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10
Section 2. Mathematics
DOI: http://dx.doi.org/10.20534/ESR-17-5.6-5-9
Milogorodskii Aleksandr Andreevich, East China Normal University, PhD student, the Department of Applied Math E-mail: amilogorodskii@mail.ru Ming Kang Ni, East China Normal University, Professor of Math, the Department of Applied Math
Shanghai, China
ON A CERTAIN PROBLEM OF NONLINEAR OPTIMAL CONTROL ON DISCONTINUOUS TRAJECTORIES
Abstract: On the example of an elementary nonlinear optimal control problem with scalar differential connection and discontinuous trajectories with a small parameter in the derivative and without limitations on control, using the methods of contrast structures in the theory of singularly perturbed boundary value problems, the possibility of the appearance of fast internal phase transitions in the optimal discontinuous trajectory was shown.
Keywords: optimal control, nonlinear functional, discontinuous trajectories, singular perturbations, phase transition, asymptotic approximations.
Introduction
Nowadays, active investigations of the phenomena of contrast structures in singularly perturbed problems are being carried out [1, 3-10; 2, 4-32; 3, 799-851; 4, 41-52], also in [5, 1381-1392; 6, 104-118], for a singularly perturbed optimal control problem, a steptype solution is considered that undergoes a sharp change in a small neighborhood of some transition point within the considered interval.
In this article it is shown the possibility of occurrence of fast phase transitions in the case when the trajectories are optimal and discontinuous, i. e. the possibility of the emergence of the process of forming the optimal solution for such type trajectories.
1. Statement of the problem
Let consider the problem
J [u ] = jF (x,t ,u )dt
• min,
dx
s— = •
dt
(1)
| A (t)+b1 (t)u, o < t < to, [A2 (t) + B2 (t)u, to < t < T, x (0,e) = x \ x (T ,s) = xT.
The analysis will be carried out taking into account the following requirements.
1) Let the function F (x ,t ,u) — double continuously differ-entiable in the domains
D1 = {(x ,t ,u ): |x| < l, u g R, l > 0,0 < t < t0}, B1 (t) > 0,
D2 = {(x,t,u): |x| < l, u g R, l > 0, t0 < t < T}, B2 (t )> 0.
2) Let the function F2 (x,t,u) > 0 when (x,t,u)g Dl,D2. Let introduce the Hamilton functions Hj (x,t,u,y) =
= -F(x,t,u) + s~iy (t)[Aj (t) + Bj (t)u], j = 1,2
where y(t) — the corresponding conjugate variable, from the necessary conditions can be obtain
sx' = A; (t) + Bj (t)u, y' = -Fx (x,t,u), (2)
sFu (x,t,u) + yBj (t) = 0, x (0,e) = x0, x (T ,e) = xT.
The 1) and 2) conditions let to find from (2) follow singularly perturbed boundary value problem for extremals
ex' = A; (t) + Bf (t)u,
FxBj (t)-Fxu [Aj (t) + Bj (t)u]-" _ -e\_Fut + FuBj (t)Bj-1 (t)]
x (0,e) = x 0,x (T ,e) = xT.
Such a representation (2) emphasizes that both the trajectory and control are fast variables, i. e. can contain fast change zones.
If in the original variational problem put £ = 0, then a degenerate problem
su' = F,
(3)
J = Jf (x ,t ,u )dt
• min,
(4)
du
dT
u = -A (t' )B- (t') _
3) Let here exist two disjoint functions X = 01 (t) X = 02 (t) such that
\F(d1 (t),t,u), 0 < t < t,
min F (x ,t,u ) = <! ,
x V ' [F(02 (t),t,u), t < t < T
and the transition point t is defined by the equation
F ( (t * ),t ',u ) = F(e2(t'),t',u ),
and the condition is satisfied
(* ),t\u ) ^( (t* ),t\u ).
From 3) it follows Fx X^6j (t),t,u) = 0, F^ ((t),t,u) > 0, 0 < t < T.
