optimization of discrete control systems with varying structure

OPTIMIZATION OF DISCRETE CONTROL SYSTEMS WITH VARYING STRUCTURE Shahlar F. Maharramov, National Academy of Science of Azerbaijan, Institute of Cybernetics, Baku, Azerbaijan ([email protected]) Stephan Dempe, TU Bergakademie Freiberg, Institute for Numerical Mathematics and Optimization Freiberg, Germany ([email protected])

Abstract In this paper a special step discrete control problem is considered. The formulation of the problem uses a parameter to control the switching point. By using Taylor’s increment methods first and second order optimality conditions (in the sense of Pontryagin‘s maximum principle) will be derived.

1.Introduction. Among the various optimal control problems the discrete control problem is characteristical. This problem arises e.g. by using numerical methods for solving continous control problems. But discrete control problems have an independent means, too. First consider an example of such a problem.

Example: (problem of a cosmic ship) Let us project a cosmic ship consisting of N steps. The ship has a starting weight G and the weight H of the rocket part. All steps have a gazeline. If one step of the rocket is pushed away the rocket takes an additional speed v , which depends on the weight of this step and the weight of the remaining part of the ship. We have to distribute the weight among the steps such that the speed of the rocket at the end of the flight is maximal. To model this problem, let u(t) denote the weight of the t-th step, t=1,...,N, where t=1 is the number of the last step. Let x(t) be the weight of the head of the rocet together with the first t steps. Then we can write x(t)=x(t-1)+u(t), t=1,...,N.

(1)

The boundary values x(0)=G and x(N)=H denote the starting weight and weight of the rocket. The additional speed gained if a step t is pushed away is

v( t )  f ( x ( t  1), u ( t )) , t=1,...,N. This additional speed depends on the weight u(t) of the t-th step and the weight x(t-1) of the re-

1

maining part of the rocket. After pushing away all steps, the common speed will be N

S( u)   f ( x( t  1), u( t ) )

(2)

t 1

The aim of the problem is now to find controls u (1); u (2),..., u ( N ) for the rocket such that (2) will be maximal subject to the conditions (1) and the boundary values.

The theory of discrete optimal problems has been investigated by Gabasov R, Kirillova F, Mordukhovich B, Mansimov K, Propoy A, Kroxotka V, Minyuk S, Ashepkov T, and others.

In the paper [2] Moyseev has investigated necessary optimality conditions for the continous control problem. The discrete anolog of Moyseev’s problem has been investigated by Mansimov K. [7]. To formulate the results of [7], consider the problem t 1 1 t1 1

S( u)    Ft , s, xt , xs  min

(3)

t t o s  t 0

subject to

x ( t  1)  A( t ) x ( t )  f ( t , u ( t )); t  T  t 0 , t 0  1,..., t 1  1, x ( t 0 )  x 0 .

(4)

Theorem (Mansimov K [7]): For optimality of an admissible control u(t) of the problem (3),(4) it is necessary that the following inequalities hold: t 1 1

    H( t)  0 for all vt  U, t  T . t t0

v t

Here H ( t )   ( t )f ( t , u ( t )) is Pontryagin’s function and  v ( t ) H ( t ) is the increment of the function H(t).

In the case of an nonsmooth objective function

S0 (u )   x t 1   min

(5)

Si (u )   i ( x ( t 1 ))  0, i  1,.., p

(6)

x ( t  1)  f ( t , x t , u ( t )); t  T  t 0 , t 0  1,..., t 1  1, x ( t 0 )  x 0 u(t)  U  R r , t  T

(7) (8)

2

the following theorem holds. Theorem (Mansimov K, 7 ): If the admissible velocities are convex along admissible processes, then for optimality in problem (5) - (8) it is necessary that the following inequalities hold: ∂   x t 1  max { i :i ∈ J u} for all v  U . ∂ l t 1 , v 

Here

t−1

l t , v =∑=t F ' t ,  v f 

Ft ,  is an n  n dimensional matrix-func-

and

0

tion, being the solution of the following problem [11]:

Ft ,   1  Ft ,  f x  , Ft , t  1  E Second order optimality conditions for control problems can be found e.g. in Gabasov R;Tarasnko N. F [5].

In all these papers one-stage optimal control problems have been investigated. But some applied problems from economy, military work, chemistry are inherently multistage problems. This means that there are several stages each being characterized by its own equations, controls, phase coordinates, constraints, etc. Usually these stages are connected by each other by additional conditions. Here problems will are considered where these connections are given by switching points which are controlled by a given parameter. These multistage processes will be called step control systems (or discrete systems with varying structure).

For example, consider a rocket with two typs of engines that work consecutively. The work of the second engine depends on the first one. Morerover, the rocket moves from one controlling area to a second one that changes all the structure (controls, functions, conditions, etc.).

This paper investigates step discrete control systems and aims to find necessary and sufficent optimality condition of first and second order.

