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:

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堠 3

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4

젠 6

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13

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--堠 15

젠 19

21

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35

蠠 35

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() 42

()

43

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䠠 66

.

- , . , , , , , . , , , , , , .

. , .

, , , , - . .

- .

.

( ) , .

 

 


-( -

 



. 1

( ) , . , , . . .

, , .

. , .

.

1. , , .

2. (, ..) , , , .

3. , . , , .. 2 , .

2 3 , , , , , . , .

. . 1 3 T, . T . 1 , .

, ( , ), , , , . , . , , , , .

.

.

n - Un. , X Un Y . X , Y - X. Un , Un.

() ,,,... , :

(1) =Y

(1) : () , , Y.

() ( ), :

(2) A(+Y)=+Y

(3) (ℷ)=ℷ(), ℷ-

, , .

.

n - = - . X1∈X , , , ∈1, =∈1. 1, , =.

, , .

- α - [1] [2]  , -∞ +∞, dx , ( α =0), 2.

x2

3

dx


2 x1

R1. , R1 , =, .. x∈R1, =∈R1.

ℷ , , .. =ℷ, R1 - , ∈R1 :

(4) =ℷ

≠0, (4) , ℷ - .

(4) , =I. :

(5) (-ℷI)=0

(5) , :

(a11-ℷ)x1+a12x2+...+a1nxn=0;

(6) a21x1+(a22-ℷ)x2+...+a2nxn=0;

.........................

an1 x1+an2x2+...+(a nn-ℷ)xn=0;

(-ℷI) (6) . . . (6) ( ) , :

(7) det(A-ℷI)+a0ℷn+a1ℷn-1+....+an-1ℷ=0

(6), :

(8) (-ℷiI)=0

, det(A-ℷiI)=0

i, . .

:

1. ( ), .

2. , , .

, , n- .

.

n - (1),...,(n), - - ;

x(1) n n

(9) = ..... = x)i) ; (x(1),...,x(n))=(x(i))

x(n) 1 1

, , .

, .

n

(10) =((i)) =((1),...,(n))

1

n .

.

) =, :

x(i)=y(i)

x(1) y(1) x(1)+y(1)

) += ...... + ...... = ........... - .

x(n) y(n) x(n)+y(n)

) - z, , +z=.

)

x(1) αx(1)

αx[3] =α=α ....... = .........

x(n) αx(n)

.

x1 y1

= 2 = 2

x3 y3

. :

(11) T=T=11+22+33

:

(12) = =(T)½ , - .

.

.

(13) 1, 2, 3,..., n

() (13) - , .

. .

=(1, 2) =(1, 2) - . , , x1= x , 1 =0 (.3)

2

y2 y

α x

y1 1

α

T=11+22= * cosα

:

α=arccos(xTy/ x y )

││=1 T , 90○, ..

T=0.

.

.

m*n , m- n-,

a11 .......... a1n

A= ...................... =[aij]

am1 .......... amn

m=n, .

=[ij] =[ij] (=) , ij=ij ij.

n- , m - Y:

(1) :→Y

,

(2) Y=-, Y.

, :

(3) (1+2)=1+2, (ℷi)=ℷ

(3) , i j :

n ___

(4) (i)= ∑ aijx(j), i=1,m , ij -

j=1 ____ ___

ij, i=1,m; ;j=1,n :

a11......a1n

A= ................ = [aij]

am1......amn

.

= :

y(1) a11......a1n x(1)

(5) .... = ............... * .....

y(n) am1......amn x(n)

. .

, α- .

(6) α=[α ij ]

α .

.

= v= - =[aij] =[ij] m*n.

, ∈ +v∈Y

(7) +v=+=(+)

(+) , :

(8) +=[aij]+[ij]

, .

.

X,Y,Z- m, r, n =, z= - Y, Y Z, =[kj] A=[aik] m*k k*n . z.

(9) Z=Cx=A(Bx)=ABx

= n*n .

n ___ ___

(10) ij= ∑ ikkj , i=1,n , j=1,m

k=1

(10) ij i- j- , :

a11...a1k 11...1m

(11) = ............ * .............

an1...ank k1....km

.

=[aij] - m*n. T=['ij] m*n, , .

'ij T ij :

(12) 'ij=ji

.

.

, aij det A.

det A :

1) ℷ det A ℷ;

2) det A ;

3) , det A=0;

4) , , det A ;

5) , det A=0;

-.

.

(13) det (A-ℷI)=a0ℷn+a1ℷn-1+...+an-1ℷ an=0

(14) a0An+a0An-1+an-1A+anI=0[n*n]

.

, n*n, -1 , :

(15) *-1=-1*=

= - =[xij]. =-1. -1 .

