BOUNDARY VALUE PROBLEM
FOR ORDINARY DIFFERENTIAL EQUATIONS
WITH APPLICATIONS TO OPTIMAL CONTROL
S. N. AVVAKUMOV AND YU. N. KISELEV
CMC DEPARTMENT, MOSCOW STATE LOMONOSOV UNIVERSITY
Abstract. Variation of parameter scheme (continuous form) in BVP for ODE
is outlined. Presented algorythm is used in the elaborated Maple program for
numerical solving of BVP. Some computational experience is described in the
paper, OC applications including.
1. Introduction
VPM (continuation method or homotopy method) may be considered as special development and modification of classical Newton’s method, see [1] - [5], [16].
The main idea of VPM admits short formulation: reducing of a given problem
to some IVP for ODE. This fundamental principle of mathematical physics is well
known, widely used and effective because IVP for ODE may be considered as a routine computational problem. The VPM-scheme outline is presented here in pressed
continuous form. It allows to combine simplicity of exposition of the main idea with
conviniencies for numerical realization on the way to computational experiments.
Some special VPM-schemes for solving BVP ODE are outlined below. It occurs convinient to produce numerical experiments within the approach in MAPLE program
environment which permits to fulfil analytical and numerical-analytical calculations
using MAPLE [15] computer algebra possibilities. By the way, instead of MAPLE,
the MAXIMA program, for example, may be used. Serious computational difficulties, arising here, are adequate to complexity of the considered problem. Intention
to solve PMP BVP arising in OC was the main impulse for providing the investigations. These results are of interest also for tutoring activity. Some numerics are
presented in the paper. The elaborated software (Program) allows to solve regular
BVP and some PMP BVP in OC, to search periodic solutions, limit cycles of ODE,
to determine unknown parameters in nonlinear ODE, etc. Note that considered
computational scheme does not coincide with previous ones.
Key words and phrases. Boundary value problem (BVP), initial value problem (IVP), ordinary differential equations (ODE), variation of parameter method (VPM), optimal control (OC),
Pontrygin maximum principle (PMP), Maple program, numerical experiments.
This work is supported by ”UR-BR”, pr.43,5199, ”RFBR” pr.99-01-01051 .
1
2. Theoretical background: short review
2.1. VPM for nonlinear vector equation in IRm . Let us consider the vector
equation
Φ(p) = 0,
(1)
m
m
where Φ : IR 7→ IR is a smooth function. It is assumed that at least one solution
of the equation (1) exists. Classical Newton’s method
pk+1 = pk − [Φ(pk )]−1 Φ(pk ),
k = 0, 1, ...
(2)
requiring nonsingularity of the matrix
m
Φ0 (p) = (∂Φi (p)/∂pj )i,j=m
(3)
along the process possesses quadratic rate of convergence in the case of convergence,
but ”enough good” starting point p0 should be taken often. This disadvantage of
Newton’s method is overcome in VPM-scheme which elaborates ”good” initial approximation to solution starting with rough one. The VPM-scheme reduces searching of a solution of the equation (1) to some IVP. It is possible to do under certain
assumptions listed below.
The VPM-scheme deals with the auxiliary equation
Φ(p) = (1 − µ)Φ(p0 ),
µ ∈ [0, 1],
(4)
m
containing parameter µ. Here p0 is some fixed point from IR , being an initial
approximation to solution of the equation (1); the accuracy of this approximation
is not assumed ”small”. The equation (4) with µ = 0 has the known solution p0 .
For µ = 1 the equation (4) coincides with the initial equation (1). In VPM the
known solution p0 is transformed into an unknown solution of the equation (1). The
transformation law is described via some IVP. Note, that the auxiliary equation (4)
may be chosen in more general form: H(p, µ) = 0, H(p0 , 0) = 0, H(p, 1) = Φ(p),