3

2. Necessary optimality condition Consider a controlling process, which is described by the following discrete system with varying structure: 3

Minimize Su , v     i x i t i 

(9)

i 1

subject to

x i t  1  f i t , x i t , u i t , t  Ti  t i 1 , t i 1  1,..., t i  1, i  1,2,3,

(10)

x 1 (t 0 )  g1 (v1 )   x i ( t i1 )  g i ( x i1 ( t i1 ), vi ), i  2,3

(11)

u i t  U i  R r , t  Ti , i  1,2,3.

(12)

Here, v i , i  1,2,3 are q-dimensional controlling parameters and Vi  R q , i  1,2,3 , i. е.

v i  Vi , i  1,2,3.

(13)

In this problem, g 1 : R  R are given at least twice continuously differentiable vector-valued q

n

functions, g i : R  R  R n

q

n

are given at least twice continuously differentiable vector-valued

functions, i=2,3, f i : R  R  R  R are given continuous vector-valued functions, which are n

r

n

at least twice continuously partially differentiable with respect to x, i : R  R are given at least n

twice continuously differentiable functions, i=1,2,3, u i (t): R

Ui  R r

are controls and

v i  Vi  R q are controlling parameters. The sets U i , Vi , are assumed to be nonempty and bounded. In this problem we consider a discrete control problem with three steps. Equations (10) describe the systems behavior within each of the steps. Here, u t  describes the control, x t  the trajectory of the system, and t the time. Equations (11) are used to fix the move of the system from one to the next step. To do this switching parameters v i are introduced and conditions (11) guarantee that the trajectory x i 1 is „continued“ by x i . In problems without these switching points, as e.g. in one-step problems, we can apply any part of the conditions obtained by using Pontryagin’s principle to derive optimality conditions. Hence, as it will be seen below, such conditions appear as part of the necessary optimality conditions for problem (9)—(13), too. But new conditions for the switching points need to be added.

4

A tuple

u t , u t , u t , v , v 1

2

3

1

2

, v 3   u t , v  with the properties (12) and (13) is

called admissible control and the corresponding solution x 1 t , x 2 t , x 3 t   x t  of the sys-



tem (9)-(13) is called admissible trajectory. For the fixed admissible control u

0

t , v

o



we in-

troduce the following notation:

H i t , x i , u , i0   i0 t   f i t , x i , u i  , /

 u H i t   H i t , x 0 t , u i t , i0 t   H i t , x i0 t , u i0 t , i0 t , i

i

H i t  Hi t , x i t , u 0i t ,  0i t  0  ,  v g1 v1   g1 v1   g1 v1  x i x i 0

1

 v g i x i01 t i1 , v 0i   g i x i01 t i1 , v i   g i x 0i1 t i 1 , v i0 , i  2, 3 i

L1 v1 , 10 t 0  1  10 t 0  1 g1 v1  , ´

 L 2 x 1 t 1 , v 2 , 20 t 1  1  20 t 1  1 g 2 x 1 t 1 , v 2   L 3 x 2 t 2 , v 3 , 30 t 2  1  30 t 2  1 g 3 x 2 t 2 , v 3  , where the unknown functions  0i are defined below in (16). Тhеоrem 1 12 : If the sets

f i t i x i0 t , U i    i :  i  f i t i x 0i t , u i , u i  U, i  1,2,3 g1 V1    4 :  4  g1 v1 , v1  V1  g i x i01 t i 1 , Vi    i :  i  g i x i01 t i 1 , v i , v i  V, i  2,3





are convex then for optimality of an admissible control u 0 t , v 0 in problem (9)-(13) it is necessary that the following conditions are true: a) Discrete maximum principle for the control u i t , i  1, 2,3 : 0

t i 1

 u   H i t   0 , for all u t  U , i  1,2,3, t  T

t  t i 1

i

it

i

i

b) Discrete maximum principle for the controlling parameter v i , i  1, 2,3 . 0

 max L x







max L1 v1 , 10 t 0  1  L1 v 01 , 10 t 0  1

v1 V1 v i Vi

i



0 i 1 t i 1

,vi , i0 t i 1  1  Li xi01 t i 1 ,vi0 , i0 t i 1  1,i  2,3

5

It is known that, for continuous-time systems, an optimal control satisfies Pontryagin´s maximum principle without the restrictive convexity assumption. But in the discrete maximum principle this does not hold in general unless a certain convexity is imposed a priory on the control system. A clear explanation of this phenomenon is given in Pshenichnyi’s book [15], where it is shown why discrete systems require a convexity assumption for the validity of the maximum principle, while continuous-time systems enjoy it automatically due to the socalled “hidden convexity“. In the control problem one of the methods to get necessary optimality conditions is the increment formula. For this we have to calculate the increment formula, to find a conjugate system for the corresponding problems and to use an anolog of needle variations in the continuous case. Then the rest of the increment formula can be estimated using the step method. Proof: Using Taylor’s formula we can write the increment of the functional at an arbitrary

  as Su t , v   Su t , v   Su

admissible pair u t , v 0

0

0

, v0 

  3 t 1 3 t 1  i x 0i  t i    0 H i t  0    i t i 1 x i  t i      u H i t    i t  1  x i t  i 1  x i i 1 t  t i 1 t  t  x i   3

i

i

i

i1

i1

  v L1 v10 , 10 t 0  1   v L 2 x 10 t 1 , v 02 , 20 t 1  1   v L 3 x 02 t 2 , v 30 , 30 t 2  1  1

