In Section 5 it is shown that the sections of the set of trajectories is continuous with respect to the Hausdor¤ metric (Proposition 5.2) and the set of trajectories is a precompact set in the space of continuous functions (Theorem 5.1). Existence of "-optimal trajectories for optimal minimization control problem is proved (Theorem 5.2).
2. Preliminaries
The control system described by a Urysohn type integral equation x( ) = f ( ; x( )) +
Z
K( ; s; x(s); u(s))ds (2.1)
is considered, where x 2 Rnis the state vector of the system, u 2 Rmis the control
vector, 2 ; Rk is a compact set, > 0 is a real number.
For given p > 1 and r > 0 we set Up;r =
n
u( ) 2 Lp( ; Rm) : ku( )kp r
o
; (2.2)
where Lp( ; Rm) is the space of Lebesgue measurable functions u( ) : ! Rm
such that ku( )kp< +1; ku( )kp=
Z
ku(s)kpds
1 p
; k k denotes the Euclidean norm.
The set Up;r Lp( ; Rm) is called the set of admissible control functions and
every function u( ) 2 Up;r is called admissible control function.
We assume that the functions f ( ) : Rn! Rn; K( ) : Rn Rm! Rn
A.the functions f ( ) : Rn ! Rn and K( ) : Rn Rm ! Rn are
continuous;
B. there exist M0 2 [0; 1); M1 0, H1 0, M2 0; H2 0, M3 0 and
H3 0 such that kf( ; x1) f ( ; x2)k M0kx1 x2k ; K( 1; s; x1; u1) K( 2; s; x2; u2) [M1+ H1(ku1k + ku2k)] k 1 2k + [M2+ H2(ku1k + ku2k)] kx1 x2k + [M3+ H3(kx1k + kx2k)] ku1 u2k for every ( 1; s; x1; u1) 2 Rn Rm; ( 2; s; x2; u2) 2 Rn Rm; C.the inequality 0 M2 ( ) + 2H ( ) p 1 p r < 1 M 0 is satis…ed, where
( ) is the Lebesgue measure of the set ;
H = max fH1; H2; H3g : (2.3)
If the function K( ) : Rn Rm ! Rn is Lipschitz continuous, then it
satis…es the conditions A and B.
Now let us de…ne the trajectory of the system (2.1) generated by a given admis-sible control function.
Let u( ) 2 Up;r: A continuous function x( ) : ! Rn satisfying the equation
(2.1) for every 2 is said to be a trajectory of the system (2.1) generated by the admissible control function u( ) 2 Up;r:
We denote by Xp;r the set of all trajectories of the system (2.1) generated by all
admissible control functions u( ) 2 Up;r: The set Xp;ris called the set of trajectories
of the system (2.1).
For each …xed 2 we set
Xp;r( ) = fx( ) 2 Rn: x( ) 2 Xp;rg : (2.4)
Now let us give a proposition which will be used in following arguments. Proposition 2.1. Let Rk be a compact set, ( ) : ! R and r( ) : ! R be continuous functions, ( ) : ! [0; +1) be a Lebesgue integrable function, Z
(s)ds < 1 and
( ) r( ) + Z
(s) (s)ds for every 2 . Then the inequality
( ) r( ) + Z r(s) (s)ds 1 Z (s)ds (2.5)
holds for every 2 .
Moreover, if r( ) = r0 for every 2 , then it follows from (2.5) that
( ) r0
1 Z
(s)ds for every 2 .
The proof of the Proposition 2.1 is similar to the proof of the Proposition 1 from [10].
3. Existence and Uniqueness of the Trajectories Denote M ( ) = M0+ h M2 ( ) + 2H ( ) p 1 p r i : (3.1)
The following theorem shows that every admissible control function generates the unique trajectory of the system (2.1).
Theorem 3.1. Let the functions f ( ) : Rn ! Rn; K( ) : Rn Rm! Rn
and the number 2 (0; 1) satisfy the conditions A - C. Then each u ( ) 2 Up;r
generates the unique trajectory x ( ) of the system (2.1). Proof. De…ne a map x( ) ! F (x( )), x( ) 2 C( ; Rn) setting
F (x( ))j( ) = f( ; x( )) + Z
K( ; s; x(s); u (s))ds; 2 ; (3.2) where C( ; Rn) is the space of continuous functions x( ) : ! Rn with norm
kx( )kC= maxfkx( )k : 2 g.