In this paper, the existence of an extremal with an internal transition layer on discontinuous trajectories on th e basis of the following statement will be established.
Theorem 1 [7, 1401-1411] Let in the boundary-value problem for the system of two first-order differential equations
dx / \ s— = G, (x ,t ,u ,s), dt lV '
du / \
s — = G2 (x ,t ,u,s), dt 2
x (0,e) = x \ x (l,e) = x1 the following requirements are completed.
4) Degenerated system
JG, (x,t,u ) = 0, I G2 (x,t,u) = 0
has two isolated roots xj = dj (t),uj = y/j (t), j = 1,2.
5) On the phase plane (x,«) points Pj ( (t(t)), j = 1,2 are the rest saddle points for ancillary system, where t — parameter and
\x = G1 (x, t ,u), I u = G2 (x, t ,u)
and this system has the first integral ^(x, t ,u) = = X( (t), t(t), which goes through the 'P..
6) Equations t ,u) = X( (t ), t ,¥j (t )) are solvable with respect to 'u : '
SPi : u(T)= V (x,0'. (t ),Vj (t),f).
7) Equation H(t) = u(_) - u(+) = 0 has the solution t = t' and dH(t*)/dt * 0 .
2. Existence of the solution
Let show that the Euler's equation system has step-type solution. In order to do it is necessary to compare the first passage to the limit theorem with optimal conditions.
It can be observed that attached system can be written as
B (t KF-1 - fxuf; [ Aj (t )+Bj (t )u ], dT = Aj (t) + B, ( t )u.
(5)
Lemma 1. Let the requirements l)-3) completed. Then attached system (5) has two special saddle type points Pj ( (t ),u) = 1,2 and u = -Aj (t )B-1 (t ).
Proof. It is obvious than there are two isolated degenerated system's solutions Bf (t )FxF-21 - FuF- [ A, (t ) + Bf (t )u ] = = 0,[Af (t ) + Bf (T)u] = 0 and when t e [0,T] is fixed on the
phase plane (x,U),these points P, (( (t ),u), j = 1,2 — special saddle-type points, because of the structure of general functional matrix for the system (5)
Giu O
\G2u G2x J
(6)
(8)
and characteristics equations X1 — (G1u + G2x G1bG2x —
— G1xG2u = 0 it follows that the eigenvalues of the matrices (6) for any special point have different signs because their multiplication is X1X2 = —B2 (t )F_c2 F^1 < 0 and these special points have saddle type.
Lemma 2. Attached system (5) has the first integral [As (t ) + Bt (t )u]Fu (x, t ,u) -Bt (t )F(x, t ,u) = Ct, (7)
when t € [0,T] and C., j = 1,2 — are constants. Proof. Let rewrite equations in (5)
F^ (x, t ,u)u' = Bf (t )Fx (x, t ,u)-
-Fu (x, t ,uAj (t) + Bj (t )u]. Using the second equation from (5) instead of (8) -dFu (x, t ,u) -Bj (t )Fx (x, t ,u) = 0. Considering x" = B. (t it can be obtained
-d[xF (x, t ,u)- Bj (t )F(x, t ,u= 0.
As the result it can be observed for (5)
[Aj (t ) + Bj (t )u]Fu (x, t ,u) -Bj (t )F(x, t ,u) = C.. Lemma 3. Let the requirements 1), 2) completed and u ^ —Aj (t )B-1 (t ) . Then the first integral (7) can be explicitly resolved when t € [0,T] is fixed.
Proof. Let Aj = Aj (t ),Bj = Bf (t ) and
f (x,t,u) = I^A'. + Btu] Fu (x, t ,u)-Bf (t )F(x, t ,u)- Cj.
By the existence theorem for an implicit function the equation f (x, t ,u) = 0 can be resolved related to u :
u = {h(x,t,Cj)(x,i )e Dj, (9)
where D' ={(x,t): |x| < Z1, i e[0,T]},j = 1,2, since ^ =
= [Af + B^ Fu2 ^ 0 because of 2) and u ± -Af (t )B-1 (t ). Obviously two separatrices SP , j = 1,2 going through saddles
Pj, j = 1,2 are existed and satisfy equations SPj : [A. + B.uJFu ■ >
■(x, t ,u)-BF(x, t ,u) = -BjF(ej (t ), t ,u).
