六分钟步行试验.ppt

Differential Equations, Vol. 38, No. 8, 2002, pp. 1081–1094. Translated from Differentsial’nye Uravneniya, Vol. 38, No. 8, 2002, pp. 1017–1029.
Original Russian Text Copyright
c 2002 by Arutyunov, Jacimovic.
ORDINARY
DIFFERENTIAL EQUATIONS
2-Normal Processes
in Controlled Dynamical Systems
A. V. Arutyunov∗ and V. Jacimovic∗∗
∗Russian University of Nations’ Friendship, Moscow, Russia
∗∗University of Montenegro (Crne Gore), Podgorica, Montenegro, Yugoslavia
Received December 5, 2000
1. STATEMENT OF THE PROBLEM
Consider the optimal control problem
x˙= f(x, u, t),t∈[t1,t2] , (1)
W (p)=0,p=(x1,x2) ,x1 = x (t1) ,x2 = x (t2) , (2)
Zt2
0
J = J(p, u)=W0(p)+ f (x, u, t)dt → min . (3)
t1
Here t ∈ T =[t1,t2]istime,t1 <t2 are given, x is the phase variable ranging in the n-dimensional
n m
space R , u =(u1,...,um) ∈ R is a control, f is an n-dimensional vector function, W0 and
f 0 are scalar functions, and W is a w-dimensional vector function (n, m,andw are positive
integers). We assume that W0 and W are twice continuously differentiable functions and f0 and f
are piecewise smooth functions; ., the closed interval T can be represented as the union of finitely
n m
many intervals [τi,τi+1] such that the restriction of f0 and f to R × R × [τi,τi+1] is infinitely
differentiable.
As the class of admissible controls, we consider the set of measurable essentially bounded func-
m
tions u ∈ L∞[t1,t2]. A pair (x(t),u(t)), t ∈[t1,t2], of vector functions is referred to as an admissible
process if u(·) is an admissible control and x(·) is the corresponding solution of Eq. (1) satisfying
the endpoint constraints (2). The problem is to find the minimum of the functional J on the set
of admissible processes.
Definition. An admissible process (x0,u0) is referred to as a finite-dimensional minimum if
m 0 0 0
for any finite-dimensional subspace R ⊂ L∞[t1,t2] containing the point u , the process (x ,u )is
a local minimum in problem (1)–(3) with