Bounded Motions of the Dynamical Systems Described by Differential Inclusions

and Applied Analysis 3 It is obvious that δ∗ μ∗, ·, ·, · ∈ Δμ∗ δ∗ · . Let us choose an arbitrary h · ∈ Δμ∗ δ∗ · . For given t0, x0 ∈ 0, θ × R, U∗, δ∗ · ∈ Upos × Δ 0, 1 , h · ∈ Δμ∗ δ∗ · , we define the function x · : t0, θ → R in the following way. The function x∗ · on the closed interval t0, t0 h t0, x0, U∗ t0, x0 ∩ t0, θ is defined as a solution of the differential inclusion ẋ∗ t ∈ F t, x∗ t , U∗ t0, x0 , x∗ t0 x0 see, e.g., 13 . If t0 h t0, x0, U∗ t0, x0 < θ, then setting t1 t0 h t0, x0, U∗ t0, x0 , x∗ t1 x1, the function x∗ · on the closed interval t1, t1 h t1, x1, U∗ t1, x1 ∩ t1, θ is defined as a solution of the differential inclusion ẋ∗ t ∈ F t, x∗ t , U∗ t1, x1 , x∗ t1 x1 and so on. Continuing this process we obtain an increasing sequence {tk}k 1 and function x∗ · : t0, t∗ → R, where t∗ sup tk. If t∗ θ, then it can be considered that the definition of the function x∗ · is completed. If t∗ < θ, then to define the function x∗ · on the interval t0, θ , the transfinite induction method should be used see, e.g., 14 . Let ν be an arbitrary ordinal number and {tλ}λ<ν are defined for every λ < ν, where tλ ∈ t0, θ and tλ1 < tλ2 if λ1 < λ2. If t∗ supλ<νtλ θ, then it can be considered that the definition of the function x∗ · on the interval t0, θ is completed. Let t∗ < θ. If ν follows after an ordinal number σ, then setting x∗ tσ xσ,we define the function x∗ · on the closed interval tσ , tν ∩ tσ , θ , where tν tσ h tσ , xσ,U∗ tσ , xσ , as a solution of the differential inclusion ẋ∗ t ∈ F t, x∗ t , U∗ tσ , xσ , x∗ tσ xσ. If ν has no predecessor, then there exists a sequence {tλi}i 1 such that tλi1 < tλi2 < · · · and tλi → tν − 0 as i → ∞. Then we set x∗ tν limi→∞x∗ tλi .Note that it is not difficult to prove that via conditions a – c , this limit exists. Since the intervals tλ, tλ 1 are not empty and pairwise disjoint then tν θ for some ordinal number νwhich does not exceed first uncountable ordinal number see, e.g., 15, 16 . So, the function x∗ · is defined on the interval t0, θ . From the construction of the function x∗ · it follows that for given t0, x0 ∈ 0, θ × R, U∗, δ∗ · ∈ Upos×Δ 0, 1 , μ∗ ∈ 0, 1 , h · ∈ Δμ∗ δ∗ · such a function is not unique. The set of such functions is denoted by Yμ∗ t0, x0, U∗, h · . Further, we set Zμ∗ t0, x0, U∗, δ∗ · ⋃ h · ∈Δμ∗ δ∗ · Yμ∗ t0, x0, U∗, h · . 2.2 The set Zμ∗ t0, x0, U∗, δ∗ · is called the pencil of step-by-step motions and each function x · ∈ Zμ∗ t0, x0, U∗, δ∗ · is called step-by-step motion of the system 1.1 , generated by the strategy U∗, δ∗ · from the initial position t0, x0 . It is obvious that for each step-by-step motion x · ∈ Zμ∗ t0, x0, U∗, δ∗ · there exists an h∗ · ∈ Δμ∗ δ∗ · such that x · ∈ Yμ∗ t0, x0, U∗, h∗ · . By X t0, x0, U∗, δ∗ · we denote the set of all functions x · : t0, θ → R such that x · limk→∞xk · , where xk · ∈ Zμk t0, x0, U∗, δ∗ · , μk → 0 as k → ∞. X t0, x0, U∗, δ∗ · is said to be the pencil of motions and each function x · ∈ X t0, x0, U∗, δ∗ · is said to be the motion of the system 1.1 , generated by the strategy U∗, δ∗ · from initial position t0, x0 . For every initial position θ, x0 we set X θ, x0, U, δ · {x0} for all U, δ · ∈ Upos × Δ 0, 1 . Using the constructions developed in 3, 4 it is possible to prove the validity of the following proposition. Proposition 2.1. For each t0, x0 ∈ 0, θ × R, U∗, δ∗ · ∈ Upos × Δ 0, 1 the set X t0, x0, U∗, δ∗ · is nonempty compact subset of the space C t0, θ ;R and each motion x · ∈ X t0, x0, U∗, δ∗ · is an absolutely continuous function. 4 Abstract and Applied Analysis Here C t0, θ ;R is the space of continuous functions x · : t0, θ → R with norm |x · | max‖x t ‖ as t ∈ t0, θ . 3. Positionally Weakly Invariant Set Let W ⊂ T × R be a closed set. We set W t {x ∈ R : t, x ∈ W}. 3.1 Let us give the definition of positionally weak invariance of the set W ⊂ T × R with respect to dynamical system 1.1 . Definition 3.1. A closed setW ⊂ T ×Rn is said to be positionally weakly invariant with respect to a dynamical system 1.1 if for each position t0, x0 ∈ W it is possible to define a strategy U∗, δ∗ · ∈ Upos ×Δ 0, 1 such that for all x · ∈ X t0, x0, U∗, δ∗ · the inclusion x t ∈ W t holds for every t ∈ t0, θ . We will consider positionally weak invariance of the set W ⊂ T × R, described by the relation W { t, x ∈ T × R : c t, x ≤ 0}, 3.2 where c · : T × R → R1. For t, x ∈ 0, θ × R, f ∈ R we denote ∂ c t, x ∂ ( 1, f ) lim sup δ→ 0 ,‖y‖→ 0 [ c ( t δ, x δf δy ) − c t, x δ−1. 