but we will consider the case (4) only.
Assumption 1 (on existence of smooth branch). The equation (4) for any
µ ∈ [0, 1] has solution
p = p(µ), 0 ≤ µ ≤ 1,
(5)
moreover, the function (5) is smooth with respect to parameter µ ∈ [0, 1], and
satisfies the initial condition
p(µ)|µ=0 = p0 .
(6)
Assumption 2 (on nonsingularity of matrix (3)) . The matrix (3) is nonsingular
along the solution (5).
Validity of the Assumptions depends on equation (1) and on choice of p0 as
well. Of course, direct verification of Assumptions 1, 2 for complicated cases is not
possible in an effective way.
Substitution of the solution (5) into the equation (4) leads to the identity
Φ(p(µ)) = (1 − µ)Φ(p0 ),
µ ∈ [0, 1].
Differentiation of this identity with respect to parameter µ implies
dp(µ)
= −Φ(p0 ).
Φ0 (p(µ))
dµ
2
(7)
So, due to (6), (7), the function (5) is defined by the IVP
dp
= −[Φ0 (p)]−1 Φ(p0 ),
dµ
p(µ)|µ=0 = p0 ,
0 ≤ µ ≤ 1.
(8)
Numerical solving of the IVP (8) allows to calculate the function p(µ), 0 ≤ µ ≤ 1;
the vector
p(µ)|µ=1
(9)
should be a solution of the equation (1). Under real calculations the vector (9) gives
new approximation for solution, and its accuracy depends on applied numerical
method for solving the IVP (8).
So one can associate one step of iteration procedure with the IVP (8) solving.
On this way the following description of iterative process to search a solution of the
equation (1) may be proposed: iterative consequence p0 , p1 , p2 , ... is produced by
p0 = p0
IV P (8),p0 =p0
=⇒
p1 = p(µ)(8) |µ=1
IV P (8),p0 =p1
=⇒
p2 = p(µ)(8) |µ=1 =⇒ ...
(10)
The outlined process (10) includes a wide variety of discrete numerical schemes
defined by invoked numerical method for solving of IVP (8), but we do not need to
specify them now. Note only, that application of Euler’s method with step ∆µ = 1
in the IVP (8) converts process (10) into Newton’s process (3).
Some modifications of ODE (8) are possible.
2.2. VPM in BVP for ODE. Let us consider the BVP for ODE
ẋ = f (t, x),
a ≤ t ≤ b,
x ∈ IRn ,
R(x(a), x(b)) = 0.
(11)
Here f (t, x) : IR1 × IRn 7→ IRn , R(x, y) : IRn × IRn 7→ IRn are smooth vector
functions. Supposing existence of solution to the BVP (11) we will discuss the
calculation questions.
Solution of BVP may be reduced to some nonlinear vector equation in IRn . Let
us choose a point t∗ ∈ [a, b] and consider the IVP
ẋ = f (t, x),
x|t=t∗ = p ∈ IRn .
(12)
Freedom in choice of this point t∗ may be very useful for numerical practice. Solution of IVP (12) is designated as
x(t, p),
a ≤ t ≤ b.
(13)
Solution (13) is assumed to be defined for all t ∈ [a, b] with any p. The initial value
parameter p ∈ IRn is to be found in such a way to satisfy the boundary condition
in the problem (12), i.e. p is a solution of the equation
Φ(p) ≡ R(x(a, p), x(b, p)) = 0.
(14)
Thus BVP (11) is reduced to the vector equation (14), with dimension n. VPMscheme, subsection 2.1, is used for solving equation (14). Some discussion on the
matrix Φ0 (p) calculation is to be done. We have:
Φ0 (p) = Rx0
∂x(b, p)
∂x(a, p)
+ Ry0
.
∂p
∂p
3
(15)
Here n × n−matrices Rx0 (x, y), Ry0 (x, y) are calculated along the solution (13), i.e.
for x = x(a, p), y = x(b, p). The notation
X(t, p) ≡
∂x(t, p)
∂p
(16)
for n × n-matrix of derivatives of the solution (14) with respect to initial value is
used. Matrix (16) is defined by the following differential equation in variations
Ẋ = AX,
where n × n− matrix
X|t=t∗ = I,
a ≤ t ≤ b,
A = A(t, p) ≡ fx0 (t, x)|x=x(t,p) ,
I is identity matrix. Main IVP has form
dp
= −[Φ0 (p)]−1 Φ(p0 ), p(0) = 0, 0 ≤ µ ≤ 1,
dµ
with
Φ(p) = R(x(a, p), x(b, p)),
Φ0 (p) = Rx0 (x(a, p), x(b, p))X(a, p) + Ry0 (x(a, p), x(b, p))X(b, p).
(17)
(18)
(19)
(20)
For simultaneous calculation of x(t, p), X(t, p) the following vector-matrix IVP may
be written