From (9) it can be obtained ut+) (t, f) = h(t) (x^f),Bj (t ),f ). Let H (t) = u(-) (0,t ) - u(+) (0,t ) = h(-) (x(t )A (t ), t ) -- h(+)(x(i),02_(i ) t), where x(")(0) = x{+\0) = x{t) =
= (01 (F ) + 0 (t ))/2.
Lemma 4. If the requirements l)-3) are completed, then the expressions can be written hx ((t), t ) =
= ± BjFx2 (), t u (t )) (), t u (t)) , j = 1,2.
Proof. From differentiating the implicit function it was obtained
du bf-(a + Bu)f
hx (x, F ) = du = j 'J ' ! x .
dx (A. + Bu ) 2
Applying the L'Hopital rule here in the neighborhood of special points it was found hx (j ),^) =
tlM ( ( ), t Û (t )) ( (t ) t ,u„ (t )) .
x (t ,e) = IV ( (t ) + n tx (t. ) + Qt}x (t)),
Was encounteredthen By > 0, F2 ( (t ), t ,u0 (t )) > 0 ,
Fu-1 ( (F), t ,u (t ))> 0. x
In the article [5, 1381-1392] were formulated and proved two lemmas 5 and 6 where the conditions l)-3) and F (t0) = 0 are completed then theequation F(dl (t0 ),t0,u) = F(2 (t0 ),t0,u) also completed and dH (t0) / dt ^ 0 expression can be used if and
only if Ft (01 (t0),10,u) * Ft (2 (t0 ),t0,u).
Lemma 7. Let the conditions l)-3) are completed. Then it is existed moment t = t0 such that in attached system (5) Separatrix thatconnectssaddles Pj ( (t'*,u )j = 1,2,w = -A, (f* ) B-1 (t*) existed. This expression follows directly from the second and the fifth lemmas.
Thus, for boundary problem (3) all the requirements of the first passage to the limit theorem are completed. It means that the problem (1) has extremal x(twith contrast step-type structure.
3. Asymptotics of the formal solution
It is necessary to construct formal asymptotics solution of the problem(1) usingdirectsolutionscheme.As itwas shown in [5, 1381-1392], asymptotic solution is found as
it ,e) =
'-(t ,e) =
u (t ,e) = fek ( (t )-
k=0 V
x (t ,e) = j>>k ( (t )
k=0
u (t, e ) = f>'k (( (t ) + Q^u (t) + rkU (r1 ))
^ u(to) + q^u(t)), + Qk+)x (t) + RkX (Tl )),
0 < t < t ',
t' < t < l,
(10)
(11)
where z = (x,u), x0 = t / S, X = (t -1')/ s,t1 =
= (t -1)/ S, n x(t0 ) — left boundary function in the t = 0 neighborhood, Rkx (t1 ) — right boundary function in the t = 1 neighborhood and Q(+^x — left and right boundary functions that form phase transition in the t = t neighborhood. Decompositions (10), (11) are substituted to (1) and the same s order elements from the left and right part are equated. To determine the coefficients of all the series written out, we must substitute (10), (11) into (1) and minimize the expansion coefficients of the functional in a series in integral powers in the elements of the asymptotics
inf J[x] = inf J0 (x0 ) + Jj£' inf J (xi) + ....
x x0 i=1 xi
To define general element {x0 (t),u0 (t)} ofthe regular asymptotics series it was found degenerated problem (4)
Then to define {Q0(+)x,Q0+^u^ general elements of boundary series in the neighborhood of phase transition point when trajectories are discontinuous were found
Q0T J = ±°(j™A f F (X (±),u (±),t o )dr dier
■ min,
- = A (to ) + Bj (t0 )
J0 (u0 ) = JF(Xo,uo,t)dt"
min,
A (t )+Bj (t = o.
