FUZZY PERTURBATION OF VECTOR FIELDS ´ LAECIO C. BARROS AND PEDRO A. TONELLI
1. Introduction Let X : Rn → Rn be a differentiable vector field. From the point of view of dynamical systems this vector field represents an infinitesimal generator of a local flow. In this case one of the main goal is to find the corresponding trajectories ϕ(t, x0 ) of the systems. Since this set up, in most cases, is the model of a real world process we are led to consider also perturbation to this system. There are already well known standard procedures to introduce perturbation in the system. Bifurcation theory, structural stability, stochastic differential equations and, affine control systems are typical branches of research where a perturbation on a driven vector field is considered. In this work we were inspired by these models and consider fuzzy perturbation of a deterministic vector field. We will study the particularities of the resulting fuzzy dynamical system. To fix some notations we set Q(Rn ) to be the space of all non-empties compacts subsets of Rn with the Hausdorff metric. Let Qc (Rn ) denote the subspace of Q(Rn ) of convex compact sets. The space of fuzzy vectors is the set F(Rn ) of the functions u : Rn → [0, 1] which are upper semi-continuous, and [u]α ∈ Q(Rn ) for all α ∈ [0, 1]; moreover E n will denote the space of convex fuzzy vectors which are the elements of F(Rn ) such that [u]α ∈ Qc (Rn ). To apply the existence theorems of differential inclusions it is convenient to restrict attention to E n . In the space F(Rn ) one considers the metric given by D(u, v) = sup h([u]α , [v]α ) 0≤α≤1
A fuzzy perturbation is a continuous map F : E n → E n and the perturbated ˆ : E n → E n the vector field is the map Xp : E n → E n , defined as follows: Let X Zadeh’s extension of the vector field X then define ˆ Xp (u) = X(u) + F (u). 2. Solutions of the Fuzzy Vector Field Xp The first problem here is to define what could be called the “flow” associated with the perturbed vector field. There are in the literature two theories of solutions for this problem. We will call them K-solution for the theory originated by Kaleva, and H-solutions for the H¨ ullermeier solutions. The K-solution uses the generalized Aumann integral. A K-solution is a map xk : [0, T ] → E n which satisfy xk (0) = u0 , u0 ∈ E n and Z t (1) xk (t) = u0 + Xp (xk (s))ds. 0
The first author was supported by CNPq. 1
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L. C. BARROS AND P. A. TONELLI
There is already a complete theory for existence and uniqueness problem for this problem. A K-trajectory in the level α of Xp is a absolute continuous curve ξα : [0, T ] → Rn such that ξα (t) ∈ [xk (t)]α where xk (t) is a K-solution of equation (1) and [u]α denotes the α-level set of a fuzzy set u. A H-trajectory in the level α of Xp is a absolute continuous curve να : [0, T ] → Rn satisfying the differential inclusion ν˙ α (t) ∈ [Xp (να (t)]α
(2)
for almost all t ∈ [0, T ]. Given u0 ∈ E n we define Aα (t, [u0 ]α ) the accessibility set as the subset of Rn {ν(t) : ν is H-trajectory and ν(0) ∈ [u0 ]α }. An H-solution is then a curve xh : [0, T ] → E n such that for almost every t ∈ [0, T ] we have [xh (t)]α = Aα (t, [xh (0)]α ). 3. Main Results The main results here concerns the viability of the deterministic trajectories. In this section we will consider the perturbations F : E n → E n such that 0 ∈ [F (u)]1 for each u ∈ E n . Theorem 1. Any deterministic solution ξ(t) of the vector field X is a K-trajectory in the level α when ξ(0) ∈ [u0 ]α for the equation (1). In others words, the deterministic solution is a viable trajectory for the problem (1) ˆ : E n → E n and Proof. Let’s first consider the just the Zadeh’s extension of X, X we show that ξ(t) ∈ [xk (t)]α . We fix α and construct the new initial condition {ξ(0)} if 1/2 < β ≤ 1 β (3) [v0 ] = [u0 ]α if 0 ≤ β ≤ 1/2 Call v(t) the unique K-solution of the initial value problem Z t ˆ (4) v(t) = v0 + X(v(s))ds. 0
According Kaleva [3] the level sets of v(t) are also the unique solution of the Hukuhara differential equations D[v(t)]β = X([v(t)]β )
(5)
β ˆ In this last equation we are using the fact that X([v(t)]β ) = [X(v(t))] . Since 1 0 α α [v(t)] = {ξt} and [v(t)] = [xk (t)] we have that ξ(t) ∈ [xk (t)] . To complete the proof we just note that using the properties of the generalized Aumann integral we have:
(6)
α
Z
[u0 ] +
t α
α
Z
[Xp (xk (s))] ds = [u0 ] + 0
0
t
ˆ k (s))]α ds + [X(x
Z
t
[F (xk (s))]α ds
0
Due the Hypothesis on F the function f (t) = 0 is a mesurable selection of [F (xk (s))]α Rt which means that 0 ∈ 0 [F (xk (s))]α ds. This concludes the proof.