(16) -1=(1/detA) [Aij]T , ij -

.

= , detA≠0. , , .

.

=, .

- ; - . :

(17) =-1**

, .

ℷ1 0 0

(18) diag[ℷ1 ℷ2 ......ℷn ]= 0 ℷ2 0

0 0 ℷn

m*n :

m n

(19) ││= ∑ ∑ │a ij │

i=1 j=1

, t.

:

a11(t) a12(t) ...... a1n(t)

(20) (t)= a21(t) a22(t) ...... a2n(t)

............................

am1(t) am2(t) ..... amn(t)

.

(t) (t) :


da11(t)/dt da12(t)/dt ...... da1n(t)/dt

(21) (t)= dA(t)/dt = ............................................................. =

dam1(t)/dt adm2(t)/dt ...... damn(t)/dt

=[daij(t)/dt]

1.3 .

- , , .

, . , , . , .

:

v1, v2,...- .

- .

(1) A(1)(v1,..., vn)=0 A,

A(2)(v1,...., vn)=0 A(j), j=1,..., m

..................... vi , i= 1,..., n

A(m)(v1,..., vn)=0 m n -

A (1) - , (), - (), A .

- , , A .

A t0 S(t0), -

(U[t0, t], Y[t0, t]), , Y[t0, t] U[t, t] S(t0).

, t0 , , , α, ∑ , (1)α, (2) - - (3) - U[t0,t]] y[t0,t].

S(t0) - t0.

[t0,t]-

--.

- , U, - [t0,t] - ∑, R[U], R[y]- .

(2) y(t)=A (α;U[t0,t]) ∀ t>t0

A- α U[t0,t]

U R[U], R[]

(2) - . - - .

(2') [t0,t]=A (α,U),

A , (t) [t0,t]

U[t0,t],[t0,t]

- - (2), U[t0,t] [t0,t]

- α ∑.

(2') :

R[y]= { A(α,U)│ α∈∑, U∈ R[U] }

:

- - - (2')

: (1) (U[t0,t],[t0,t]), (U,) ( U∈R[U], ∈R[y]),

(U,y)=0, (U,y) (2') , α0 ∑ ,

(3) = A (α0,U[t0,t]),

(2) (U,y), (2') α, ∑ [t0,t], - A.

:

, ∑ A , : α ∑ ( A t0) U[t0,t] A, 䠠 t α U[t0,t], U y , t0, .. t0 y(t) t>t0, α U[t0,t].

. .

- - (U[t0,t],[t0,t]), (U[t,t1], [t,t1]), t0<t≤t1, U[t,t1] [t,t1] U[t0,t], [t0,t] . α ∑ -, α.

(UU',yy') - -, - - (2') α=α0. : yy'= A (α0,UU'), U',y' [t,t0]

, (U',y') - - (2'), , Q α ∑, y=A(α;U') α Q.

A t α Qt (α0,U) α0- t0 , , A.

A t S(t).

, S(t0) A, - - :

(4) y(t)= A(S(t0);U[t0,t])

S(t0) , S(t) ∑, .. t, R[S(t)]=∑.

, , S(t) S(t0) U[t0,t] S(t) S(t0) U[t0,t] - -:

y(t)= A( S(t0); U[t0,t]), α0=S(t0) (4)

S(t) S(t0) U[t0,t] .

(5) S(t)= S (S(t0);U[t0,t]),

S- ∑.

(5) - - (4).

∑ . ∑ , , .

∑ , A .

∑ , .

- t y(t) . -- (4), -- , y[t0,t] S(t0) U[t0,t] .

.


U1 1 U1 S 1


Uꠠ ꠠ Uꠠ

-

1.4

.

:

(1) Ś(t)= A(t)S(t)+B(t)U(t)

(2) (t)= C(t)S(t)+D0(t)U(t)+D1(t)U(1)(t)+...+D(t)U()(t)

.

A- [n*n]

B- [m*n]

C- [L*m]

D- [L*m]

- , - :

L+Kŷ=Mu, L,K,M- , u, - , ŷ - .

- -.

- - , , .

(2) L(p)y=u, L(p)=anpn+...+a0, an≠0, R.

, . , , .

R- , - (2), :

n t

(3) y(t)= ∑ y(ℷ-1)(t0-)ℷ(t-t0)+ ⌡ h(t-ℰ)U(ℰ)dℰ t≥t0,

ℷ=1 t0

堠 h(t)=Z {1/L(S)}= R

(4) H(S)=1/L(S)= R,

ℷ=Z-1{(anSn-ℷ+...+aℷ)/L(S)}, ℷ=1,...,n

1,...,n

L(p)ℷ(t)=0, ℷ=1,...,n

ai, i=1,..,n (2) x(t0-) - R t0-.