3



2

t i 1



i 1 t  t i1

3

 u Hi t  1 3  2  i x 0i  t i  1 3 t 1  2 H i t  x i t    x i t i  x i  t i     x i t  x i t   x i 2 i 1 x 2i 2 i 1 t  t x i2 i

i

i 1

 v L2 x10 t 1 , v 02 ,  02 t 1  1 L2 x10 t 1 , v02 ,  02 t 1  1  x1 t 1    x 1 t 1   x1 x1 2

1  2 L 2 x10 t 1 , v 02 ,  02  t 1  1 L3 x 02 t 2 , v03 ,  03 t 2  1  x1  t 1  x1 t 1    x 2 t 2   2 x 12 x 2 

 v L3 x 02 t 2 , v03 ,  03 t 2  1 3

x 2

 2 L3 x 02 t 2 , v03 ,  03 t 2  1 1 x 2  t 2   x2 t 2  x 2  t 2   2 x 22

 1 u 0 , v 0 ; u , v  ,

(14)

where by definition



1 u 0 , v 0 ; u, v   oi x i t i  3

i 1

2

   o  x t    o  x t    o  x t   t 1 1

t  t0

2

4

1

6

t 2 1 t  t1

2

5

2

t 3 1 t t2

2

6

3

 2  u H i t  1 3 t 1 2 2 1    xi t  x i t   o7 x1 t 1   o8 x 2 t 2   x2  t 2   2 2 i 1 t  t x i 2



i

i

11



 2  v L 2 x 01  t 1 , v 02 ,  02 t 1  1 2

x

2 2

 



 2  v L3 x 02 t 2 , v 03 ,  03 t 2  1 1 x 2 t 2   x 3 t 3  x 3 t 3  . 2 x 32 3

(15)

Here o i , i  1,..,8 are defined by the expansions

i x 0i t i  1  2  i x 0i t i  i x i t i   i x t i   x i t i   xt i  x i t i   x i 2 x 2i 0 i





 o i x i t i  , i  1,2,3, 2

Hi t , x 0i t , u i t ,  0i t  H i t , x i t , u i t ,  t   H i t , x t , u i t ,  t   x i t   x i 0 i

0 i

0 i

1  2 H i t , x 0i t , u i t ,  0i t  2  xi t  x i t   oi  3 x i  t  , i  1,2,3. , 2 2 x i





L2 x10 t 1 , v2 ,  02 t 1  1 L 2 x1 t 1 , v 2 ,  t 1  1  L2 x t 1 , v 2 ,  t 1  1  x 1 t 1   x1 0 2

0 1

0 2

1  2 L2 x10 t 1 , v2 ,  02 t 1  1 2  xt 1   x  t   o  x  t  , 1 1 7 1 1 2 x 12





L3 x 02 t 2 , v3 ,  03 t 2  1 L 3 x 2  t 2 , v3 ,  t 2  1  L 3 x 2  t 2 , v3 ,  3 t 2  1  x 2 t 2   x 2 0 3

1  2 L 2 x 02 t 2 , v3 ,  03 t 2  1 2  x2  t 2  x 2 t 2   o8 x 2 t 2  . 2 2 x 2





Now taking i t , i  1,2,3, as solutions of the following linear difference equations 0

H i t   , i  1,2,3, t  Ti  x i  1 x10 t 1  L 2 x10 t 1 , v 02 ,  02  t 1  1  0 1 t 1  1     x1 x1  0 0 0 0   2 x 2 t 2  L 3 x 2 t 2 , v3 ,  3 t 2  1 0 2 t 2  1     x 2 x 1 0   x t  30 t 3  1   3 3 3  x  i0 t  1 

the increment formula (14) reduces to a simpler one:

7

(16)

3 t 1   Su  t , v   S u 0 t , v 0   Su 0 , v0      u Hi t  i 1 t  t   i

i

i 1

  v L1 v10 , 10 t 0  1   v L 2 x 10 t 1 , v 02 , 20 t 1  1   v L 3 x 02 t 2 , v 30 , 30 t 2  1  1

2

 u Hi t  1 3  2 i x 0i  t i  1 3 t 1  2 H i t   x i t    xi t i   x  t    x  t    i x 2 x i t  i i x i 2 i 1 x 2i 2 i 1 t  t i

t i 1

3



i



i

i 1 t  t i1

 

