Selçuk J. Appl. Math. Selçuk Journal of

Vol. 9. No. 2. pp. 83 93, 2008 Applied Mathematics

First Order Necessary Optimality Conditions For The Systems With Three-Point Boundary Conditions

Shamo Djabrailov, Yaqub Sharifov Baku State University, Azerbaijan e-mail:Sharifov22@ rambler.ru

Received: October 09, 2008

Summary. Optimal control problem with three point boundary conditions is considered in the paper. Using the increment formula for the …rst order functional the necessary conditions of optimality is obtained in the form of Pontryagin’s maximum principle. It is shown that for some particular cases the obtained necessary condition is su¢ cient at the same time.

Key words: Three-point boundary conditions, maximum principle, necessary conditions.

2000 Mathematical Subject Classi…cation: 49K15, 49K99, 34B15 1. Introduction

The role of the Pontryagin maximum principle is crucial for any research re-lated to optimal processes that have control constraints. The simplicity of the principle’s formulation together with its meaningful and bene…cial directness has become an extraordinary attraction and one of the major causes for the appearance of new tendencies in mathematical sciences. The maximum prin-ciple is by nature a necessary …rst-order condition for optimality since it was born as an extension of Euler-Lagrange and Weierstrass necessary conditions of variational calculus. That is why the application of the Pontryagin maximum principle provides information on extremal controls, called sometimes Pontrya-gin extremals, among which an optimal control is to be searched. On the other hand, there is always a fair chance that such extremals may cease to be optimal, and it should be emphasized that in this note, we will discuss only necessary conditions for optimal control. Moreover, they will be derived under an essen-tial qualitative assumption that optimal control does exist, that is, we will not discuss the problem of existence itself.

Proved initially for time-optimal linear systems, the maximum principle later on was generalized and extended for many other classes of control problems. The circle of problems solvable by means of the maximum principle is broad-ening every year. In this paper, we demonstrate the maximum principle for

nonlinear control problem with three-point boundary conditions (Section 4). A distinctive feature of this particular problem is related to the system of di¤er-ential equations that governs the dynamic process. Namely, it is supposed that one group of the state variables is speci…ed in the initial time while the other groups are speci…ed in the intermediate and terminal time and these groups take share and share alike in the boundary condition assigned to the dynamic system. This means that there are no phase or terminal constraints of any type. In other words, the boundary conditions are linked to the ODE system (instead of traditional initial conditions) and mostly serve for determining its particular solution. On the other hand, a mere replacement of initial conditions by the boundary ones is not acceptable. Therefore, our starting point will be to ex-tract …rst-order dominant term in increment of objective functional and then to formulate the necessary condition for optimality in the traditional form of the maximum principle.

2. Problem statement

The objects under investigation are the optimal control problems for the system
of the …rst order ordinary di¤erential equations with boundary conditions
(1) x = f (x; u; t) ; _{x (t) 2 R}n; _{t 2 [t}0; t1] = T;

(2) A x (t0) + B x ( ) + C x (t1) = D:

Here f (x; u; t) is the given n- dimensional vector function, continuous over the
set of variables together with its partial derivatives with respect to x, A; B; C 2
Rn n_{;} _{D 2 R}n 1_{are constant matrices, u (t) is r- dimensional measurable and}

bounded vector of control in‡uence in the interval T = [t0; t1], 2 (t0; t1) is a

…xed point.

It is supposed that the control in‡uence satis…es inclusion type limitation:

(3) _{u (t) 2 V; t 2 T;}

almost everywhere in T , where V is a compact from Rr_{.}

The aim of the work is a minimization of the functional

(4) J (u) = ' (x (t0) ; x ( ) ; x (t1)) +

Z

T

F (x; u; t) dt;

on the solutions of the boundary value problem (1), (2) with the admissible controls, satisfying the condition (3). Here it is supposed that the scalar func-tions ' (x; y; z) and F (x; u; t) are continuous over their arguments and have continuous partial derivatives with respect to (x; y; z) up to …rst order.