, , :

Sn-1/L(S),...,S/L(S),1/L(S)

x(t0) :

(5) x1(t0-)=any(t0-),

x2(t0-)=any(n-1)(t0-)+a1y(t0-)

....................................................

xn(t0-)=any(n-1)(t0-)+...+an-1y(t0-)

y(ℷ-1)(t0-) (3) , x(t0-), (3)

t

(6) y(t)=<(t-t0),...,x(t0-)>+ ⌡ h(t-ℰ)U(ℰ)dℰ, t≥t0

t0

h- R

(t)=(1(t),...,n(t)); :

ℷ(t)= Z-1{ (anℷn-1+...+ aℷ)/L(S) },

<(t-t0), x(t0-)>

(t-t0) x(t0-).

(6) - - R.

.

, :

(11) x(t)= A(t)x(t)+B(t)U(t),

A(t)- n, , t; B(t)- [n*r]; x(t) - , U- .

A(t) n, - . :

(12) X= A(t)X(t), X(t0)=C,

C , :

(13) X(t)= (t,t0)C

, (12), (11). , (13) n n (11).

th. A(t) n, . (t,t0) n,

(14) d/dt (t,t0)=A(t)(t,t0), (t,t0)=I

(15) x(t)= A(t)x(t), x(t0)=x0,

x(t,x0,t0),

(16) x(t,x0,t0)=(t,t0)x0 ∀t, ∀x0

(t,t0) .

(16) : (t,t0)

, x0 t0 x(t) t.

.

t

1. t ⌡ A(Ʈ)dƮ A(t) ,

t0

t

(t,t0)= exp ⌡ A(Ʈ) dƮ

t0

(t,t0) (11), (14), :

t

(17) det (t,t0)= exp ⌡ a(Ʈ) dƮ ,

t0

n

a(Ʈ) ≜ ∑ aiƮ(Ʈ) ≜ trA(Ʈ).

i=1

2. -. , () .

f(A)= ∑ CiAi ,

0

n . =exp{At}

n

(18) (t)= eAt= ∑ eℷitFi ,

i=1

n

F=Ϡ (A-ℷiI)/(ℷi-ℷj)

j=1

j≠i

3. q=Aq, , , .

(19) (t)= eAt≜ ∑ Aiti/i!= I+At+A2t2/2!+...

i=1

.

4. n ℷi .

.

(20) A= KK-1 ,

- , K≜[K1,K2,...,Kn], :

, (20),

f(A)=Kf()K-1

(21) (t)=KetK-1

, ℷi, exp{t}

eℷ1t......0

et=

0......eℷnt

, .. . .

5 (19)

p-1

(22) (t)= ∑ Ai ti/t!+Rp

i=0

t, e.

6. , .

- -

U=[U1, U2,...,Um]T

x=[x1,x2,...,xm]T

q=[q1,q2,...,qm]T

( )

(23) q(t)= Aq(t)+Bu(t)

(24) x(t)= Cq(t)+Du(t)

n- :

(25) q(t)=Aq(t)+bu(t)

(26) x(t)=CTq(t)+du(t)

(27) q1 = a11 a12 q1 + b1 U; n=2

q2 a21 a22 q2 b2

(28) x=|C1 2| q1 + dU

q2

, (25) n

(29) q = a11q1+a12q2+b1U;

q = a21q1+a22q2+b2U.

(30) x= c1q1+c2q2+dU

.

xv(t), Uv(t)

(31) Uv(t)=U(V)dV δ(t-V)

U(V)dV -

δ(t-V)- t=V

, g(t-V), U(V)d .

(23), (24), :

t

(32) q(t)= (t)q(0)= ⌡ (t-Ʈ) BU(Ʈ)dƮ= q(t)+q(t)

0

Uδ(t)=δ(t)

t

(33) qδ(t)= ⌡ (t-Ʈ) bδ(Ʈ) dƮ

0

δ(Ʈ), Ʈ=0, -0

(t)b , t≥0

(34) qδ(t)=

0, t<0

(34) (26)

(35) q(t)=xδ(t)=CTqδ(t)+dUδ(t)= CT(t)b+dδ(t) t≥0

xδ(t), (31), ().

(36) xv(t)=U(V) dV g(t-V)=U(V) dV[CT(t-V)b+dδ(t-V)]


U(t)=U(V)dV δ(t-V)

U

x(t)=U(V)dVq(t-V)


V Ʈ=t-V

t

.

.

A h(*) .

, h.

(37) H(S) ≜ ⌡ e-st h(t) dt

-∞

H A.

, , ,

, q=0.

H(S);

(i,j)- hij(t), .. i- , j- t=0.

- , H(S)- . y U,

(38) Y(S)= H(S) V(S)

Y V - y U.

