. , : .
, .
u=u0(t), (4.1), (4.2) X0 (t) I0=I[u0(t)], (4.3).
u(t)=u0(t) + du(t), "tÎ[t0,tk], (4.4)
du(t) , .
, ,
, tÎ[t0,tk]; (4.5)
dX(t0)=0. (4.6)
t= t0 t= tk - l(t)=(l1(t), l2(t),, ln(t))
(4.7)
,
(4.8)
(4.8) (4.7)
, - l(t) :
"tÎ[t0,tk]; (4.9)
l(tk)=0. (4.10)
(4.4)
(4.11)
H 0=H(t, X0, u0, l)= - f0 (t, X0, u0) + < l (t), f0 (t, X0, u0)>. (4.12)
, du = (du1(t), du2(t),, dur(t)), dI (4.11), :
"tÎ[t0,tk]; (4.13)
, u(t)=(u1(t), u2(t), , ur(t)), "tÎ[t0, tk ], (4.4), dui, i=1,...,r (4.13).
, dui, i=1,...,r, "tÎ[t0, tk ] (4.3) . , dI, DI. DI=I - I0 , DI³0, |dui|, i=1,...,r.
.
, .
1.
u0(t)=(u10 (t), u20 (t), , ur0 (t)), "tÎ[t0, tk ].
2. t= t0 t= tk (4.1)
= fi (t, X(t), u0(t)), i=1,...,n,
(4.2) - h. X0(t)=(x10 (t), x20 (t), , xn0 (t)), "tÎ[t0, tk ] xi0 (tk), i=1,...,n t=tk.
|
|
3. (4.3)
,
,
,
h= .
(4.1)
(t)=f0 (t, X0(t), u0(t)), xn+1 (t0)=0.
I[u0(t)] = x0n+1 (tk).
4. , i=1,...,r, 5, 9.
5. t= tk t= t0 (4.1), (4.9)
xi(tk)= xik, i=1,...,n; li(tk)= 0, i=1,...,n; xik, i=1,...,n 2 tk .
6.
, i=1,...,r, "tÎ[t0, tk ].
7.
dui(t)=q× , i=1,...,r, "tÎ[t0, tk ],
q - .
8.
ui(t)=ui0(t) + dui(t), i=1,...,r, "tÎ[t0,tk]
q , 2 ( ).
9. I=I[u0+q ] I0=I[u0(t)].
I0I,
q=b×q, bÎ(0,1)
u(t)=u0(t) + q I , I0>I.
I0<I,
q=a×q, a>1
, .
10.
I[u0+du]-I[u0] ½£e,
e- , .
, ; , , 2.