Notice that when A = E (E is n n dimensional unit matrix) B = 0, C = 0 the problem (1) – (3) turns to the well known optimal control problem with free right end (control by Cauchy problem). This problem have a wide class of applications and e¤ective enough methods has been developed for its solution (see for examp. [1] p.108; [2]) .

For many physical-technical problems it is essential that the controlled dynam-ical process is described by the boundary value problem (1), (2). For instance, the optimization of the multi-layer constructions under periodical temperature in‡uence [3] is reduced to the problem (1) –(4).

During the control by Cauchy problem existence and uniqueness of the solution of this problem by the chosen control is solved in enough simply way. But the solvability of the problem (1), (2) is not so simple as Cauchy problem. So in the present paper we assume that under certain conditions the boundary value problem (1), (2) has a unique solution x (t; u) for each admissible control u (t) 2 V , t 2 T .

Notice that it is easy to check the solvability of the problem in linear case

(5) _x = L (t) x + f (u; t)

(6) Ax (t0) + Bx ( ) + C (t1) = D ; rang [A; B; C] = n:

The solvability conditions have the form

(7) det [A + B ( ) + C (t1)] 6= 0;

where n n - dimensional matrix function = (t) satis…es the equation

(8) = L (t) ; (t0) = E:

The condition (7) follows from the representation of the solution of the problem
(5), (6) in the form
(9)
x (t; u) = G (t; ; t1) D +R
to
N (t; ; t; )f (u; ) d +
+R
T
M (t; ; t1; ) f (u; ) d +
t
R
t0
(t) 1_{( ) f (u; ) d}
where
(10)
G (t; ; t1) = (t) [A + B ( ) + C (t1)] 1;
N (t; ; t1; ) = G (t; ; t1) B ( ) 1( ) ;
M (t; ; t1; ) = G (t; ; t1) C (t1) 1( ) ;

following from the Cauchy formula.

The admissible process fu (t) ; x (t; u)g, being the solution of the problem (1)-(4), i.e. giving minimum to the functional (4) under the conditions (1)-( 3) is said to be an optimal process and u (t) an optimal control.

3. The formula for the increment of the functional

Investigation of the optimal control problem (1) - (4) may be carried out by use
of the various formulas for the increment of the aim functional on two
admissi-ble processes fu; xg and f~u = u + u; x = x +~ x = x (t; ~_{u)g. L.I.Rosonoer}
classical method [4] allows one to get necessary optimality condition of type of
Pontrayagin [5] maximum principle. In proof it is essential the local property
of the formula of increment –the reminder terms are estimated by the
quan-tity characterizing smallness of the measure of the domain of the needle-shaped
variation of the control.

Necessary conditions of optimality for the optimal control problems described by the systems of ordinary di¤erential equations with non-local conditions have been obtained also in [6-8].

Let fu; x (t; u)g and f~u = u + u; x = x +~ x = x (t; ~_{u)g be two admissible}
processes. Then the boundary increment problem for (1), (2) may be de…ned
as:

(11) x:= f (x; u; t) ; _{t 2 T}

(12) A x (t0) + B x ( ) + C x (t1) = 0;

where

f (x; u; t) = f (~x; ~u; t) f (x; u; t)

denotes the total increment of the function f (x; u; t). For partial increments we use the denotations u~f (x; u; t) = f (x; ~u; t) f (x; u; t).