H(S) g(t), .

1.5 .

, , , ,

() . , .

.

.

Y

(1) x= Ax + Bu;

(2) y= Cx + Du,

A,B,C,D (n*n), (n*r), (p*n) (p*r)-

.

, ࠠ ( ) (kT,(k+1)T), k=...,-1,0,1...

 

S

- (1),(2)

 


U y


.1

1 , , U Y.

α(t) , α0

α0(t)=α(kT), kT<t≤(k+1)T

, T , T- . {Uk}, Uk=U(kT+).

T , {xk}, {yk}, xk= x(kT+), yk= y(kT+), x(t), y(t) t.

{xk},{yk} . xk+1 yk+1 xk Uk . :

(3) F=exp AT,

T

(4) G=( ⌡ [exp(AƮ)]dƮ)B,

0

(5) xk+1= Fxk+Cuk

(6) yk+1= Cxk+1+Duk+1

(5),(6) , , {uk}, {xk}, {yk} . A,B,C,D , .

(5) xk x0 {Ui}r-1

k-1

(7) xk=Fkx0+ ∑ FiGUk-i-1, k=1,2,3,...

i=0

.

, t1,t2 . .

t= nT- , n- , T- , .

f[nT]

f[nT]

f(t)-

堠  , t=nT+ℰT (0≤ℰ≤1). 젠 頠 f(nT+ℰT)

-4T -3T -2T -T 0 T 2T 3T 4T nT

, ℰT, (ℰ+1)T, (ℰ+2)T,.... . f(nT+ℰT)=f[nT,ℰT]

(8) f (n-1)T,T = f[nT,0]

.

Δf[n]=f[n+1]-f[n] (9) f[n]

Δ2f(n)=Δ f[n+1]- Δf[n]-

Δkf(n)=Δk-1f[n+1]- Δk-1f[n]- -

l :

l

(10) f[n+l]= ∑ (kl) Δkf[n]; (kt)=l!/k!(l-k)

k=0

.

, x[n] K:

(11) [n, x[n], Δ x[n],.., Δkx[n] =0, . (11) :

(12) [n,x[n],x[n+1],x[n+2],...,x[n+k]=0, K.

.

(13) Δ3x[n]+ Δ2x[n]+2Δx[n]+2x[n]=f[n]

(13) x[n+3]-2x[n+2]+3x[n+1]=f[n], m=n+1, :

(14) x[m+2]-2x[m+1]+3x[m]=f[m-1]

, (13) .

x[n], , . () , () x[n+k], . :

(15) x[n+K]= F[n,x[n],x[n+1],...,x[n+k-1]]

n=n0: x[n0]=x0, x[n0+1]=x1,..., x[n0+K-1]=xk-1

(15) n=n0+K. x[n0+K], x[n0+K+1], x[n0+K+2] x[n] n≥n0+K.

(15) x[n]= ℰ[n,x0, x1,...,xk-1].

(15) C0,C1,..,Ck-1

(16) x[n]=ℰ[n,C0,C1,...,Ck-1]

:

(17) a0[n]Δrx[n]+a1[n]Δr-1x[n]+....+ar[n]x[n]=f[n]

r≥K, f[n], a0[n], a1[n], ... ,ar[n] - . , f[n]≠0, .

ℰ1[n], ... , ℰl[n] :

x[n+K]+b1[n]x[n+K-1]+ ... +bk[n]x[n]=0,

l

ℰ[n]= ∑ Ciξi[n], (i=1,2, ... ,l) - ,

i=1

.

.

n≥n0 ℰ1[n],...,ℰk[n] , :

k

ℰ[n]= ∑ Ciℰi[n]

i=1

:

x[n+K]+b1[n]x[n+K-1]+ ... +bk[n]x[n]=f[n]

ψ[n] -, ..

k

x[n]=ψ[n]+ ∑ Ciℰi[n]

i=1

Ci - , Ei[n] - , :

W(E1[n0],...,Ek[n0])≠0 ().

Z - .

S

 

..

 


U y t


. 3.

, , .3, Z-. ( .3 U ).

Z-. U(0;∞) U Z :

(18) U(z)=Z(U)= ∑ U(nT)Z-n ,

n=0

- .

: U kT, (18) .

U(nT)=U(nT+), n=0,1, ...

,.. , , 0 t<0, , U(nT-) U(nT+).

: z-

1(t) 1/(1-z-1)

[4]  e-αt 1/(1-z-1e-αt)

(18) U(z) z-1. z |z|=Ru,

Ru=lim SVp √ |U(nT)|

n→∞

, .