Let us show that F (x( )) 2 C( ; Rn). Choose arbitrary 2 and " > 0: Since
x( ) 2 C( ; Rn), then from condition A it follows that there exists 1= 1("; x( )) >
0 such that for every 2 Bk( ; 1) \ the inequality
kf( ; x( )) f ( ; x( ))k " 2 (3.3) is veri…ed where Bk( ; 1) = 2 Rk : k k 1 . Denote 2= " 2 hM1 ( ) + 2H1 ( ) p 1 p r i :
Condition B and Hölder’s inequality imply that for every 2 Bk( ; 2) \ the inequality Z K( ; s; x(s); u (s))ds Z K( ; s; x(s); u (s))ds Z [M1+ 2H1ku (s)k] k k ds h M1 ( ) + 2H1 ( ) p 1 p r i 2 " 2 (3.4)
is satis…ed. Let = min f 1; 2g : (3.3) and (??) yield that for every 2 Bk( ; )\
the inequality
kF (x( ))j( ) F (x( ))j( )k "
holds. This means that the function ! F (x( ))j( ); 2 , is continuous at : Since 2 is arbitrarily chosen, we obtain that F (x( )) 2 C( ; Rn).
Let x1( ) 2 C( ; Rn) and x2( ) 2 C( ; Rn) be arbitrarily chosen functions. From
condition B, (2.3), (3.1), (3.2) and Hölder’s inequality it follows that F (x2( ))j( ) F (x1( ))j( ) M0kx2( ) x1( )k + Z [M2+ 2H2ku (s)k] kx2(s) x1(s)k ds h M0+ M2 ( ) + 2 H r ( ) p 1 p i kx2( ) x1( )kC = M ( ) kx2( ) x1( )kC
for every 2 E, and consequently
kF (x2( ))j( ) F (x1( ))j( )kC M ( ) kx2( ) x1( )kC: (3.5)
According to the condition C we have M ( ) < 1. (3.5) implies that the map F ( ) : C( ; Rn) ! C( ; Rn) de…ned by (3.2) is contractive, and hence it has a unique …xed point x ( ) 2 C( ; Rn) which is unique continuous function satisfying the equation
x ( ) = f ( ; x ( )) + Z
K( ; s; x (s); u (s))ds; 2 :
4. Boundedness of the Set of Trajectories
In this section we will show that Conditions A - C guarantee boundedness of the set of trajectories Xp;r. We set
0= maxfkf( ; 0)k : 2 g;
1= maxfkK( ; s; 0; 0)k : 2 ; s 2 g
Proposition 4.1. Let the functions f ( ) : Rn! Rn and K( ) : Rn
Rm! Rn satisfy the conditions A and B. Then
kf( ; x)k 0+ M0kxk
kK( ; s; x; u)k 1+ M3kuk + [M2+ 2H kuk] kxk
for every ( ; s; x) 2 Rn, where the constants M
0, M2 and M3 are given in
condition B, H is de…ned by (2.3). Denote = 0+ 1 (E) + M3 (E) p 1 p r 1 M ( ) ; (4.1) where M ( ) is de…ned by (3.1).
Theorem 4.1. Let the conditions A - C be satis…ed. Then for every x( ) 2 Xp;r
the inequality
kx( )kC
holds.
Proof. Let x( ) 2 Xp;r be an arbitrary trajectory, generated by the admissible
control function u( ) 2 Up;r. From Proposition 4.1, Hölder’s inequality and (2.2)
we obtain kx( )k 0+ M0kx( )k + Z [ 1+ M3ku(s)k + (M2+ 2H ku(s)k) kx(s)k] ds 0+ M0kx( )k + 1 ( ) + M3 ( ) p 1 p r + Z (M2+ 2H ku(s)k) kx(s)k ds
for every 2 . Since M02 [0; 1), then we have from the last inequality
kx( )k 0+ 1 ( ) + M3 ( ) p 1 p r 1 M0 + 1 M0 Z [M2+ 2H ku(s)k] kx(s)k ds (4.2)
for every 2 . Since u( ) 2 Up;r, then (3.1), (4.1), (4.2), Condition C and Proposition 2.1 yield kx( )k 0+ 1 ( ) + M3 ( ) p 1 p r 1 M0 1 1 1 M0 Z [M2+ 2H ku(s)k] ds 0+ 1 ( ) + M3 ( ) p 1 p r 1 M0 1 1 1 M0 h M2 ( ) + 2H ( ) p 1 p r i = for every 2 , and hence kx( )kC .