3

i 1

 v L2 x 01  t 1 , v 02 ,  02 t 1  1 2

x1  v L3 x 02 t 2 , v03 ,  03 t 2  1 3

x 2

1  2 L 2 x10 t 1 , v02 ,  02 t 1  1 x1  t 1   x1  t 1  x 1 t 1  2 x12

 2 L3 x 02 t 2 , v03 ,  03 t 2  1 1 x 2  t 2   x2 t 2  x 2  t 2   2 x 22

 1 u 0 , v 0 ; u , v  .

u

Let

o

t , v

o

(17)

be an optimal pair, and assume that the sets of admissible velocities are convex

along the process u t , v, x t  , i. e., the sets

f i t , x i0 t , U i    i :  i  f i t, x i0 t , u i , u  U i , i  1,2,3, g 1 V1    4 :  4  g 1 v1 , v1  V1 , g i x i01 t i 1 , Vi    i 3 :  i 3  g i x 0i 1 t i 1 , v i , v  Vi , i=2,3, are convex. Let   0,1 be an arbitrary number. Denote the increment of the optimal pair by

u i t;    u i t;    u i0 t  , t  Ti , i  1,2,3,

(18)

v i    v i    v i0 , i  1,2,3 . Then,

by

convexity,

for

each

u i t   U i , v i  V ,

t  Ti ,

i  1,2,3 ,

there

are

u i t,    U i , v i   Vi , i  1,2,3 such that  u   f i t    u   f i t , i  1,2,3 , i t:

i t

 v   g1 v10    v g1 v10  , 1 

1

 v   g i x i01 t i 1 , v i0    v g i x 0i 1 t i 1 , v i0 , i  2,3 . i 

i

The increment (18) introduces an increment of the solution x i t  which is denoted by

x t;  , i  1,2,3 . i

Using the step methods we can prove x i t;    Z11 , t  Ti  t i , i  1,2,3 . Using these esti-





mates in (17) it can easily be seen that the necessary optimality condition S u 0 , v 0  0 implies 8

the conditions of the theorem.

q.e.d.

It should be noted that the system of linear difference equations (16) is the conjugate system for the problem (9)-(13). 3. Special cases Under additional assumptions the formulas in Theorem 1 reduce to easier accessible forms. As example consider the situation where some of the controls are fixed, i. e. consider

the case

u 1 t   0 , u i t   0 , i  2,3 , v i  0 , i  1,2,3. Other cases can be treated in an analogous way. Then the increment formula in the proof of Theorem 1 takes the form:

S u u 0 , v 0   Su 1 , u 02 , u 30 , v 0   Su 0 , v 0   1

1 3  2  i x 0i t i  1 3 t 1  2 H i t       u Ht    x i t i  x i t i     x i t  x i t  t t 2 i1 x 2i 2 i 1 t  t x 2i t1 1

i

1

i 1

0

 2 L 3 x 02 t 2 , v03 , 30 t 2  1 1  2 L 2 x 01 t 1 , v02 , 20 t 1  1 1  x 1  t  x 1 t 1   x2  t 2  x 2  t 2  2 x 2i 2 x 22 t1 1

 t t 0

 u H1 t  x1  t   1 u 0 , v0 , u 1 , x1

(19)

1

where by definition,

1 u 0 , v 0 , u1   1 u 0 , v0 , u1 ,0,0 ,0  . Considering (10), (11) x i t , i  1,2,3, is a solutions following linearizing equations:

x1 t  1  x 1 t 0   0 ,

x 2 t  1 

f 1t  x 1 t    u f 1 t   1  t; u 1  x1 1

(20)

f 2 t  x 2 t    2 t ; u1  , x 2

g2 x10 t 1 , v02  x 2 t 1   x1 t 1    3 u1  , x 2

9

(21)

x 3  t  1 

f 3t  x 3 t    4 t ; u1  , x 3

g3 x 02 t 2 , v 03  x 3  t 2   x 2 t 2    5 u1  x 2

(22)

where by definition

 u f1t  x1  t   o 9  x1 t  ,  2 t; u 1   o10  x 2 t  , x 1

1 t , u 1  

1

 3 u 1   o11  x 1 t 1  ,  4 t; u 1   o12  x 3 t   ,  5 u 1   o13  x 2 t 2  . Here o i , i  9,..,13 are determined, respectively, from the expansions:

f 1 t , x 10 t , u1 t  f 1 t , x1  t , u1 t   f 1 t , x  t , u1 t   x 1 t   o9  x1 t   , x1 0 1

f 2 t , x 02 t , u 02 t  f 2 t , x 2 t , u 2 t   f 2 t , x 2 t , u t   x 2 t   o10  x 2  t   , x 0 2

g2 x10 t 1 , v 02  g 2 x1 t 1 , v t   g 2 x t 1 , v   x1 t 1   o11  x 1 t 1   , x1 0 2