3.3 Let us formulate the theorem which characterizes positionally weak invariance of the set W given by relation 3.2 with respect to dynamical system 1.1 . Theorem 3.2 17 . Let ε∗ > 0, and let the set W ⊂ T × R be defined by relation 3.2 where c · : T × R → R1 is a continuous function. Assume that for each t, x ∈ 0, θ × R such that 0 < c t, x < ε∗, it is possible to define u∗ ∈ P such that the inequality sup f∈F t,x,u∗ ∂ c t, x ∂ ( 1, f ) ≤ 0 3.4 holds. Then the set W described by relation 3.2 is positionally weakly invariant with respect to the dynamical system 1.1 . Abstract and Applied Analysis 5 Theorem 3.3. Let ε∗ > 0, and let the setW ⊂ T×Rn be defined by relation 3.2 where c · : T×Rn → R1 is a continuous function. Assume that for each t, x ∈ 0, θ × R such that 0 < c t, x < ε∗, the inequalityand Applied Analysis 5 Theorem 3.3. Let ε∗ > 0, and let the setW ⊂ T×Rn be defined by relation 3.2 where c · : T×Rn → R1 is a continuous function. Assume that for each t, x ∈ 0, θ × R such that 0 < c t, x < ε∗, the inequality inf u∈P sup f∈F t,x,u ∂ c t, x ∂ ( 1, f ) ≤ 0 3.5 is verified. Then for each fixed t0, x0 ∈ W and ε ∈ 0, ε∗ it is possible to define a strategy Uε, δε · ∈ Upos ×Δ 0, 1 such that for all x · ∈ X t0, x0, Uε, δε · the inequality c t, x t ≤ ε holds for every t ∈ t0, θ . For t, x, s ∈ T × R × R we denote ξ t, x, s inf u∈P sup f∈F t,x,u 〈 s, f 〉 . 3.6 Here 〈·, ·〉 denotes the inner product in R. The function ξ · : T × R × R → R is said to be the Hamiltonian of the system 1.1 . We obtain from Theorem 3.3 the validity of the following theorem. Theorem 3.4. Let ε∗ > 0, and let the setW ⊂ T×Rn be defined by relation 3.2 where c · : T×Rn → R1 is a differentiable function. Assume that for each t, x ∈ 0, θ ×Rn such that 0 < c t, x < ε∗, the inequality ∂c t, x ∂t ξ ( t, x, ∂c t, x ∂x ) ≤ 0 3.7 holds. Then for each fixed t0, x0 ∈ W and ε ∈ 0, ε∗ it is possible to define a strategy Uε, δε · ∈ Upos ×Δ 0, 1 such that for all x · ∈ X t0, x0, Uε, δε · the inequality c t, x t ≤ ε holds for every t ∈ t0, θ . 4. Boundedness of the Motion of the System Consider positionally weak invariance of the setW ⊂ T × R given by relation 3.2 where c t, x 〈E t x − a t , x − a t 〉 − 1, 4.1 E · is a differentiable n×n matrix function, a · : T → R is a differentiable function. Then the set W is given by relation W { t, x ∈ T × R : 〈E t x − a t , x − a t 〉 − 1 ≤ 0}. 4.2 If the matrix E t is symmetrical and positive definite for every t ∈ T, then it is obvious that for every t ∈ T the set W t ⊂ R is ellipsoid. 6 Abstract and Applied Analysis Theorem 4.1. Let ε∗ > 0, and let the set W ⊂ T × R be defined by relation 4.2 where E · is a differentiable n×n matrix function, a · : T → R is a differentiable function. Assume that for each t, x ∈ 0, θ × R such that 0 < 〈E t x − a t , x − a t 〉 − 1 < ε∗ the inequality 〈[ dE t dt x − a t − ( E t E t )da t dt ] , x − a t 〉 ξ ( t, x, [ E t E t ] x − a t ) ≤ 0 4.3 holds. Then for each fixed t0, x0 ∈ W and ε ∈ 0, ε∗ it is possible to define a strategy Uε, δε · ∈ Upos ×Δ 0, 1 such that for all x · ∈ X t0, x0, Uε, δε · the inequality 〈E t x t − a t , x t − a t 〉 − 1 < ε 4.4 holds for every t ∈ t0, θ . Here E t means the transpose of the matrix E t . Proof. Since the function c · given by relation 4.1 is differentiable and ∂c t, x ∂x [ E t E t ] x − a t , ∂c t, x ∂t 〈[ dE t dt x − a t − E t da t dt − E t da t dt ] , x − a t 〉 4.5 then the validity of the theorem follows from Theorem 3.4. We obtain from Theorem 4.1 the following corollary. Corollary 4.2. Let ε∗ > 0, and let the set W ⊂ T × R be defined by relation 4.2 where E · is a differentiable n × n matrix function, a · : T → R is a differentiable function and E t is a symmetrical positive definite matrix for every t ∈ T. Assume that for each t, x ∈ 0, θ × R for which 0 < 〈E t x − a t , x − a t 〉 − 1 < ε∗, 4.6