From Theorem 4.1 it follows the validity of the following corollary.
Corollary 4.1. The inclusion Xp;r( ) Bn( ) holds for every 2 ; where the
set Xp( ) is de…ned by (2.4), the number > 0 is de…ned by (4.1), Bn( ) = fx 2
Rn: kxk g.
5. Precompactness of the Set of Trajectories and Existence of "-Optimal Trajectories
In this section precompactness of the set of trajectories and existence of "-optimal trajectories are studied. Denote
D1= Bn( ); !0( ) = max kf( 2; x) f ( 1; x)k : k 2 1k ; ( 1; x) 2 D1; ( 2; x) 2 D1 ; (5.1) ' ( ) = 1 1 M0 n !0( ) + h M1 ( ) + 2H1 ( ) p 1 p r i o : (5.2)
By virtue of condition A, we have !0( ) ! 0; '( ) ! 0 as ! 0+.
The Hausdor¤ distance between the sets U Rn and V Rn is denoted by
h(U; V ) and de…ned as
h(U; V ) = maxfsup
u2U
d(u; V ); sup
v2Vd(v; U )g;
where d(u; V ) = inf fku vk : v 2 V g :
Proposition 5.1. Let the conditions A - C be satis…ed. Then for every x( ) 2 Xp;r; 12 ; 22 the inequality
holds and hence
h (Xp;r( 2); Xp;r( 1)) ' (k 2 1k)
where Xp;r( 1) and Xp;r( 2) are de…ned by (2.4).
Proof. Let x( ) 2 Xp;rbe an arbitrarily chosen trajectory of the system (2.1). Then
there exists u( ) 2 Up;r such that
x( ) = f ( ; x( )) + Z
K ( ; s; x(s); u(s)) ds; 2 :
Now let 1 2 and 2 2 : Since x( ) 2 Xp;r; u( ) 2 Up;r; then from (5.1),
Condition B, Theorem 4.1 and Hölders inequality we have
kx( 2) x( 1)k kf( 2; x( 2)) f ( 1; x( 2))k + kf( 1; x( 2)) f ( 1; x( 1))k + Z kK ( 2; s; x(s); u(s)) K ( 1; s; x(s); u(s))k ds !0(k 2 1k) + M0kx( 2) x( 1)k + Z [M1+ 2H1ku(s)k] k 2 1k ds !0(k 2 1k) + M0kx( 2) x( 1)k + h M1 ( ) + 2H1 ( ) p 1 p r i k 2 1k :
Since M02 [0; 1), then the last inequality and (5.1) complete the proof.
Since '( ) ! 0 as ! 0+, then Proposition 5.1 yields the validity of the
following propositions.
Proposition 5.2. Let the conditions A - C be satis…ed. Then the set valued map ! Xp;r( ); 2 ; is continuous, where Xp;r( ) is de…ned by (2.4).
Proposition 5.3. Let the conditions A - C be satis…ed. Then the set of trajectories Xp;r is a family of equcontinuous functions.
Now, from Theorem 4.1 and Proposition 5.3 it follows precompactness of the set of trajectories.
Theorem 5.1. Let the conditions A - C be satis…ed. Then the set of trajectories Xp;r is a precompact subset of the space C ( ; Rm) :
Now, consider minimization of the lower semicontinuous functional (x( )) : C( ; Rn) ! R on the set of trajectories Xp;r: Denote
I = inf
x( )2Xp;r
(x( )):
Since Xp;r C( ; Rn) is nonempty and precompact set and (x( )) is a lower
semicontinuous functional, we have that jI j < +1.
Let " > 0 be a given number. A trajectory x"( ) 2 Xp;r satisfying the inequality
Theorem 5.2. Let the conditions A - C be satis…ed and (x( )) : C( ; Rn) ! R be
a lower semicontinuous functional. Then for every " > 0 there exists an "-optimal trajectory.