From the condition 3) was found
-ftitjitr;.;<t)B-,<t».
dT 1
X} (o) = x(t o ), X(t o ).
After substitution
T = A (to )+B (tO )t+}
it was found that the functional does not depend on t :
m.m(t„) a(t)f(x(T),u(T),t0)
Q0T) J = t -die ^ min .
Q0 fli(t0)^x(t0)] Aj (t. ) + Bj (t. )uè(T) u(.(i(.)
The necessary condition for a minimum of integrand is
[ Aj (t o ) + Bj (t o ))] Fu - Bj (t o )F (T),u (T),t o ) =
= -F (j (t o ), u'o,t o ).
Using (9) was obtained that u= h(x(+),to,0j (t. ) ismin-imum point because
o
(F)[A. (f0)+ B, (t0)ü^
^)&[A, (t0 ) + B, (t0 )ü<*J
+
+2 % 'F )) [ A' (t • ) + B, (t. y>J
dü
= [Af (t0 ) + Bj (t0 ) u(t)J |J A, (t0 ) + Bj (t0 ) U(T)] F,]> 0.
The last is true because on the plane (x,U) when x changes u from x(t0 ) to G, (t0 ) in (12) the functions At (t0 ) + Bj (t0 )u(+), , = 1,2 are positive.
Thus, to define Q0(+)x in the neighborhood of transition point differential equations were obtained
= B, (t 0 )h ( 0 ) + Q0T)x ,t 0 ).
As it was formulated in [5, 1381-1392] the conditio n 8) tells that the equations (12) with initial conditions Q0(+'x(0) = x(t0 ) — fy (10 )have continuously differentiable solutions Qf)x(t), 0 < t0 < T, -œ < t < œ.
To define {R0x (t),R0u (t)}
R0J = jV(02 (T) + R,x,u* (T) + R0u,T) ^ niin,
— R0x = A2 (T ) + B2 (T )[u; (T ) + Ru ], R,x (0 ) = xT -02 (0 ),R,x (00) = 0.
Th define {n0x(4n0u(t)}
n0 J = RF(01 (0) + n0x,u0* (°) + nU,0) ^ mm, ¿n 0 x=Ai (0 )+Bi (0)[u; (0 )+n 0u ],
EL x (0 ) = x0-01 (0 ), n 0 x (-«)=0.
As it was formulated in [5, 1381-1392] condition 9) tells that boundary conditions x0 —01 (0),xT —02 (T) in the problems H0 J, R0J are located in the neighborhood of origin.
Thus, it can be written minimum value for functionals
J0,n0J', Q0J(T)*,R0J ■■
J0 (ü0 ) = iF(XO'ü0*'t)dt,
( X ,ü , 10
M. )]«2 (t. ) A(T)F ( 0 J= f 0 _
0 *(,)L)] A (to ) + B, (to )ü(T)
J. _^r0) A0F(X',ü ,0) d n0J _ i Ai (0) + Bi (0)ü-dX' r j. _ XfT A(T)F(( ,U\T) KJ _fl2|f)Ai(f) + Bi(f)udX■
dX,
As it was formulated in [5, 1381-1392] if the conditions l)-3), 8), 9) are completed then when S > 0 a formal asymptotic approach to step-type for (1) extremals x (t, £ ) existed and has such asymptotic expression:
01 (t) + n0x* (t ) + Qt}x' (t) + O(s), t < t0,
x(t,s) = l x(J0 ),j = ^
02 (t ) + R0x* (T1 ) + Q^x' (t) + O (s), t > t, u (t ) + ft, " T ) + Qi-V (t ) + O (s),t < t0,
(t,e) = | X'(t0 ),t = t0,
_ "0* (t) + R0W (t ) + Q0V (t) + O(s),t > t
and J' [u ] = J0 + g [ n0 r + QVJ' + Q0+)/ + R0 J' ] + O g ). 4. Example
Let consider the problem
J [ü w
Here
1 x4 -1 xcost -1 x2 +12 - ü2 dt ^ min, 8 2 4 J "
dx _J-4t + 2ü, 0 < t <n/2,
dt [ 4t + 2ü, 0 < t <n,
x (0,e) = 0, x (n,s) = 1.