Introduction
Consider the dynamical system, the behavior of which is described by the differential inclusion ẋ ∈ F t, x, u , 1.1 where x ∈ R n is the phase state vector, u ∈ P is the control vector, P ⊂ R p is a compact set, and t ∈ 0, θ T is the time.It will be assumed that the right-hand side of system 1.1 satisfies the following conditions: a F t, x, u ⊂ R n is a nonempty, convex and compact set for every t, x, u ∈ T × R n × P ; b the set valued map t, x → F t, x, u , t, x ∈ T × R n , is upper semicontinuous for every fixed u ∈ P ; c max{ f : f ∈ F t, x, u , u ∈ P } ≤ c 1 x for every t, x ∈ T × R n where c const, and • denotes Euclidean norm.
given in the form 1.1 see, e.g., 1-3 and references therein .The investigation of a conflict control system the dynamic of which is given by an ordinary differential equation, can also be reduced to a study of system in form 1.1 see, e.g., 3-5 and references therein .The tracking control problem and its applications for uncertain dynamical systems, the behavior of which is described by differential inclusion with control vector, have been studied in 6 .In Section 2 the feedback principle is chosen as control method of the system 1.1 .The motion of the system generated by strategy U * , δ * • from initial position t 0 , x 0 is defined.Here U * is a positional strategy and it specifies the control effort to the system for realized position t * , x * .The function δ * • defines the time interval; along the length of which the control effort, U * t * , x * will have an effect on.It is proved that the pencil of motions is a compact set in the space of continuous functions and every motion from the pencil of motions is an absolutely continuous function Proposition 2.1 .
In Section 3 the notion of a positionally weakly invariant set with respect to the dynamical system 1.1 is introduced.The positionally weak invariance of the closed set W ⊂ T × R n means that for each t 0 , x 0 ∈ W there exists a strategy U * , δ * • such that the graph of all motions of system 1.1 generated by strategy U * , δ * • from initial position t 0 , x 0 is in the set W right up to instant of time θ.Note that this notion is a generalization of the notions of weakly and strongly invariant sets with respect to a differential inclusion see, e.g., 5, 7-11 and close to the positional absorbing sets notion in the theory of differential games see, e.g., 3-5 .In terms of upper directional derivatives, the sufficient conditions for posititionally weak invariance of the sets W { t, x ∈ T × R n : c t, x ≤ 0} with respect to system 1.1 are formulated where c • : T × R n → R is a continuous function Theorems 3.2 and 3.3 .In Section 4, the boundedness of the motions of the system is investigated.Using the Hamiltonian of the system 1.1 , the sufficient condition for boundedness of the motions is given Theorem 4.3 and Corollary 4.4 .