The increment of the functional (4) may be written as:

(13) J (u) = J (~u) J (u) = ' (x (t0) ; x ( ) ; x (t1)) +

Z

T

F (x; u; t) dt

Now we do some standard procedures: in the right side of the formula (13) we will perform a few necessary transformations, namely,

- add zero terms Z

T

and

h ; A x (t0) + B x ( ) + c x (t1)i;

where (t) 2 Rn_{; t 2 T ;} _{2 R}n _{are some not de…ned yet vector function and}

constant vector; h ; i is a scalar product in Rn_{;}

- introduce Pontryagin’s function

H ( ; x; u; t) = h (t) ; f (x; u; t)i F (x; u; t) ; - apply Taylor formula for ', with the terms of …rst order:

(14) ' (x (t0) ; x ( ) ; x (t1)) =
D
@'
@x(t0); x (t 0)
E
+D_{@x( )}@' ; x ( )E+
+D_{@x(t}@'
1); x (t1)
E
+ o'(k x (t0)k ; k x ( )k ; k x (t1)k) ;

- by use of the integrating by parts introduce the identity

(15)
R
T
h (t) ; x:_{(t)idt =}R
to
h (t) ; x (t)idt +
t1
R
h (t) ; xidt =
= h (t1) ; x (t1)i + h ( 0) ( + 0) ; x ( )i
(t0) ; x (t0)i
R
T
h (t) ; x (t)idt:

Considering (14), (15), the increment of the functional

J (u) = ' (x (t0) ; x ( ) ; x (t1)) +R T F (x; u; t) dt+ +R T h (t) ; x f (x; u; t)idt + h ; A x (t 0) ; B x ( ) + C x (t1)i

may be represented in the form
J (u) = R
T
h (t) ; x (t)idt R
T
~
x~uH ( ; x; u; t) dt+
+Dh_{@x(t}@'
0) (t0) + A
0 i_{;} _{x (t}
0)
E
+
(17)
+Dh_{@x( )}@' + ( 0) ( + 0) + B0 i_{;} _{x ( )}E_{+}
+Dh_{@x(t}@'
1)+ (t1) + C
0 _{i}i_{;} _{x (t}_{1}_{)}E _{+ o}_{'}_{(k x (t}_{0}_{)k ; k x ( )k ; k x (t}_{1}_{)k) ;}

where ’–means transpiration and

~

x ~uH ( ; x; u; t) = H ( ; ~x; ~u; t) H ( ; x; u; t) :

(18) x ~~uH ( ; x; u; t) = x~H ( ; x; ~u; t) + u~H ( ; x; u; t) ~ xH ( ; x; ~u; t) = D @H( ;x;~u;t) @ x ; x(t) E + oH(k x (t)k) @H( ;x;~u;t) @ x = u~ @H( ;x;u;t) @ x + @H( ;x;u;t) @ x

Putting (18) into (17), for the increment of the functional we get:
(19)
J (u) = R
T
~
uH ( ; x; u; t) dt R
T
D
~
u@H( ;x;u;t)_{@x} ; x (t)
E
dt
R
T
D
(t) +@H( ;x;u;t)_{@x} ; x (t)Edt +Dh_{@x(t}@'
0) (t0) + A
0 i_{;} _{x (t}_{0}_{)}E_{+}
+Dh_{@x( )}@' + ( 0) ( + 0) + B0 i_{;} _{x ( )}E_{+}
+Dh_{@x(t}@'
1)+ (t1) + C
0 i_{;} _{x (t}
1)
E
+
+o'(k x (t0)k ; k x ( )k ; k x (t1)k) R
T
oH(k x (t)k) dt

Now we de…ne unde…ned vector function (t) and constant vector as a solution of the following linear boundary value problem (stationary condition of the Lagrange function over state):

(20) (t) = @H ( ; x; u; t)
@ x ;
(21) (t0) =
@ '
@ x (t0)
+ A0 ;
(22) ( + 0) ( 0) = @ '
@ x ( )+ B
0 _{;}
(23) (t1) =
@ '
@ x (t1)
C0 :

The system of equations (20) - (23) is called a conjugate system. To …nd a
solution of (20) - (23) it is necessary to …nd the vectors (t) 2 Rn _{and} _{2 R}n_{,}

satisfying the system of di¤erential equations (20) and boundary conditions (21) - (23). The condition (22) shows that the solutions of the conjugate system (20) have a …rst order discontinuity at the point t = in the general case.

