eu ON THE SMOOTNESS OF FUNCTION OF REACHABILITY FOR AUTONOMOUS DIFFERENTIAL EQUATIONS

A function of reachability � + is investigated for a wide class of autonomous systems of differential equations and relatively wide class reachable sets. The sufficient conditions for differentiability of the function � + are found.


Assume that:
-The domain G ⊂ R n is a phase space of autonomous system of differential equations; -The initial point x 0 ∈ G; -The set Φ = ∅; -The curve γ, γ ⊂ G, is a trajectory of the autonomous system which: -at the moment t 0 = 0, passes through the initial point x 0 ; -crosses the set Φ, i.e. γ ∩ Φ = ∅.
url: www.acadpubl.eu We say that Θ + is a function of reachability if this function assigning to the initial point x 0 the moment θ at which the trajectory γ crosses Φ for the first time.The articles [7], [8] and [14] are devoted on the qualities of function of reachability.The sufficient conditions under which the function is continuous and bounded are found there.In this paper, the studies of function qualities are continued.The sufficient conditions for differentiability of Θ + in respect to the initial point x 0 are given.
The function of reachability has a fundamental role in the studies on the existence of periodic solutions of the systems of differential equations with variable moments of the impulsive effects.(see [6] and [13]).In the definitions of impulsive equations, the set of reachability is a set of impulses too.The impulsive effects take place when the trajectory of impulsive system reaches the set Φ.The trajectory instantaneously "bounces" at the moment of these impacts.Thus, the study of function of reachability is equivalent to the investigation of impulsive moments of the systems of differential equations with impulsive effects.
The dynamic processes that have interrupted nature can be modeled by the impulsive systems of differential equations (see [1]- [5] and [10]- [11]).The processes in which the impulsive effects are not predefined, and they are performed when the trajectory reaches the above-mentioned impulsive sets are particularly important.In these cases, the function of reachability is an important feature of such processes and their corresponding models.
The considerations above justify the research presented in this paper.
The closure and contour of the set X are denoted by X and ∂X, correspondingly.Euclidean distance between the sets X and Y will denote by ρ(X, Y ).
The main object of investigation is the following initial value problem where: the function f : G → R n ; set G ⊂ R n , G = ∅ and G is a domain (an open and connected set); the initial point x 0 ∈ G.The solution of problem (1) will be denoted by x(t; x 0 ).The trajectory of system (1), closed between the points x(0; x 0 ) = x 0 and x(θ; x 0 ), where θ ∈ R will be denoted by γ(θ, x 0 ).We have In particular: Definition 1. [8] Assume that: 1.The sets 2. For any initial point x 0 ∈ X + 0 , the solution of problem ( 1) is defined and unique in the infinite interval [0, ∞); 3. There exists a function Θ + : X + 0 → R + , which assigns to any point Then, we say that: 1. Φ is a positive reachable set from the set X + 0 through the system (1); 2. X + 0 is a positive initial set for Φ through the system (1); 3. Any initial point x 0 ∈ X + 0 is named positive initial point for Φ through the system (1); 4. If X + 0 = G, then Φ is a totally positive reachable set through the system (1);

The function Θ
The following concepts are defined in a similar manner: 1. Negative reachable set; 2. Negative initial set X − 0 ; 3. Negative initial point; 4. Totally negative reachable set; 5. Negative function of reachability Θ − : Let the reachable set has the form where the function ϕ : G → R.
We introduce the following conditions: H1.There exists a constant C f > 0 such that H2.There exists a constant C Lip > 0 such that H3.For any point x 0 ∈ G, the solution of problem (1) exists and is unique in R.
In other words, the function ϕ(x) = a, x + a 0 , x ∈ G.
In connection with this particular case, we introduce the following condition.H8.There exists a constant C a,f > 0 such that As a consequence of the previous theorem (in the case where the reachable set is a part of the hyperplane), we formulate the following statement.
Then the assertions of Theorem 1 are satisfied.
Proof.We will check whether the conditions of Theorem 1 are valid, which is the evidence of this corollary.It is clear that the function ϕ(x) = a, x +a 0 ∈ C 1 [G, R].Furthermore, by using the condition H8, we find i.e. the condition H4 is fulfilled.The conditions H5 and H6 are checked trivial.In this case, we have by which the condition H7 C ϕ = a is established.