Motion of the System
Now let us give a method of control for the system 1.1 and define the motion of the system 1.1 .
A pair U, δ • ∈ U pos × Δ 0, 1 is said to be a strategy.Note that such a definition of a strategy is closely related to concept of ε-strategy for player E given in 12 .
Now let us give a definition of motion of the system 1.1 generated by the strategy At first we give a definition of step-by-step motion of the system 1.1 generated by the strategy U * , δ * • ∈ U pos × Δ 0, 1 from initial position t 0 , x 0 ∈ 0, θ × R n .Note that step-by-step procedure of control via strategy U * , δ * • uses the constructions developed in 3, 4, 12 .
For δ * • ∈ Δ 0, 1 and fixed μ * ∈ 0, 1 , we set The function x * • on the closed interval t 0 , t 0 h t 0 , x 0 , U * t 0 , x 0 ∩ t 0 , θ is defined as a solution of the differential inclusion ẋ * t ∈ F t, x * t , U * t 0 , x 0 , x * t 0 x 0 see, e.g., 13 .If t 0 h t 0 , x 0 , U * t 0 , x 0 < θ, then setting x 1 and so on.Continuing this process we obtain an increasing sequence {t k } ∞ k 1 and function x * • : t 0 , t * → R n , where t * sup t k .If t * θ, then it can be considered that the definition of the function x * • is completed.If t * < θ, then to define the function x * • on the interval t 0 , θ , the transfinite induction method should be used see, e.g., 14 .
Let ν be an arbitrary ordinal number and {t λ } λ<ν are defined for every λ < ν, where Note that it is not difficult to prove that via conditions a -c , this limit exists.
Since the intervals t λ , t λ 1 are not empty and pairwise disjoint then t ν θ for some ordinal number ν which does not exceed first uncountable ordinal number see, e.g., 15, 16 .So, the function x * • is defined on the interval t 0 , θ .
From the construction of the function x * • it follows that for given The set Z μ * t 0 , x 0 , U * , δ * • is called the pencil of step-by-step motions and each function x • ∈ Z μ * t 0 , x 0 , U * , δ * • is called step-by-step motion of the system 1.1 , generated by the strategy U * , δ * • from the initial position t 0 , x 0 .
It is obvious that for each step-by-step motion is said to be the pencil of motions and each function x • ∈ X t 0 , x 0 , U * , δ * • is said to be the motion of the system 1.1 , generated by the strategy U * , δ * • from initial position t 0 , x 0 .
For every initial position θ, x 0 we set X θ, x 0 , U, δ Using the constructions developed in 3, 4 it is possible to prove the validity of the following proposition.Proposition 2.1.For each t 0 , x 0 ∈ 0, θ × R n , U * , δ * • ∈ U pos × Δ 0, 1 the set X t 0 , x 0 , U * , δ * • is nonempty compact subset of the space C t 0 , θ ; R n and each motion x • ∈ X t 0 , x 0 , U * , δ * • is an absolutely continuous function.
Here C t 0 , θ ; R n is the space of continuous functions x • : t 0 , θ → R n with norm |x • | max x t as t ∈ t 0 , θ .

