On stochastic inverse problem of construction of stable program motion

2.1 The set of vector functions Q ( y ) that do not explicitly depend on time

Theorem 1. Let in the neighborhood Λ ε there exist a Lyapunov function V ( λ ; y , t ) of the integral manifold Λ with the properties

Then, the program motion λ ≡ y − φ ( t ) = 0 of system (2.3) is asymptotically ρ -stable in probability with respect to an arbitrary s -dimensional vector function Q ( y ) , which is continuous in the neighborhood Λ ε for 1 ≤ s ≤ n .

Proof. Let us consider the difference x = Q ( y ) − Q ( φ ( t ) ) by the definition of Q -stability of motion [2]. The existence of Lyapunov function V ( λ ; y , t ) with properties (2.4), (2.5) by the condition of the theorem ensures the asymptotic ρ -stability in probability of the program motion λ = 0 [19]

And from the continuity of the vector function Q ( y ) and from condition (2.6), it follows that

The fulfillment of the latter gives the asymptotic stability of the motion λ ≡ y − φ ( t ) = 0 of system (2.3) with respect to the vector function Q ( y ) .

2.2 The set of vector functions Q ( y , t ) depending on y and t

Theorem 2. Let in the neighborhood Λ ε there exist a Lyapunov function V ( λ ; y , t ) of the integral manifold Λ with the properties (2.4), (2.5). Then, the program motion λ ≡ y − φ ( t ) = 0 of system (2.3) is asymptotically ρ -stable in probability with respect to an arbitrary continuous in y and t s -dimensional vector function Q ( y , t ) satisfying the condition

Here, x = Q ( y , t ) − Q ( φ ( t ) , t ) .

Proof. The existence of the function V ( λ ; y , t ) with properties (2.4), (2.5) implies asymptotic ρ -stability in probability of motion λ ≡ y − φ ( t ) = 0 uniform with respect to { t 0 , y 0 } [19], that is

and

follows from (2.6) and (2.7).

Consequently, λ ≡ y − φ ( t ) = 0 is asymptotically stable in probability with respect to the vector function Q ( y , t ) .

2.3 Program motion λ ( y , t ) = 0 and a set of comparison vector functions Q ( λ , t )

Let us define the program motion in the following form:

Here, λ ∈ R k , y ∈ R n , k ≤ n .

Suppose that it takes place rang ∂ λ ∂ y = k for all y ∈ Λ h , t ≥ t 0 in the neighborhood Λ h ( t ) ∈ R n :

The set of equations of perturbed motion can be represented as

following the Erugin’s method [22]. Here, A ( λ ; y , t ) is a vector function and B ( λ ; y , t ) is a n × k -dimensional Erugin-type matrix such that A ( 0 ; y , t ) ≡ 0 , B ( 0 ; y , t ) ≡ 0 .

Consider the continuous s -dimensional vector functions Q ( λ , t ) for which the inequality

holds. Here, x = Q ( λ ( y , t ) , t ) − Q ( 0 , t ) , 1 ≤ s ≤ n .

Theorem 3. Suppose that in neighborhood (2.9) of the integral manifold (2.8) there exists a Lyapunov function V ( λ ; y , t ) with properties (2.4) and

Then, the integral manifold Λ ( t ) (2.8) is asymptotically stable in probability with respect to an arbitrary s -dimensional vector function Q ( λ , t ) , which is continuous with respect to λ and t and satisfying condition (2.11) for 1 ≤ s ≤ n .

The proof of Theorem 3 is carried out similarly to the proof of Theorem 2.

2.4 A set of n -dimensional vector functions of the form C ( t ) λ

Assume that the perturbed motion equation (2.3) in the first approximation has the form

Let us consider a Lyapunov function V ( λ ) = ( λ , λ ) and a n -dimensional vector function Q ( y , t ) = C ( t ) λ . Here, λ ≡ y − φ ( t ) . In this case, x = Q ( y ) − Q ( φ ( t ) ) has the form

Assume that

(i) the matrix A 1 T ( t ) + A 1 ( t ) is definitely negative, and the vector function A 2 satisfies the condition ∥ A 2 ∥ = o ( ∥ λ ∥ ) ;

(ii) the matrix

Then, taking into account properties (i), (ii) and Theorem 2, the following theorem holds.

Theorem 4. Let A 1 ( t ) and C ( t ) be continuous matrices such that conditions (i) and (ii) are satisfied.

Then, the motion λ ≡ y − φ ( t ) = 0 of the system (2.3) is stable in probability with respect to the arbitrary vector functions Q ( y , t ) = C ( t ) λ .

Remark. In Theorem 4, the weaker condition (2.15) is required instead of condition (1.4).