Note that the system (20) - (23) is called a conjugate system in di¤erential form. It is possible to get the conjugate system also in the integral form. To do this one can write the formula for the increment of the functional in the form:

(24)
J (u) = R
T
~
uH ( ; x; u; t) dt
R
T
D_{@H( ;x;u;t)}
@x ; x (t)
E
dt
R
T
D
~
u@H( ;x;u;t)@x ; x (t)
E
dt +R
T h (t) ; x (t)i dt+
+D_{@x(t}@'
0); x (t0)
E
+D_{@x( )}@' ; x ( )E+D_{@x(t}@'
1); x (t1)
E
+
+h ; A x (t0) + B x ( ) + C x (t1)i+
+o'(k x (t1)k ; k x ( )k ; k x (t1)k)
R
T
oH(k x (t)k) dt

Let’s transform the second term in the right hand side of (24) as follows:
(25)
R
T
D
@H( ;x;u;t)
@x ; x (t)
E
dt = R
T
d
dt
t1
R
t
@H( ;x;u;s)
@x ds; x (t) dt =
=
t1
R
t
D
@H( ;x;u;s)
@x ds; x (t)
E
t=t1
t=to +
R
T
t1
R
t
@H( ;x;u;s)
@x ds; x
:_{(t)} _{dt+}
= R
T
@H( ;x;u;t)
@x ; x (t0) +
R
T
t1
R
t
@H( ;x;u;s)
@x ds; x
:_{(t)} _{dt}

It is easy to check that the relations

(26) x ( ) = x (t0) + Z T ( t) _x (t) dt; (27) x (t1) = x (t0) + Z T _x (t) dt;

are true, where (s) = 0 •{ge s 0 1 •{ge s > 0:

Considering (25)-(27) in (24) for the increment of the functional we get
(28)
J (u) = R
T
~
uH ( ; x; u; t) dt +R
T
(t)
t1
R
t
@H( ;x;u;s)
@ x ds+ ( t)
@'
@x( )+
+_{@x(t}@'
1)+ B
0 _{(} _{t) +C}0 _{;} _{x (t)i dt +} R
T
@H( ;x;u;t)
@ x dt + A0 +
@'
@x(t0)+
+B0 _{+} @'
@x( )+ C0 +
@'
@x(t); x (t0)i + o'(k x (t0)k ; k x ( )k ; k x (t1)k)
R
T
D
~
u@H( ;x;u;t)_{@x} ; x (t)
E
dt R
T
oH(k x (t)k) dt

Now let any arbitrary vector function (t) 2 Rn _{and constant vector} _{2 R}n

(29)
(t) =
t1
Z
t
@H ( ; x; u; s)
@x ds ( t)
@'
@x ( )
@'
@x (t1)
+ B0 ( t) C0
(30)
Z
T
@H ( ; x; u; t)
@x dt + A
0 _{+} @'
@x (t0)
+ B0 + @'
@x ( )+ C
0 _{+} @'
@x (t1)
= 0

Consider (20)-(23) in (19) or (29), (30) in (28). Then …nally we get the formula for the increment of the functional

(31)
J (u) = R
T
~
uH ( ; x; u; t) dt R
T
D
~
u@H( ;x;u;s)_{@ x} ; x (t)
E
dt+
+o'(k x (t0)k ; k x ( )k ; k x (t1)k) R
T
oH(k x (t)k) dt

4. Necessary optimality conditions

Let’s consider the increment formula for the aim functional on the needle-shaped variation of the admissible control. As a variation parameter we take the point 2 (t0; ) [ ( ; t1], the number " 2 (0; t0], and the vector v 2 V . The

variation interval ( "; ] belongs to T . Needle-shaped variation u = u (t) we give in the form

(32) "u (t) = v u (t) ; t 2 ( "; ]

0 _{t 2 T n (} "; ]

Let ~u (t) = u"(t) = u (t) + "u (t) and "x (t) = x (t; u") x (t; u).