U ,

U= ∑ U(kT)δ(t-kT)

k=0

U(S)= ∑ U(kT)e-srT

k=0

(18) , ,

U(z)|z=esT =U(S)

Th. , . 3. H(z) Z- h. U, t=0.

:

(19) Y(Z)=H(Z)U(Z) , |Z|>max(Ru,Rk)

(19) Y(S)=H(S)V(S), , U . H(Z) , Z-.

(20) H(Z)U(Z)= ∑ ylz-e=Y(Z), |Z|>max(Rh, Ru)

l=0

{y(kT)}, .. .

Z-.

1. .

Z(αf)=αZ(f ) ∀ α, ∀|Z|>Rf

Z(f+g)=Z(f)+Z(g) ∀|Z|>max (Rf,Rg)

2.

f(nT)=1/2∏j ⌡ F(Z)Z-1 dZ, n=0,1,...,

- , |Z|=R>Rf.

3. .

f(0+)= lim F(Z)

Z→∞

4. .

F(Z) Z- {f0,f1,f2,...}, Z-1F(Z) Z- {0,f0,f1,f2,...}.

1.6. .

: : , .

:

(1) x=Ax+Bu

(2) y=Cx+Du

A,B,C,D- (n*n), (n*r), (p*n) (p*r)- ;

x- n- , ;

u- r- , - p- .

, Ӡ , A B x0 t0 u[t0,t0+T], x0 0 t0+T.

. , (1) , 0∈ℰN x0 t=0 T(T>0) U[0,T]

, :

(3) x(T;x0;0;U[0;T])=0

. 1 , (1), , U[0,T] , :

x(T;x1;0;U[0;T])=0

.

. , , , , , . , , Y[0,t] .

: , (1) (2) , , >0 (0), Ѡ Y[0,t] , x(0).

h: , Y (1), (2) , , np =[* ,* * ,..,*(n-1) * ] ℇ . ( *, *,. , ,. . )

.

h: Y , (1), , ,,..,B(n-1) Q≜[,,...,(n-1)] Y. . , . , :

(6) =+u

n, -n- , u- .

k⋜ n-1, , (6), .

, =y, , (-1)=J, J- . =(-1), (6) :

(7) y=Jy+eU

Th. , J=diag(ℷ1,...,ℷN). , (6), , e=-1 .

1.7. .

( ), , ( ), , .

:


t t

; . U(t) ( ).

, . , .

, , , .

, . . .

, , , .

( ) . . , . , : . , . () . U1,...,Um , :

(1)     xg=x(∞)=lim x(t)=f(U1,...,U v)

t→∞

(∞).

. , .

, , , ( )

(2) xn +An-1 xn-1+...+A1 x+A0 x=Bm Um+...+B0 U

U(t)- , x(t) .

x=q1, x=q2, xn-1=qn :

(3) q(t)=Aq(t)+Bu(t)

x(t)=cTq(t)+du(t)

C .

:

us(t)=uδ(t)= 0, t<0

u0, t⋝0

δ(t) :

δ(t)≜ 0, t<0

1, t⋝0

, u(t) .

u(t)


1.


t

. , s(t) :

xs(t)≜q us(t)=q U0δ(t) (4)

h(t) :

(5) h(t)≜xs(t)/U0=q(t)

, . .

.

. , u(t) 2


1/u u→0


u u

2 3

, u . 2, 3 , . (t) :

(6)     x↑(t)≜q*u↑(t)=q*A*δ(t)

(* - u(t) q(t) ); δ(t)- ; - u↑(t). q(t) :

q(t)≜ x↑(t)/A=q*δ(t)

q(t) , , .

( 3) ,

, . : .

, , . , .


x(t)=cu(t) 렠 x(p)=G(p)U(p)

x(t)=f{u(t)}

4

, , . , ( 4) :

x(t)=Cu(t) x(t)=F{u(t)}

- ,F , .

.

u u u


) t ) t ) t

u

-

-

. 5

) t

, 5, , , t, . , t , .. .

, 5. , , . , , . . - , , .

.

, , , .

, , .

, 蠠 .

- , - , .

, (, , ) , , , , . , .

1.8. .

, , . , , . , .

X,Y,Z,.., x, y, z,...,. , . =(ω).

, , , .

() .

, , , , . .

G () n - m - n- m- , ( 1)

 

 
g1 x1

g2 x2

X G

gn xn

 

GT=[g1,g2,...,gn] XT=[x1,x2,...,xn]

X,Y,Z, . n x1( ), x ( ),.., x ( ) . , :

(1) XT(ω)=[x1(ω),...,xn(ω)]

, . , , -, , .

() .

. , . , . , .

[] , :

(2) mX=M[X]=⌡ xf(x)dx

-∞

f() - , - .