0 1

0 2

f 3t , x 30 t , v 03  f 3 t , x 3 t , v   f 3 t , x t , v   x 3 t   o12  x 3 t   , x 3 0 3

0 3

0 3

g3 x 02  t 2 , v03  g 3 x 2  t 2 , v   g 3 x t 2 , v   x 2 t 2   o13  x 2 t 2   . x 2 0 3

0 2

0 3

Interpreting systems (20)-(22) as linear nonuniform difference equations in x i t  (cf. the discrete analog of Cauchy problem), respectively, we obtain the following result, using the representation formulas for the solutions to linear nonuniform difference equations 11 t 1

t 1

t 1

 t 0

  t0

x1 t    F1t ,   u f 1     F1t , 1 , u1    F1t ,   u f 1     6  t, u1  ,   t0

1

1

(23)

g2 x10  t 1 , v 02  x 2 t   F2 t , t 1  1 x 1 t 1   F2 t, t 1  1 3 u1   x1 t 1

  F2 t ,  2 , u1  ,

(24)

 t 1

x 3  t   F3t , t 2  1

g 3 x 02 t 2 , v03  x 2  t 2   F3t , t 2  1 5 u 1   x 2

10

t 1

  F3t ,  4 , u 1  .

(25)

 t 2

Here Fi t ,  , i  1,2,3, are n n  -matrix functions being solutions of the following problem

f   Fi t ,   1  Fi t ,   i  x i  Fi t , t  1  E, i  1,2,3,  where E denotes the unit matrix. Inserting (23) into (24), we derive

 g2 x10 t 1 , v02  x 2 t    F2 t , t 1  1 F1 t 1 ,   u f1     7 t; u 1  ,  t x1 t 1 1

1

(26)

0

where by definition t 1 1

 7  t; u1    F2t , t 1  1  t 0

g 2 x10 t 1 , v 02  F1 t 1 ,  1 ; u1   F2 t , t 1  13 u 1  x 1

t 1

  F2 t ,  2 ; u1  .  t 1

Now, inserting (26) into (25) the following formula is obtained:

x 3 t  

g 3 x 02 t 2 , v 03  g 2 x10 t 1 , v 02  F3t , t 2  1 F2 t 2 , t 1  1 F1  t 1 ,  u f 1    8 t; u1  ,   t x 2 x1 t 1 1

1

0

where by definition

 8 t; u1   F3t , t 2  1

g3 x 02 t 2 , v 03   7 t 2 ; u1   F3t, t 2  1 5 u1   x 2

t 1

  F3t ,  4 , u 1  .  t 2

Using the abbreviations

g 2 x10 t 1 , v 02  R 1  t,    F2 t , t 1  1 F1 t 1 ,   , x1 g 3 x 02 t 2 , v30  g2 x10  t 1 , v02  R 2 t ,   F3t , t 2  1 F2 t 2 , t 1  1 F1 t 1 ,    x 2 x1  F3t , t 2  1

 g3 x 02 t 2 , v 03  R 1 t 2 ,  , x 2

the formulas (23), (26) ,(27) reduce to

11

(27)

t 1  x1 t    F1t ,  u f    6 t ; u1    t  t 1 x 2 t    R1 t ,   u f 1    7 t ; u1   t  t 1 x 3  t    R 2 t ,  u f 1     8 t ; u1    t 1

0

1

1

0

1

2

0

4. Necessary optimality conditions using the linearizing principle If the functions f i , g i have also partial derivatives with respect to u i , v i , respectively, and the sets U i and Vi are convex, then another necessary optimality condition can be obtained using the linearizing maximum principle of Pontryagin. The proof of the next theorem is to a large extend similar to the proof of Theorem 1 and is omitted. For it the interested reader is referred to the thesis

12 . Тhеоrem 2 12 (Linearizing maximum princple). If the sets U i , Vi are convex,then ,for the opti-



mality of the pair u t i 1

1)



t  t i 1

0

t , v

, it is necessary that following inequalities hold:

Hi t  u i t   u 0i t   0 ui

for all u i t   U i , t  Ti , i  1,2,3 .

L1 v10 , 10 t 0  1 v1  v10   0 , 2) v1 for all v 1  V1 .

Li x 0i1 t i 1 , vi0 , i0  t i 1  1 vi  v0i   0, for all vi  Vi , i  2,3. 3) vi

If the sets U i , Vi , i  1,2,3 are open, also using Euler’s equation can be used to derive necessary optimality conditions:

Theorem 3 12 (An analogue of Euler equation): If the sets U i , Vi are open, then for optimality of

12



the pair u 1)

0

t , v

 , it is necessary that following equations hold

H i t   0, t  Ti , i  1,2,3 . u i

L1 v10 , 10 t 0  1 0 , 2) v1 L i x 0i1 t i1 , vi0 , i0  t i 1  1  0, for all i  2,3. 3) vi The proof is again omitted. The interested reader can find it in the thesis [12].