x|t=t∗ = p,
 ẋ = f (t, x),
(21)
Ẋ = fx0 (t, x)X, X|t=t∗ = I,

a ≤ t ≤ b.
The IVP (19) is said to be the external problem, the IVP (21) is said to be the
internal problem. In general nonlinear case the internal problem (21) depends on
p.
Thus iterative process (10), on the base of the external IVP (19) and the internal
IVP (21), is proposed.
The outlined scheme has been used to elaborate MAPLE Program for numerical
solving BVP (11). For composition of matrix fx0 the MAPLE analytical possibilities
were involved. For solving IVP, arising here, various MAPLE procedures may be
invoked.
Numerical expirience with the created software (Program) is presented in section 3.
2.3. Smoothing technique used in PMP BVP in OC. PMP BVP in OC containes discontinuous or unsmooth functions, for example, signature function ( sign),
saturation function (sat), dead zone function (dez), etc. So the previous approach
can not be used in a direct way to BVP in OC. Another reason for smoothing : in
problems with bang-bang type controls the inverted matrix may occur singular in
some domains. Certain preparing work is neccesary. Some technique for smoothing
optimal control problems were presented in [6] - [11]. These smoothing procedures
deal with changing of control dimension. Regularization of the problem may be
reached sometimes without changing of control dimension. Some simple smoothing
formulas are mentioned in this subsection.
4
The signature function sign(s) is approximated by
p
SIGN U M 1(s, ν) = s/ ν + s2 , SIGN U M 2(s, ν) = tanh (s/ν).
The saturation function
sat(s) =
is approximated by
SAT (s, ν) =
The dead zone function
s,
sign(s),
|s| ≤ 1,
|s| > 1
p
1 p
ν + (s + 1)2 − ν + (s − 1)2 .
2
dez(s) =
is approximated by
1
DEZ(s, ν) =
2
0,
sign(s),
s+1
p
ν + (s + 1)2
|s| < 1,
|s| > 1,
+p
s−1
ν + (s − 1)2
!
.
Smoothing parameter ν is some small positive number. Corresponding smoothing
formulas for extremal controls, while applying maximum principle [12], may be
got under appropriate ”small” perturbation of the cost for some classes of OC
problems. The animation pictures for these approximating functions with respect
to parameter ν may be found in http://www.cs.msu.su/˜asn/Maple/movies.htm
3. Numerical Experiments
Example 1. Two bodies’ boundary value problem [3]:
x
y
ẍ = − 2
, ÿ = − 2
,
2
3/2
(x + y )
(x + y 2 )3/2
x(0) = a1 , y(0) = a2 ,
x(T ) = b1 , y(T ) = b2 .
To apply the Program this BVP is rewritten in the form :