, , - :

n n

(3) M[X]=x ∑ pk δ(x-xk)dx= ∑ xk pk

k=1 k=1

k , pk- , .

(3) , , , .

, , ; , .

, f()dx . Y, Y=Ψ(x) , Y k=Ψ(k) pk; Y=Ψ() (3).

n

(4) M[Ψ(x)]= ∑ Ψ(xk)pk

k=1

- , , Y=Ψ() Ψ(x) f()d. (4) :

(5) M[Ψ(x)]= ⌡ Ψ(x)f(x)dx

-∞

, (5) Ψ()=n .

( ) , n- . αn ..

(6) αn=M[Xn]= ⌡ xn f(x)dx

-∞

, , .. :

* * * * **** x1

0 m

* *** *** * x2

0 m

2 , 1, 2. M[x1]=M[x2]=m, , 2 , 1.

, , .

D[]. :

n

(7) D[X]= ∑ (xk-M[X])2 pk

k=1

, , . (6) , .

, (6) :

(8) D[X]=M(X-M[X])2= ⌡ (x-M[X])2 f(x)dx

-∞

○.

(9) X○=X-M[X]

n- n- ○, ..

(10) μn=M[(X○)n]=M[X-M[X]n]= ⌡ (x-M[X]n) f(x)dx

-∞

(8), (9) , . , , , , . .

(11) δx=√ D[X] = √ μx

.

, , . n- - 1, 2,...,n

k1+k2 +,...,+kn , 1, .. , n

(12) αk1, k2,..., kn=M[x1k1, x2k2,..., xnkn]

k1, k2,..., kn , 1 ,..,n (x1○)k1(x2○)k2..(xn○)kn ..

(13) μk1, k2,..., kn=M[(x1○)k1(x2○)k2...(xn○)kn]

X

(14) α0,..,0,1,0,..,0=M[(x1)○...(xi-1)○ xi (xi+1)○...(xn)○=M[xi]

, , n- , .

, , ..

(15) M[X]T=M[x1]...M[xn]

, i, i. . (13) :

(16) μ1,1=M(xi○yj○)

ij.

, rij, Kij i, j . . .

Kij

(18) rij= √D[xi]D[xj]

1, 2,.., n. , :

k11 k12 ......k1n

k21 k22 ......k2n

(19) K= ...................... =M[X○(X○)T]=[M[Xi○Xj○]]

kn1 kn2 ......knn

. , ij=ji, .. :

(20) T=

, , ..

(21) Y=

(21) Y y=xT (22)

.

, , , 1,...,n, , . n- , , . 1,..., n, :

X1 __ __

(23) X= ....... =(X1,...,Xn)T

Xn

:

x1 __ __

(24) X=M[X]= ...... =(x1,...,xn)T

xn

, :

(25) δ2xi=M[(xi-M[x])2] , i=1,...,n

(26) cov(xixj)=M[(xi-M[xi])(xj-M[xj]) ,i, j=1,...,n, i≠j

:

(27) Pxx=M[(X-M(X))(X-M(X)T)]

. .

.

, , . , , , , , , . , , , t, .

t- , . ω=ω0, (t,ω0) t. , .. , .

t, .. t=tk, , , (tk,ω).

(t), n- : Fn(x1,...,xn; t1,..,tn), n 1,...,n t1,...,tn, fn.

. , , ;

(1) M[X(t)]= ⌡ xf1[x,t]dx=mx(t)

-∞

(2) D[X(t)]= ⌡ [x-mx(t)]2f1(x,t)dx=D1(t)

-∞

:

∞ ∞

(3) Kx(t1,t2)=M[X○(t1)X○(t2)]= ⌡ ⌡ (x1-mx(t1))(x2-mx(t2))

-∞ -∞

f2(x1,x2;t1,t2)dx1dx2,

(4) X○(t)=X(t)-M[X(t)], .

t , (1) (2) t, (3) t1 t2. x(t1,t2) (t).

(t) 2, , , .

(t1) (t2)- t=t1 t=t2.

x(t)

m(x)

t

2

, , 2 .

m(t) (t1,t2) (t), n- , x(t1),...,x(tn) t1, t2,...,tn.

(5) mT=[m1, m2,..., mn]

K(t1,t1) K(t1,t2) ..... K(t1,tn)

K(t2,t1) k(t2,t2) ..... K(t2,tn)

(6) K= ........................................

........................................

K(tn,t1) K(tn,t2) K(tn,tn)

.

1. , ..

K(t,t)=D(t)

2. - , ..