5. Analysis of singular control 5.1. Necessary optimality conditions using Pontryagin’s maximum principle If the first order necessary optimality condition degenerates in the sense that we have an equation for all admissible controls, the admissible controls are singular in the sense of Pontryagin’s maximum principle. Then, the first-order necessary optimality conditions should be replaced by second-order ones. To derive these conditions suitable changes in the increment formula are helpful.



Definition. An admissible control u t , v 0

0

 is called singular in the sense of Pontryagin‘s maxi-

mum principle if the following relations hold: t i 1



t  t i 1

ui t 

Hi t   0 , for all u i t  U i , t  Ti , i  1,2,3

 v L1 v10 , 10 t  1  0 for all v1  V1 1

 v L i x i01 t i 1 , v i0 , i0 t i 1  1  0 for all v i  Vi , i  2, 3 i

It is clear that the necessary optimality conditions for problem (9)-(13) are degenerate in the singular case. Therefore, they cannot detect nonoptimality of an admissible pair (u(t),v(t)). Thus, there is some need for new necessary optimality conditions. Singular cases for various control systems were studied by many authors (see e.g. [5,7]). However, to the best of our knowledge, they have not been studied for step systems (both with continous and discrete time).

13

The proof of the following theorem uses the ideas of the proof of Theorem 1. It starts with the increment formula for the objective function, too. To explain the increment formula consider the special case of Section 3 i.e. let the controls

u 2 , u 3 , v i , i  1,2,3 be fixed and u 1 ( t )  U1  t  T1 . Then, the increment formula reads as

 u Su , v     u H1 t  0

t 1 1

0

1

t t 0

1

t 1 t 1  H  t   1 t 1 t 1   f    K   , s   f  s    u 1 1    u 1 F1 t ,  u f 1    2 u 0 , v 0 , u1  . u 1 2  t s  t t  t   t x1 1



1

1

1

1

0

1

1

0

0

0

Here we used (n , n ) -dimensional matrix functions K 1 , s  and F1 t ,   defined via K 1 , s  

t 1 1

 

t  max  , s 1

t 3 1

  R 2  t,   t  t2

F1t ,  

t 1  2 H 1 t   2 H 2 t   F  t , s   R  t ,   R 1 t , s    1 1 x 12 t t x 22 2

1

 H 3 t    1 x10  t 1   2 L 2 x10 t 1 , v 02 ,  02 t 1  1  R  t , s   F  t ,    2 1 1   F1 t 1 , s   x 32 x12 x 12   2

2

  2  2 x 02 t 2   2 L 3 x 02 t 2 , v 03 ,  30 t 2  1   2  3 x 03 t 3    R 1 t 2 ,    R  t , s   R  t ,   R 2 t 3 , s ,  1 2 2 3 x 22 x 22 x 23  

and

F1 t ,   1  F1t , 

f 1  , F1 t , t  1  E, x1

g 2 x10 t 1 , v02  R 1  t,    F2 t , t 1  1 F1 t 1 ,  . x1 If the controls u 2 , u 3 , v i , i  1,2,3 are not fixed but restricted to the respective sets, the increment formula needs to be adapted by adding terms for these functions. These terms can be expressed using similar matrix functions K i , s  and Fi t ,  for i=2,...6. These formulas can be found in the Appendix. The increment formula above in the general case is obtained by adding five times the first three terms on the right hand of the formula (using the correct indices). Тhеоrem 4 12 . In the case of singular control and if the convexity assumptions of Theorem 1 are satisfied then the following conditions are necessary for optimality of an admissible singular control for the problem (9)-(13):

1)

t i 1



 t i 1

t 1  t1  u Hi t     f    K   , s   f  s   2 Fi  t,   u f i   0,    u i i u i  s t t  t   t x i  t i 1

i

i

i

i 1

i

i

i 1

14

i 1

for all u i t   U i , i  1,2,3 .

 

   0 , for all v  V g x t v K  g  x t v   0 , for all v  V , i  2,3.