ẋ1 = x3 ,
x1 (0) = a1 , x1 (T ) = b1 ,


 ẋ = x ,
x (0) = a , x (T ) = b ,
2
4
2
ẋ3 = −x1 (x21 + x22 )−3/2 ,



ẋ4 = −x2 (x21 + x22 )−3/2 .
2
For the case
T = 7, a1 = 2., a2 = 0., b1 = 1.0738644361,
choosing t∗ = 0, with the starting points
p0 1 = [2., 0., −0.5, 0.5]
and
2
2
b2 = −1.0995343576,
p0 2 = [2., 0., 0.5., −0.5]
two different solutions of the BVP were got with defining answer-vectors correspondently
ans1 = [2., 0., 0.0000004834, 0.5000000745] and
ans2 = [2., 0., 0.4510782034, −0.2994186665].
5
1.2
1
0.8
0.6
0.4
0.2
0
–0.2
–0.4
–0.6
–0.8
–1
–1.2
x2
10
1
3
8
2
1
S
0.5
1
1.5
2
x2
6
4
2
x1
2.5
–10 –8 –6 –4 –2 0
–2
x1
2
4
6
8
10
–4
–6
F
2
–8
–10
Fig.1.
Fig.2.
Corresponding trajectories 1 and 2 (with the starting point S and the final point
F ) of the system in x1 x2 plane are shown on Fig.1. Here and in the following
examples rkf-45 methods for solving IVP were used. Chosen accuracy: for the
external problem 10−4 , for the internal problem 10−6 . Number of iteration steps
= 3.
Example 2. Calculation of limit cycles in Eckweiler system [13]
ẋ1 = x2 ,
ẋ2 = −x1 + sin (x2 ).
This system has a countable set of limit cycles. Some of them are calculated together with unknown period T . Using different starting vectors p0 the Program finds
different limit cycles. Limit cycle searching is reduced to the following BVP:

ẋ1 = x3 x2 ,




 ẋ2 = x3 (−x1 + sin (x2 )),
ẋ3 = 0,


ẋ4 = 0,



x1 (0) = x4 (0), x2 (0) = 0, x1 (1) = x4 (1), x2 (1) = 0.
Auxiliary variables: x3 = T , for period, and x4 = x1 (0), for abscissa of limit cycle
intersection point with axis x1 , are introduced. Choosing point t∗ = 0, and the
starting vectors
p0 1 = [2., 0., 2π, 2.],
p0 2 = [6.5, 0., 2π, 6.5],
p0 3 = [9., 0., 2π, 9.]
the following defining answer-vectors have been calculated:
ans1 = [3.9655467678, 0., 6.4661401325, 3.9655467678],
ans2 = [7.1078664573, 0., 6.3387892836, 7.1078664573],
ans3 = [10.2456910360, 0., 6.3101121791, 10.2456910360].
Corresponding limit cycles 1, 2, 3 are shown on Fig.2. This problem is solved
without any difficulties.
6
14
12
10
8
6
4
2
–2
–1
–2
–4
–6
–8
–10
–12
–14
x2
1
x1
14
12
10
8
6
4
2
0
–2
–4
–6
–8
–10
–12
–14
2
x1,x2
x1(t/T)
x2(t/T)
0
0.2
Fig.3.
0.4 t/T 0.6
0.8
1
Fig.4.
Example 3. Calculation of limit cycle period of van der Pol equation
ẋ1 = x2 ,
ẋ2 = −λ(x21 − 1)x2 − x1 ,
(22)
for λ = λ1 = 10, λ = λ2 = 2.7721992263175, λ = λ3 = 0.95893843253661.
Using similar technique as in Example 2 for period T (λ) and solving corresponding BVP one has got
T (λ1 ) = 19.0783730753, T (λ2 ) = 8.56600769087, T (λ3 ) = 6.6337646662.
Calculations with λ = 10 were not succesful for some starting vectors p0 , but there
exists a wide enough domain for p0 with stable convergence of the process; limit
cycle of the system (22) for λ = 10 is shown on Fig.3, dependence of coordinates
on time is shown on Fig.4. It is interesting to mention that the system is stiff.
Example 4. Determination of parameter λ in van der Pol equation (22) under
initial conditions x1 (0) = 2, x2 (0) = 0 and ”mesurement” x1 (8) = 1.161168571601.
This problem may be reduced to solving the equation
h(λ) ≡ x1 (8, λ) − 1.161168571601 = 0.
Using MAPLE the graph of function h(λ) is plotted, Fig.5, and its roots are calculated: see λ1 , λ2 . λ3 from Example 3. These results may be calculated also using
the Program through solving the following BVP