______

K(t1; t2)=K(t1, t2)

3. :

1) K(t1, t2) ≤√ D(t1)D(t2)

4. . R(t1, t2) ;

(t1, t2)

(5)        R(t1, t2)=

√ D(t1)D(t2)

, :

_____ ______

R(t, t)=1 , R(t2,t1)=R(t1,t2) , R(t1,t2)≤1

, 0, . . , :

(6) M[X(t)=0

(7) K(t1, t2)=G(t) δ(t1-t2)

G(t) . - , 0, 0, , , .

n :

(8) X1(t),X2(t),...,Xn(t)

. (8).

Xi(t) Xi(t), :

(9) Kxixj(t1,t2)=M[Xi○(t)Xj○(t)]

, , , .

i(t) Yj(t) :

________

(10) Kxy(t1, t2)=Kxy(t1, t2)

(11) Kxy(t1, t2) ≤ √Dx(t1)Dy(t2)

(t) Y(t) , ..

(12) Kxy(t1, t2)=0

:

Kxy(t1,t1)

(13) Rxy(t1, t2)=

√ Dx(t1)Dy(t1)

Ƞ .

, :

1. .

X(t), Y(t). :

M[X(t)],M[Y(t)], Kx(t1,t2),Ky(t1,t2) Kxy(t1,t2)

:

(15) Z(t)=X(t)+Y(t)

:

(16) M[Z(t)]=M[X(t)]+M[Y(t)]

.. .

(15) (16), :

(17) Z○(t)=X○(t)+Y○(t)

(t)+Y(t). : _____ ____________

(18) Kz(t1, t2)=M[Z○(t1)Z○(t2)]=M[(X○(t1)+Y○(t2)*(X○(t1)+Y○(t2))]=

=Kx(t1, t2)+Kxy(t1, t2)+Kyx(t1, t2)+Ky(t1, t2)

, .

2. .

Y(t) (t) t, :

X○(t+h)-X○(t) 2

(19) lim M -Y○(t) =0

h→0 h

, , . X(t) , :

(20) lim X(t)=X(t○)

h→0

dX○(t)/dt=Y○(t) :

d2K(t1,t2)

(21) Ky(t1,t2)=

dt1dt2

○(t) :

(22) Kxy(t1,t2)=dK(t1,t2)/dt2

:

dn+mKx(t1,t2)

(23) Kx(n)x(m)(t1,t2)=

dt1n dt2n

x(n)(t) x(m)(t)- n- m- (t).

3. .

(Ʈ) q(t, Ʈ), Ʈ (, ). (, ) Ʈ○=,Ʈ○,...,Ʈn=, n :

n

(24) ∑ X○(Ʈi) q(t,Ʈi)(Ʈi-Ʈi-1)

i=1

Ʈi Ʈi-1≤Ʈ≤Ʈi. :

n→∞ max[Ʈi-Ʈi-1]→0

n

(25) lim ∑ X○(Ʈi) q(t, Ʈi)(Ʈi-Ʈi-1)

n→∞ i=1

,

X○(t)

q(t,Ʈ) :

b n

(26) Y= ] X○(Ʈ) q(t,Ʈ) dƮ = lim ∑ X○(Ʈi) q(t,Ʈi) ΔƮi

a n→∞ i=1

Y(t). :

b

(27) Y(t)= ⌡ X(Ʈ) q(t,Ʈ)dƮ

a

Y(t):

b

(28) M[Y(t)]= ⌡ mx(Ʈ) q(t,Ʈ) dƮ, mx=M[X(t)]

a

(28) , , (t) .

.

, . .

, n- , .

1.10 .

. , , . . - , , .

, , , , .

- : , , , .

, , .

, , .

, () .

, .

. , . . , , , .

, , .

, : , , , , , .

, (), , .

.

1. .

, .

, Q ℰ(U) U∈Q, , . ; , q.

, , U :

(1) q=q(,U)

:

(2) G=(X,Q,q)

(2) *∈X, q, .. :

(3) *={∈ՠ q(,U)=min}

.

, . Q U0, 1. .

(4) q=q(x)=q(x(1),...., x(n))

, :

≤⠠ ____

(5) fi(x(1),..., x(n)) =⠠ , i=1,m

, , , fi(x(1),...,x(n)) q() .

.

, , .. , .

. , (5) (4) (1),..., (n). (1),..., (n), .

(6) q(x(1),...,x(n))= ∑ Cjx(j)

j

:

(7) ∑ aijx(j)≤i, i=1,m

j

, , . , x(1),..., (n), (6), (7), , , . . , Q , U, ξ(U) Q.

(8) q (x)= ∑ ξ(U) q(x,U)

U∈Q

q () , (1),...,(n), (5) (8), .

, , , . , . - *Y, ={1,..., n} - ; Y - . :

(9) q=q(x,y)

*Y.

.

, .

(10) q=qV[x(t),u(t)]

(10)- ,

(11) q(t)=u(t)/x(t)

(10) , , 0 .