2)  v1 g1 v1 K 4  v1 g1 v 0

3)  vi

0 i 1

i

0 i

i 1

0

1

i 3

vi

i

0 i 1

1

0 i

i 1

i

i

5.2. Necessary optimality condition using linearizing maximum principle If the linearizing maximum principle degenerates for the problem (9)-(13) in the sense that we have an equation for all admissible controls in the conditions of Theorem 2 then the admissible controls are called quasisingular. In that case the first order necessary conditions in Theorem 2 should be replaced by second order ones. For this we need twice continuously partial differentiability of f i with respect to controls. We drop the proof since it uses mainly the ideas of the proofs of Theorems 2 and 4. Theorem 5 12 : If the sets



V i and

U i are convex, then for optimality of the quasisingular



control u 0 t , v 0 , it is necessary that the following inequalities hold:

 f  f s   u  t  u  t  u K , s u u s  u s   f K, s f s  t i 1

1)

t i 1

 t i 1 s t i 1

0 i

i

i

i

i

i

i

0 i

ui

i

ui

i

i

t 1  t 1  f     2 H i t   2    u i t   u 0i t  Fi t ,   i u i    u 0i   t t u i x i u i   t  2 t 1   Hi t    u i  t   ui0 t  ui t   u 0i t   0, 2 t t u i i

i 1

i 1

i

i 1

for all u i t   U i , i  1,2,3. 2 0 0  g1 v10  g1 v10  0 0   L1 v1 , 1 t 0  1 K4 v1  v1   v1  v1  v1  v10   0 , 2) vi  v  0 v1 v1 v1 0 i

for all v1  V1 . 3) for all v i  Vi i  2,3 : 0 ∂ g 1  x i−1 t i−1  , v i0  ∂ g i  x i−1 t i−1  , v i  v i −v ' K i3 v i −v i0  ∂ vi ∂ vi 0 i

2

0 i

v i −v '

0

0

∂ Li  x i−1 t i−1  , v i , i t i−1−1 ∂v

2 i

v i −v i0 ≤0

Conclusion. In this paper have given some results for necessary optimality conditions for discrete control

15

systems with varying structure. These problems have important applications, one of them is the control of a rocket described at the beginning of the paper. The conditions formulated in the case of regular control include conditions being based on Pontryagin’s maximum principle, the linearizing maximum principle and one using Euler’s equations. After that we investigated singular and quasisingular controls in the sense of Pontryagin‘s maximum principle. The step control problem discussed in this paper is one possible (and from the point of view of applications interesting) generalization of control problems. Other generalizations are not investigated as e.g. problems with an unknown switching time. In the continuous-time case necessary optimality conditions can be found in [18] but in the discrete-time case they seem not to be discussed, yet. Summing up (discrete-time) step control problems rise challenging questions and need further investigation in the future. References 1..Magerramov Sh.F, Mansimov.K.B.: Optimization of a class of discrete step control systems Comput. Math. Math. Phys. 41(2001)3, 334-339. translation from Zh. Vychisl. Mat. Mat. Fiz. 41, No.3, 360-366 (2001). 2.Moyseev H.: Elementi teorii optimalnikh sistem.Moskow Nauka,1975. (in russian) 3.Propoy A.: Elementi teorii optimalnikh diskretnikh prosesov. Mokow Nauka, 1969. (in russian) 4. Mansimov, K.B., Maharramov, Sh.F.: Necessary conditions of optimality for discrete system with variable structure and rolling right end of the path.Second Joint Seminar On Applied Mathematics in IRAN Republic, Zanjan University, 2000,October 4-6, pages 53.. 5.Gabasov, R; Tarasenko, N.: Necessary high-order conditions of optimality for discrete systems. Autom.Remote Control

1971, 50-57, translation from Avtom. Telemekh. 1971, No.1, 58-65

(1971). 6.Gabasov R.,Kirillova F., Mordukhovich B.: Diskretniy prinsip maximuma. Dokl. AN SSSR1973.T 213,1, pp. 19-22, (in russian) 7. Mansimov, K.B.: Diskretnie systemi. Baku State Unversity publish.2002,134 p. (in russian) 8. Pham Khiu Shak: Ob optimal’nom ypravlenii diskretnimi sistemami s zapazdivaniem. Avtom.i Telemekh., 1970, No. 7, 40-49. (in russian) 9. Maharramov, Sh.F.: Analysis of the quasisingular control for the a step control problem. Azerbaijan republic,“Tahsil“ Society ,“Bilgi” publish. Physical, Mathematics,Earth Sciences, 2003, No. 1, 44-50. (in russian) 10. Maharramov, Sh.F.: Investigation of singular controls in one discrete system with variable structure and delay. Proceeding of Institute of Mathematics and Mechanics, Baku, 2001, vol. 14, 16916