ẋ1 = x2 ,



ẋ2 = −x3 (x21 − 1)x2 − x1 ,
ẋ3 = 0,



x1 (0) = 2, x2 (0) = 0, x1 (8) = 1.161168571601
with different initial vectors p0 .
7
h
2
0
λ1
6 λ
4
8
10
λ3
λ2
–1
–2
h(λ)
–3
Fig.5.
Example 5. Time optimal control problem for the simple entity
Let us consider time optimal problem ([12], example 1)
ẋ1 = x2 + u1 ,
T → min
ẋ2 = u2 ,
x1 (0) = x2 (0) = 2, x1 (T ) = x2 (T ) = 0,
(23)
with interval U0 = {u1 = 0, |u2 | ≤ 1} as control domain. At first we will apply
control domain smoothing technique. The set U0 is approximated by smooth convex
set Uµ = {u21 /µ + u22 ≤ 1}, depending on smoothing parameter µ > 0 and bounded
by narrow (for small µ) ellipse. The problem (23) is reduced to PMP BVP which
is presented as

ẋ1 = x5 (x2 + √ µx23 2 ),


µx3 +x4


x4

√
ẋ
=
x
,

2
5

µx23 +x24

ẋ3 = 0,
(24)


ẋ
=
−x
x
,
4
5
3





 ẋ5 = 0,
x1 (0) = 2, x2 (0) = 2, x1 (1) = 0, x2 (1) = 0, x3 (1)2 + x4 (1)2 = 1.
Auxiliary variable x5 = T is introduced, new time interval is [0, 1], and x3 = ψ1 ,
x4 = ψ2 are conjugate variables. Starting with t∗ = 0 and the initial point p0 =
[2, 2, 0, −1, 4] the Program solves BVP (24),µ = 0.5, and the received answer-vector
is used as p0 for solwing BVP (24), µ = 0.1, and its answer-vector, at last, is used
as p0 for solving BVP (24), µ = 10−8 . The resulting answer-vector:
ans = [2., 2., −0.4472136002, −1.7888543875, 5.9999999428]
It is interesting to make comparison with the exact solution :
T = 6.0, ψ(0) = (− √15 , − √45 ),
√1 = 0.447213595499..., √4 = 1.788854381999...
5
5
Optimal trajectories and controls u2 (t) of smoothed problem for µ = 0.5, 0.1, 10−8
are presented on Fig.6,7. Optimal control u2 (t) for small µ has practically bangbang character while |u1 (t)| is small negligibly.
8
One dimensional smoothing of control in this example leads to the following PMP
BVP:

ẋ1 = x5 x2 , 




ẋ2 = x5 tanh xµ4 ,



ẋ3 = 0,

ẋ4 = −x5 x3 ,




ẋ5 = 0,



x1 (0) = 2, x2 (0) = 2, x(1) = 0, x(1) = 0, x3 (1)2 + x4 (1)2 = 1.
Here similar numerical results may be be reached. Of course, special programs for
time optimal problem, see [10], [11], are more effective in comparison with the
Program, which permits to treat a wider class of problems.
2
1
(2,2)
x2
0.8
2: µ=1*10–1
0.6
3: µ=1*10–8
0.4
1
2
3
4
1
2
3
4
5
t
–0.4
2
1
–0.6
–0.8
3
–2
2: µ=1*10–1
3: µ=1*10–8
0
–0.2
1
–1
1: µ=5*10–1
0.2
x1
0
u1
u2
u3
1
1: µ=5*10–1
2
3
–1
Fig.6.
Fig.7.
Example 6. Minimization of the energy cost for 3-tiple integrator
Let us consider the control problem

ẋ1 = x2 ,




 ẋ2 = x3 ,
ẋ3 = u, |u| ≤ 1,


 x(0) =R(1, 0, 0), x(T ) = (0, 0, 0),


T
L = 21 0 u(t)2 dt → min .
(25)
T = 3.275,
For the control problem
ẋ = Ax + bu, |u| ≤ 1, x(0) = x0 , x(T ) = 0,
RT
L(u) = 12 0 u2 dt → min
T > 0 − fixed,
with one dimensional bounded control the PMP BVP has form:
ẋ = Ax + b · sat(b∗ ψ),
ψ̇ = −A∗ ψ,
9
x(0) = x0 , x(T ) = 0.
(26)
In special case (25), using smoothing technique for saturation function (with smoothing parameter µ ), this PMP BVP takes the form