FUZZY PERTURBATION OF VECTOR FIELDS
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Another question is in what sense the H-solutions are narrower (or stronger) than K-solutions. The following Theorem gives an insight in this way. Theorem 2. If xh (t) is a H-solution for Xp with xh (0) = u0 then [xh (t)]α ⊂ [xk (t)]α . Where xk (t) is a K-solution with the same initial fuzzy condition. Proof. We take an absolute continuous curve ν : [0, T ] → Rn which is a H-trajectory in the level α of Xp . It satisfies the differential inclusion ν(t) ˙ ∈ [Xp (ν( t)]α
(7)
and as a consequence of the definition of the Aumann integral we have: Z t Z t α α (8) ν(t) ∈ ν(0) + [Xp (ν( s)] ds ⊂ [u0 ] + [Xp (ν( s)]α ds 0
0
where the right side integrals are the Aumann integrals. Again using properties of the Aumann integral we conclude that ν(t) ∈ U (t) where U (t) is the convex set valued function which is the solution of Z t (9) U (t) = [u0 ]α + U (s)ds 0
Take into account we are under the hypothesis of uniqueness of the solution of equation (9). We use the Peano technique like in [3] to write this solution as: (10)
U (t) = limn→∞ On (ν(·))(t)
where the operator O is defined by: (11)
α
Z
O(f (·))(t) = [u0 ] +
t
f (s)ds 0
for f taking values in the compact convex set of Rn . Also from (8) we have (12)
ν(t) ∈ O(ν(·))(t) ⊂ On (ν(·))(t).
This shows that ν(t) ∈ U (t). But U (t) is by definition [xk (t)]α . Since it is obvious that ν(t) also belongs to [xh (t)]α , the proof is done. The next point is the viability of the deterministic solution of X with respect to H-solutions. Theorem 3. If 0 ∈ [F (u)]1 for each u ∈ E n and x(t) is a deterministic solution of the vector field X, consider xh (t) a H-solution of Xp with x(0) ∈ [xh (0)]α then we have x(t) ∈ [xh (t)]α . Proof. This assertion follows fast immediatelly from the definition, since we have (13)
α ˆ x(t) ˙ = X(x(t)) ⊂ [X({x(t)})] + F ({x(t)}) = [Xp (x(t))]α
This means that x(t) is an H-trajectory in level α and for this reason x(t) ∈ [xh (t)]α .
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4. Comments and final remarks In this work we proposed a definition of a fuzzy perturbation of a deterministic vector field. We have used already existing theories of solutions for fuzzy differential equations. A major difficulty we were faced to is that the main hypothesis to apply the Aumann integral is convexity. In general the Zadeh’s extension of a vector field don’t preserve convexity. This is only true for affine vector fields, for them this theory applies. The fact that the perturbation is far more general garantees, in our point of view, the interest of the results. We will be improving this point. We consider also important to treat together both concepts of solution of fuzzy vector field, H-solutions and K-solutions, althought it is clear that, as was already showed in [1], H-solutions are prefered if one want to study limit sets or attractors and looks for a classification of flows. But on the other hand, if one is concerned just with viability, then it is easier to find viable trajectories in K-solutions. References [1] Phil Diamond, Time Dependent Differential Inclusions, Cocycle Attractors and Fuzzy Differential Equations, IEEE Trans. On Fuzzy Systems - Vol. 7 - pp. 734-740. (1999). [2] E. H¨ ullermeier, An Approach to Modeling and Simulation of Uncertain Dynamical SystemsInt. J. Uncertainty, Fuzziness, Knowledge-Bases Syst. Vol. 5, pp. 117-137 (1997). [3] O. Kaleva, The Cauchy Problem for Fuzzy Differential Equations- Fuzzy Sets and Syst. Vol. 35, pp. 389-396 (1990). IMECC-UNICAMP, C.P. 6065, 13081-970 Campinas, S. P., Brazil E-mail address:
[email protected] IME-USP C. P. 66281, 05315-970, Sao Paulo, SP- Brazil E-mail address:
[email protected] 