175. 11. Gabasov R., Kirillova F.: Optimisazija lineinikh sistem. Minskiy Izd. BGU, 1973. (in russian) 12. Maharramov, Sh.F. Uslovija optimal’nosti dlja odnogo klassa diskretnikh zadach optimal’nogo upravlenija. PhD thesis, Akademija Nauk Azerbajdshana, Inst. Kibernetika, 2003. (in russian) 13. Maharramov , Sh.F.: Optimization of a class discrete step control system. French - German Polish Conference on Optimization. September 9 - 13 , 2002, Cottbus, Germany.p.53 14. Dempe, S.: Foundations of Bilevel Programming. Kluwer Acdemic Publishers. 2002. 15.Pshenichnyi, B.N.: Necessary Conditions for an Extremum, Marcel Dekker, New York (1971) 16. Mordukhovich, B.S., Shvartsman, I.: Discrete maximum principle for nonsmooth optimal control problems with delay, Cybernet.Systems.Anal.,38(2002) ,pp.255-264. 17. Mordukhovich, B.S.: Approximation Methods in Problems of Optimization and Control, Nauka, Moscow, 1988 (in russian) 18. Medvedov V.A., Rozova V.N.: Optimalnoe upravlenie stupenchatnimi sistemami. Avtomatika i Telemekhanika,1972, No. 3, 15-23 19.GurmanV.I, Theory of optimum discrete processes. Autom. Remote Control 34, 1082-1087 (1973); translation from Avtom. Telemekh. 1973, No.7, 53-58 (1973). 20.Kharatashvili G.L, TadumadzeT.A: Regular perturbations in optimal control problems with variable delays and a free right end. Sov.Math.Dokl 42, No.2, 399-403 (1991); translation from Dokl. Akad. Nauk SSSR 314, No.1, 151-155 (1990). 21.Boltayanski V.G. Diskretniy prinsip maksimuma. Moscow Nauka.,1973.(in russian) 22. Alejnikova ,Z.M., Zabello,L.E.: About an investigation of the second variation in controlled discrete systems. Dokl.Akad.Nauk.BSSR 27, 879-882 (1983). (in russian) 23.ZakharovG.K.: Optimization of step control systems. Autom. Remote Control

42, 1001-1004

(1982); translation from Avtom. Telemekh. 1981, No.8, 5-9 (1981). 24. Nikolski M.S. Ob odnoy variasionnoy zadache s peremennoy strukture.Vestnik Moskow State University, ser.Vich.mat. i kibern., 1984, No 1, 71-77 (in russian)

17

Appendix

K 2 , s  

t 2 1



t  max   ,s 1

F2 t , 

t 1  2 H 2 t   2 H 3 t   F2 t , s    R 3 t ,   R 3 t , s  x 22 t t x 23 3

2

 2 3 x 03 t 3    2 2 x 02 t 2   2 L 3 x 02 t 2 , v30 ,  03  t 2  1  R  t , s   F  t ,    3 3 2 2   F2 t 2 , s , 2 2 x 22  x  x  2 2  t 1  2 H 3 t   2 3 x 03 t 3  K 3 , s   F3 t,  F3 t , s   F3 t 3 ,  F3 t 3 ,  , t  max  ,s 1 x 23 x 23 t 1 t 1 t 1  2 H1 t   2 H 2 t   2 H 3 t    K 4   F1t , t 0  1 F  t , t  1     t    t     t   2 t     1 0 1 1 2 tt x12 tt x 22 tt x 32  R 3  t 3 ,  

3

1

2

0

3

1

2

  1 x t 1   L 2 x t 1 , v ,  t 1 1   2 2 x 02 t 2  F1t 1 , t 0  1    F1 t 1 , t 0  1  1 t 2  x 12 x12 x 22     2 L 3 x 02 t 2 , v03 ,  03 t 2  1  2 3 x 03  t 3     2  t 3 ,  1 t 2    2 t 3  x 22 x 32  2

t 2 1

K 5   F2 t , t 1 1 t  t1

0 1

2

0 1

0 2

0 2

t 1  2 H 2 t   2 H 3 t   F  t , t  1     t   3 t , s   2 1 3 x 22 t t x 23 3

2

  2 2 x 02 t 2   2 L3 x 02 t 2 , v03 ,  03 t 2  1  2  3 x 03 t 3    F2 t , t 1  1  F  t , t  1     t   3  t 3 , 3 3  2 1 2 2 2  x  x  x  2 2  3

 2 H 3 t  2 3 x 03  t 3  K 6   F3 t , t 2  1 F3 t , t 2  1  F3 t 3 , t 2  1 F3  t 3 , t 2  1, t t x 32 x 23 t 3 1

2

Here by definition

g 2 x10 t 1 , v02  R 1  t,    F2 t , t 1  1 F1 t 1 , , x1 g 3 x 02 t 2 , v30  R 2 t ,   F3 t , t 2  1 R 1 t 2 ,  , x 2 g3 x 02 t 2 , v 03  R 3 t ,    F3 t , t 2  1 F2 t 2 , , x 2 g 2 x10 t 1 , v 02  1 t   F2 t , t 1  1 F1  t 1 , t 0  1, x1 g3 x 02 t 2 , v03   2 t   F3  t, t 2  1 1 t 2 , x 2

18

g 3 x 02  t 2 , v03   3  t   F3 t , t 2  1 F2  t 2 , t 1  1, x 2 Fi t , , i  1,2,3 are n , n -matrix functions and solutions of the problem f  Fi t ,   1  Fi t ,  1 , x1 Fi t , t  1  E, i  1, 3

19