ẋ1 = x2 ,




ẋ2 = x3 ,p


p

ẋ3 = 12 ( µ + (x6 + 1)2 − µ + (x6 − 1)2 ),
ẋ4 = 0,




ẋ = −x4 ,


 5
ẋ6 = −x5
with boundary conditions extracted from (25). This BVP was solved by the Program
(t∗ = T, µ = 10−10 ) with resulting answer-vector
ans=[0,0,0,-2.9850435834, 4.8880088678,-2.9083874537]
Dependence of optimal xi (t), i = 1, 2, 3, and control u(t) on t is shown on Fig.8,9.
1
0.8
0.6
x1
x2
x3
1
x1(t)
x3(t)
u
ψ3
2
1
0
t
0.4
0.2
0
3
–1
0.5
1
1.5
2
2.5
3
- u(t)
- ψ3(t)
t
–0.2
–2
–0.4
–0.6
x2(t)
Fig.8.
Fig.9.
Example 7. Minimization of the ”fuel”-type cost
RT
The cost L1 (u) = 0 |u(t)| dt is said to be ”fuel”-type cost. In the problem
(26) with cost L1 (u) the extremal control is u∗ (ψ) = dez(b∗ ψ). The cost L1 (u) is
approximated by
L1,m (u) =
ZT
0
1
1+ m+1
|u|
dt,
m = 0, 1, 2, ...
when m is ”large”. Extremal control in the case of cost L1,m is u∗ (ψ)=sat(b∗ ψ)m+1 .
This idea is used in the problem


 ẋ1 = x2 + u1 , |u1 | ≤ 1,

ẋ2 = −x1 + u2 , |u2 | ≤ 1,
6, x2 (0) = 2, x1 (T ) = 0, x2 (T ) = 0, T = 8,

 x1 (0)R =

T
J = 0 (x21 /4 + x22 /4 + |u1 |1+1/(m+1) + |u2 |1+1/(m+1) ) dt → min .
Using smoothing technique for saturation function with smoothing parameter µ =
10−8 , choosing parameter m = 5 and applying the Program optimal solution was
calculated. Optimal controls to the problem are shown on Fig.10.
10
u1,u2 1
0.8
0.6
0.4
0.2
0
–0.2
–0.4
–0.6
–0.8
u2(t)
u1(t)
–1
1
2
3
4
5
6
7
t 8
Fig.10.
For three next examples only models are mentioned.
Example 8. Time optimal control for nonlinear pendulum [14].
Example 9. Time optimal control for two dimensional inertial entity with friction.
α1 0
ẍ = −
ẋ + v, x, v ∈ IR2 , kvk ≤ 1.
0 α2
Example 10. Time optimal control for two dimensional inertial oscillating entity
with friction.
ẍ = −
α1
0
0
α2
ẋ −
β1
0
0
β2
x + v,
x, v ∈ IR2 ,
kvk ≤ 1.
References
[1] D.F. Davidenko, Ob odnom novom metode chislennogo resheniya system nelineynyh uravneniy, Doklady AN SSSR, 1953, 88, pp. 601 – 602 (in Russian).
[2] V.S. Kiriya, Trudy Tbilisskogo Un-ta, 44, 1951, (in Russian).
[3] R.E. Bellman, R.E. Kalaba, Quasilinearization and nonlinear boudary-value problems, NY.
1965.
[4] V.E.Shamansky, Metody chislennogo reshenya kraevih zadach na ECVM, Kiev. Naukova
Dumka. 1966, (in Russian).
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S. N. Avvakumov and Yu. N. Kiselev,, CMC Department, Moscow State Lomonosov
University,, Moscow 119899, Russia
E-mail address: [email protected] and [email protected]
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