Nec-essary optimality condition-Pontryagin’s maximum principle will follow from the increment formula (31), if we can show that on needle-shaped variation ~

u (t) = u (t) + "u (t) increments of the state "x (t) is of order ". We’ll

repre-sent the boundary value problem (11), (12) in increments in the form:

(33) x:= x~f (x; ~u; t) + u~f (x; u; t) ;

(34) A x (t0) + B x ( ) + C x (t1) = 0:

Introduce the matrix

L (t) = 1 Z 0 @f (x (t) + x (t) ; ~u (t) ; t) @ x d :

The matrix L (t) allows one to represent the partial increment as [2 c.92] .

~

xf (x; ~u; t) = L (t) x (t) :

Using matrix L (t) , we construct the linear boundary value problem

(35) z:(t) = L (t) z + vt (x; u; t) ;

(36) A z (t0) + Bz ( ) + Cz (t1) = 0:

The boundary value problem (35), (36) connects some state z (t) 2 Rn _{with}

admissible controls v = v (t) . As follows from the form of the problem (35), (36) its solution is z (t) 0, when v (t) = u (t) and z (t) x (t) when v (t) = ~u (t) .

Thus the boundary value problem (35), (36) is always solvable for any admissible control v = v (t) under the solvability condition of the boundary value problem (11), (12). It follows from the solvability condition (7) and formula (9), (10). Representation of the solution of boundary value problem (35), (36) from the same formulas it follows the estimation

(37) _{k x (t)k} M

Z

T

k ~uf (x; u; t)k dt; M = const > 0:

Now let u (t) = u"(t) in (37). Then

(38) _{k} "x (t)k K"; t 2 T; K = const > 0:

The estimation (38) demonstrates that for ~u (t) = u"(t) we have

(39)
R
"
D
v@H( ;x;u;t)_{@ x} ; "x (t)
E
dt + o (k "x (t0)k ; k "x ( )k ; k "x (t0)k)
R
T
oH(k "x (t)k) dt o (")
and
(40) "x (t) = x (t; u") x (t; u) ":

Considering (38) - (40) in the increment formula (31), of the functional we …nally get:

(41) "J (u) = J (u") J (u) = vH ( ; x; u; ) " + o (") 8v 2 V; 8 2 (t0; ) [ ( ; t1]

Let fu ; x = x (t; u )g be an optimal process. If we take u (t) = u (t) in (32) the obtained necessary condition has the form of Pontryagin’s maximum principle. Thus the following theorem is proved.

Theorem 1. _{Let the admissible process fu ; x = x (t; u )g be optimal for the}
problem (1) - (4) and (t) = (t; u ) be a solution of the conjugate problem
(20) - (23) (or (29), (30)). Then for any v 2 V the inequality

(42) vH ( ; x ; u ; t) 0:

is valid.

Note that in the linear convex case of the problem (1) - (4), i.e. when

(43) f = A (t) x + g (u; t) ;

F = F1(x; t) + F2(u; t) ;

' (x; y; z)and F1(x; t) are convex over (x; y; z), the Pontryagin’s maximum

prin-ciple is also su¢ cient condition of optimality.

Theorem 2. Let the condition (43) be satis…ed in the optimal control prob-lem (1) - (4). Then the condition (42) is necessary and su¢ cient condition of optimality of the process fu ; x (t; u )g .

The proof of this theorem follows from the increment formula (31)

J (u) = R T ~ uH ( ; x; u; t) dt+ +o'(k x (t0)k ; k x ( )k ; k x (t1)k) + R T oF1(k x (t)k) dt ; where o'(k x (t0)k ; k x ( )k ; k x (t1)k) 0; oF1(k x (t)k) > 0: References

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3. Babe G.B., Kanibolotskiy M.A., Urjumtsev Yu.S. Optimization of the multilayer constructions under periodic temperature in‡uence. Doklady SSSR, 1983.- v.269, N 2. p.311-314. (in Russian)

4. Rozonoer L.I. Pontryagin’s maximum principle in the theory of optimal systems.
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