The proof of the theorem follows from precompactness of the set of trajectories Xp;r, i.e. from Theorem 5.1 and lower semicontinuity of the functional (x( )).
6. Conclusion
Nonlinear control systems arise in di¤erent problems of theory and applications. Integral constraint on control functions appears if the control resource is exhausted by consumption. The precompactness property of the set of trajectories is a useful tool to study the existence of approximately optimal trajectories in the optimal control problems with semicontinuous payo¤ functionals. Note that control system described by an integral equation with geometric constraints on the control func-tions can be studied in the framework of integral inclusions. For control systems with integral constraint on the controls, the situation is di¤erent. The matter is that integral boundedness of the function does not guarantee geometric bounded-ness. Note that extending the system dimension, it is possible to write the control system described by integral equation with integral constraint on the controls in the form of integral inclusion with unbounded right hand side and with phase state constraint. But in this case, the new system turns out more complex than the original one. Therefore studying the considered system in its original form is more preferable, than the reduced one and it is one of the actual problems of control systems theory.
References
[1] Appell, J.M., A.S. Kalitvin, A.S. and Zabrejko, P.P., Partial integral operators and integro-di¤erential equations, M. Dekker Inc., New York, 2000.
[2] Balder, E.J., On existence problems for the optimal control of certain nonlinear integral equations of Urysohn type, J. Optim. Theory Appl. 42 (1984), 447-465.
[3] Browder, F.E., Nonlinear functional analysis and nonlinear integral equations of Hammerstein and Urysohn type. Contributions to nonlinear functional analysis, Academic Press, New York, 1971, 425-500.
[4] Gohberg, I.G. and Krein, M.G., Theory and applications of Volterra operators in Hilbert space, Amer. Math. Soc., Providence, R. I., 1970.
[5] Guseinov, Kh.G., Approximation of the attainable sets of the nonlinear control systems with integral constraint on controls, Nonlinear Anal. TMA, 71 (2009), 622-645.
[6] Heisenberg, W., Physics and philosophy: The revolution in modern science, George Allen & Unwin, London, 1958.
[7] Huseyin, A., On the approximation of the set of trajectories of control system described by a Volterra integral equation, Nonlin. Anal. Model. Contr. 19 (2014), 199-208.
[8] Huseyin, A. and Huseyin, N., Precompactness of the set of trajectories of the controllable system described by a nonlinear Volterra integral equation, Math. Model. Anal. 17 (2012), 686-695.
[9] Huseyin, A. and Huseyin, N., Dependence on the parameters of the set of trajectories of the control system described by a nonlinear Volterra integral equation, Appl. Math. Praha, 59 (2014), 303-317.
[10] Guseiin, N., Guseiin, A. and Guseinov, Kh.G., Approximation of the set of trajectories of a control system described by the Urysohn integral equation. Tr. Inst. Mat. Mekh. 21 (2) (2015), 59-72.
[11] Infante, G. and Webb, J.R.L., Nonlinear non-local boundary-value problems and perturbed Hammerstein integral equations, Proc. Edinb. Math. Soc. 49 (2) (2006), 637-656.
[12] Joshi, M.C. and George, R.K. Controllability of nonlinear systems, Numer. Funct. Anal. Optim. 10 (1989), 139-166.
[13] Krasnoselskii, M.A. and Krein, S.G., On the principle of averaging in nonlinear mechanics, Uspekhi Mat. Nauk, 10 (1955), 147-153. (In Russian)
[14] Krasovskii, N.N., Theory of control of motion: Linear systems, Nauka, Moscow, 1968. (In Russian)
[15] Polyanin, A.D. and Manzhirov, A.V., Handbook of integral equation, CRC Press, Boca Ra-ton, FL, 1998.
[16] Subbotin, A.I. and Ushakov, V.N., Alternative for an encounter-evasion di¤erential game with integral constraints on the players controls, J. Appl. Math. Mech. 39 (1975), 367-375. [17] Ukhobotov, V.I., One dimensional projection method in linear di¤erential games with integral
constraints, Chelyabinsk State University press, Chelyabinsk, 2005. (In Russian)
[18] Urysohn, P.S., On a type of nonlinear integral equation, Mat. Sb. 31 (1924), 236-255. (In Russian)
Current address : Cumhuriyet University, Faculty of Science, Department of Statistics, 58140 Sivas, TURKEY