F(x,ü,t) =1 x4 -1 xcost -1 x2 +12 - ü2 8 2 4
ü = ■
and
min
F (x ,t ) =
|2t,0 < t <n/2, -2t,0 < t <n
- - +—cost ,0 < t <n/2, 82
-1 -—cost ,0 < t <n. 8 2
General element of the transition point t0 =n /2 is defined by the equation cost 0 = 0. Where
ddtF (01 (t 0 ),u ,t 0 ddtF (02 (t 0 ),u ,t 0),
because ( (t0),M,t0) = -:^'d[tF( 0)'w0) = ^ In such case the first integral (7) is
4(u + 2t )(t -u)-1 x4 + xcost +1x2 = Cj, j = 1,2,
where Cj — constants. Separatrix Sp which go through the saddles Pj(t ) and when value of t e [0,^ will be Sp :u(+' =
Because X(_) (0) =
1d
-(x 1)(xx (t) +1)2.
= x(+) (0) = 0 was found H (t0) = u(_) (0,t 0) - u(+) (0,t0) = 0.
To construct left part of transition lay in the zeroth approximation was obtained next variational problem:
o0-)J=j1 O0-)2X
2 o0-)2x -1
- 0<o)ü [2n + o0-)ü ] dT ^ min,
with solution
—Q0(-)x = 2Q0(-)u,
dT
Q0-)x (0 ) = 1, Q(-)x (-») = 0
s
s
Q-X * (t) = -
1 + O ■
(
-X
1 + O 2
V )
QtJ - =—t< 0.
4
Similarly it can be found
{x* (T),Q0+)U* (T)},{n0X* (T0 ),n0" T )}, {{x * (Ti ),^u * (t )}.
s
--T
Q0+)X * (t ) = ' Q0+U ' (T) =
--T
1 + O 2
n 0 X' T ) =
•J2 V2o 2 T
•J2 ^ 2
1 + 0 2 T
s
e2
q0+)j * =At> 0,
04
s
n0U* T ) = -
1 + O
S
(
1 + O 2 T
v j
R0X* fa ) =
n 0 J ' =f t > 0,
S S
O 2 " „ * ( ) 7o 2 " ,R0U ( T1 ) =
5-o ^
( -¡Lr\ 5 - o2' v j
R0 j * ^ < 0.
As a result, the formal asymptotic representation takes the form
S S
:(t ,e) =
-1-
O
+ 0 s + O (s),t <n /2,
s 1 s
1 + 0 2 T0 1 + O 2 '
0,t = n/2
2
S
--T
2
1+
l(t ,s) =
2t-
S J2
5 - 0 ^ 1 + 0 2'
Vi s
v2o 2 T° v2o 2'
- + -
' S Y ( S Y
- —T
1 + 0 2
V )
2t +
1 + 0 2 "
V )
7o 2 t
+ O(s),t > n /2. + O (s), t <n/2,
( Y (
5 - 0 2 T
+ . \ , + O(s),t >n/2
V2 A
1+e
J V
and J* [«] = —1 + e + O(e2).
For example, when S = 0.01, the functional will have value
J' [u]« -0.09705.
Conclusion
Using the example of an elementary control problem, it is shown that by constructing a quality functional for the control problem of an object with fast motions, one can achieve presence in the optimal discontinuous trajectories along with boundary fast dynamics, boundary layers on the boundary of fast internal dynamics as well such transition layer in the neighborhood of the inner point of the control interval.
References:
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4. Ni Ming Kang, Dmitriev M. G. Contrast structures in the simplest vector variational problem and their asymptotics // Avtomatika i telemehan. - 1998. - No. 5. - P. 41-52.
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