The stability for many problems in real dynamical systems is actual asymptotical stability, although may not be mathematical asymptotical stability. The mathematical asymptotical stability demands that trajectories from all initial states in the neighborhood of zero solution must approach the origin as t. If there are only a small part or even a few of the initial states from which the trajectories do not approach the origin as t , the zero solution is not mathematically asymptotically stable. However, when the probability of occurrence of an event is zero, it means the event does not occur actually. If the probability of occurrence of the event that the trajectries from the initial states are that they do not approach zero when t, is zero, the stability of zero solution is actual asymptotical stability though it is not mathematical asymptotical stability. In order to analyze the asymptotical stability of the equilibrium point of such systems, the pragmatical asymptotical stability theorem is used.
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Let X and Y be two manifolds of dimensions m and n (m<n), respectively, and
be a differentiable map from X to Y, then (X) is subset of Lebesque measure 0 of Y [59]. For an autonomous system
( ,1 , n)
dx f x x
dt (B-1) where x
x1, ,xn
T is a state vector, the function f
f1, , fn
Tis defined onDRn and x H 0. Let x=0 be an equilibrium point for the system (B-1).
Then
(0) 0
f (B-2) For a non-autonomous systems,
x f x( ,...,1 xn1) (B-3) where x[ ,...,x1 xn1]T , the function f [ ,...,f1 fn]T is define on DRnR,here txn1R. The equilibrium point is
f( 0 ,xn1 ) . 0 (B-4)
Definition The equilibrium point for the system (B-1) is pragmatically asymptotically stable provided that with initial points on C which is a subset of Lebesque measure 0 of D, the behaviors of the corresponding trajectories cannot be determined, while with initial points on D－C, the corresponding trajectories behave as that agree with traditional asymptotical stability [60,61].
Theorem Let V [ ,x1 ,xn]T: D→R+ be positive definite and analytic on D, where x x1, 2,...,xn are all space coordinates such that the derivative of V through Eq.
(A-1)or(A-3), V , is negative semi-definite of [ ,x x1 2, ,xn]T.
For autonomous system, Let X be the m-manifold consisted of point set for which x 0, V x( )0 and D is a n-manifold. If m+1<n, then the equilibrium
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point of the system is pragmatically asymptotically stable.
For non-autonomous system, let X be them1-manifold consisting of point set of which x 0, ( ,V x x1 2,...,xn)0and D is n1-manifold. If m 1 1 n 1, i.e.m 1 nthen the equilibrium point of the system is pragmatically asymptotically stable. Therefore, for both autonomous and non-autonomous system the formula
1
m nis universal. So the following proof is only for autonomous system. The proof for non-autonomous system is similar.
Proof Since every point of X can be passed by a trajectory of Eq. (B-1), which is one- dimensional, the collection of these trajectories, A, is a (m+1)-manifold [60, 61].
If m+1＜n, then the collection C is a subset of Lebesque measure 0 of D. By the above definition, the equilibrium point of the system is pragmatically asymptotically stable.
If an initial point is ergodicly chosen in D, the probability of that the initial point falls on the collection C is zero. Here, equal probability is assumed for every point chosen as an initial point in the neighborhood of the equilibrium point. Hence, the event that the initial point is chosen from collection C does not occur actually.
Therefore, under the equal probability assumption, pragmatical asymptotical stability becomes actual asymptotical stability. When the initial point falls on D C ,
( ) 0
V x , the corresponding trajectories behave as that agree with traditional asymptotical stability because by the existence and uniqueness of the solution of initial-value problem, these trajectories never meet C.
In Eq. (5-8) V is a positive definite function of n variables, i.e. p error state variables and n-p=m differences between unknown and estimated parameters, while V e CeT is a negative semi-definite function of n variables. Since the number of
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error state variables is always more than one, p>1, m+1<n is always satisfied, by pragmatical asymptotical stability theorem we have
lim 0
t e
(B-5) and the estimated parameters approach the uncertain parameters. The pragmatical adaptive control theorem is obtained. Therefore, the equilibrium point of the system is pragmatically asymptotically stable. Under the equal probability assumption, it is actually asymptotically stable for both error state variables and parameter variables.
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