Positionally Weakly Invariant Set
Let W ⊂ T × R n be a closed set.We set Let us give the definition of positionally weak invariance of the set W ⊂ T × R n with respect to dynamical system 1.1 .Definition 3.1.A closed set W ⊂ T × R n is said to be positionally weakly invariant with respect to a dynamical system 1.1 if for each position t 0 , x 0 ∈ W it is possible to define a strategy U * , δ * • ∈ U pos × Δ 0, 1 such that for all x • ∈ X t 0 , x 0 , U * , δ * • the inclusion x t ∈ W t holds for every t ∈ t 0 , θ .
We will consider positionally weak invariance of the set W ⊂ T × R n , described by the relation

3.3
Let us formulate the theorem which characterizes positionally weak invariance of the set W given by relation 3.2 with respect to dynamical system 1.1 .
Then the set W described by relation 3.2 is positionally weakly invariant with respect to the dynamical system 1.1 .Then for each fixed t 0 , x 0 ∈ W and ε ∈ 0, ε * it is possible to define a strategy U ε , δ ε • ∈ U pos × Δ 0, 1 such that for all x • ∈ X t 0 , x 0 , U ε , δ ε • the inequality c t, x t ≤ ε holds for every t ∈ t 0 , θ .

Boundedness of the Motion of the System
Consider positionally weak invariance of the set W ⊂ T × R n given by relation 3.2 where Then the set W is given by relation If the matrix E t is symmetrical and positive definite for every t ∈ T, then it is obvious that for every t ∈ T the set W t ⊂ R n is ellipsoid.
Here E T t means the transpose of the matrix E t .
Proof.Since the function c • given by relation 4. Then for each fixed t 0 , x 0 ∈ W and ε ∈ 0, ε * it is possible to define a strategy U ε , δ ε • ∈ U pos × Δ 0, 1 such that for all x • ∈ X t 0 , x 0 , U ε , δ ε • the inequality holds for every t ∈ t 0 , θ .Now let us give the theorem which characterizes boundedness of the motion of the system 1.1 .
Using the results obtained above, we illustrate in the following example that the given system has bounded motions.

Theorem 3 .2 17 .
Let ε * > 0, and let the set W ⊂ T × R n be defined by relation 3.2 where c

Theorem 3 . 3 .
Let ε * > 0, and let the set W ⊂ T ×R n be defined by relation 3.2 where c • :

Theorem 4 . 1 .
Let ε * > 0, and let the set W ⊂ T × R n be defined by relation 4.2 where E holds.
t * sup λ<ν t λ θ, then it can be considered that the definition of the function x * • on the interval t 0 , θ is completed.Let t * < θ.If ν follows after an ordinal number σ, then setting x * t σ x σ , we define the function x * • on the closed interval t σ , t ν ∩ t σ , θ , where t ν t σ h t σ , x σ , U * t σ , x σ , as a solution of the differential inclusion ẋ * t ∈ F t, x * t , U * t σ , x σ , x * t σ x σ .If ν has no predecessor, then there exists a sequence {t λ Let ε * > 0, and let the set W ⊂ T × R n be defined by relation 4.2 where E • is a differentiable n × n matrix function, a • : T → R n is a differentiable function and E t is a symmetrical positive definite matrix for every t ∈ T. Assume that for each t, x ∈ 0, θ × R n for which R n → R is defined by 4.11 .It is obvious that t, x ∈ W if and only if t ∈ T and x ∈ B a, r .