, ..

T

(12) J(u)= ⌡ qU[x(t),u(t)]dt

0

(5), V, :

T

(13) ⌡ H [x(t),u(t)]dt ≤ k = const

0

u*(t), V, , x=qv(u, x), x(0)=C, (12) (13).

.

x=(x(1),.., x(n)) , min q(), .. *∈X, :

(14) q(x*) ≤ q(x) ∈

* , q*() . , .. () , , ∈X. =* , q(x*)<q(x) x≠x*. , . max q(x) min q(x), . . , .. , q()=q(*), .

- , .

=* (), Qξ(*) *, ∈Qξ(x*) q(*)≤q().

q() , , q(x)

___

(15) dq(x)/dx(i)=0, i=1,n

.

q(), =((1),..., ()) - R() :

___

(16) fi(x)=0, i=1,m, m<n

(16) , q() . (16) , , q().

, (16) m - . :

(17) q(x(1),...,x())=q1(y(1),..., y()),

(1),..., () . (1),...,() q1 , .. .

.

q() x, .. , (). , , . , . .

f() - , , 1, x2 - , =ℷ1+(1-ℷ)2; 0≤1≤1; - , 1 2. z=ℷf(1)+(1-ℷ)f(2), f(1) f(2) f().

f() , , 1 2 0≤ℷ≤1 z , f(1) f(2) , .

.

.

=((1),..., (n)), q() , : ___

(20) fi(x)≤0, i=1,n

(j), .

.

q() (20) , . .

(21) Qk(x)=q(x)+rkΨ(x), r=1,2,..

Ψ()- , ( ) , rk, =1,2,....... - . (20), :

m

(22) Ψ()= ∑ (fi+(x))2

i=1

fi+() - fi(x), , fi()≤0 fi(), fi()≥0.

:

) 0 r→∞

) R=1,2,.., k-1 Qk(), xk .

.

q(x)

___

(24) fj(x)=0, j=1,m ___

(i), i=1,n. :

(25) L(x,ℷ)=q(x)

- , . , ≥0, .. , (i)>0, (i)=0.

(i)>0, , (i). - , dq(x)/dx(i)-0

(i) , .. (i)=0, dq(x)/dx(i)>0. , - :

dL(x,ℷ) =0, x(i)>0; ___

dx(i) >0, x(i)=0, i=1,n

____

(27) dL(x; ℷ)/dℷj=0, j=1,m

().

, . :

(28) q(x)=Cx+xTdx=min,

Ax≤, x≥0

.

q(), grad q(x), , q(), . grad q(x), q(), q().

.

, :

1)  q()

2)  .

().

, , , q().

- q() , , q(), .

.

, , , , q() ;

q(x)=q(x)*+½ ∑ ∑ akj Δx(k) Δx(j) X=X -X -

k j .

.

, ij , , . , q() = aij .

1.11. .

m ,..., :

a11x1+...+a1nxn=1

(1) ............................

am1x1+...+a1mxn=n ___

i≥0, i=1,m.

, , , .. m<n. , (1) :

(2) q=c1x+...+anx

:

q(x)=cx=min

(3) Ax=B (B≥0), x≥0

- m*n, m - , c n - . - (3).

.

n R1 ( n=2 - ). ___

(Ax)i≥bi, ( i=1,m)

x≥0 (j=1,n)

.

1) .

- , .

() :

(4) min<c, x> R0 ={x; Ax=b, x≥0}

x∈R0

(4)-, .. ∈R0. ', ; <C, x'> < <C, x> x - . , m . =[,D], =[a1, a2,..., am].

D=[an+1, an+2,..., an]

xT=(x, x), C=(C, C)

- , - .

2) .

: >0, =0, det()≠0, =

: =(ℷ)= x-ℷbak-1 ______

ℷ k=(m+1,n)

0

ℷ - .

>0, ℷ>0; k≥0, .. k=, k∈R○ ℷ>0. :

<C1 xk>=<C1 x>-ℷ[<C, B-1ak>-Ck]=<C, x>-ℷ∆k

∆ - =1,n, =1,m;

∆=<C, B-1ak>-Ck=<C, C>C=C-C=0

; <C1 x>=<C1x>-ℷ∆x (k=1,n)

3) .

∆ (-1ak); 3 . ___

a) =1,n ∆=0, - . _____

) ≥m+1 , ∆>0 蠠 -1ak≤0, R0 <1> R0.

) ≥m+1 i≤m, ∆>0 (-1ak )>0.

4) .

k - , <C1k>=<C1x>-ℷ0 ∆k < <C1 x>. , 1, 2,..., s, s+1, am a1, a2,..., as-1, as+1,..., am, ak , , *, .

: 19
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2012 , , .