The Partial Averaging of Fuzzy Differential Inclusions on Finite Interval

Abstract

The substantiation of a possibility of application of partial averaging method on finite interval for differential inclusions with the fuzzy right-hand side with a small parameter is considered.

1. Introduction

In 1990, Aubin [1] and Baidosov [2, 3] introduced differential inclusions with the fuzzy right-hand side. Their approach is based on usual differential inclusions. In 1995, Hüllermeier [4–6] introduced the concept of 𝑅-solution similar to how it has been done in [7]. Further in [8–20], various properties of solutions of fuzzy differential inclusions and their applications at modeling of various natural-science processes were considered.

The averaging methods combined with the asymptotic representations (in Poincare sense) began to be applied as the basic constructive tool for solving the complicated problems of analytical dynamics described by the differential equations. After the systematic researches done by N. M. Krylov, N. N. Bogoliubov, Yu. A. Mitropolsky, and so forth, in 1930s, the averaging method gradually became one of the classical methods in analyzing nonlinear oscillations.

In works [21, 22], the possibility of application of schemes of full and partial averaging for differential inclusions with the fuzzy right-hand side, containing a small parameter, was proved. By proving these theorems, the scheme offered by Plotnikov et al. for a substantiation of schemes of an average of usual differential inclusions [23–28] was used. In this work, the possibility of application of partial averaging method for fuzzy differential inclusions without passage to reviewing of separate solutions is proved; that is, all estimations are spent for 𝑅-solution corresponding fuzzy systems.

2. Preliminaries

Let comp(𝑅^𝑛)(conv(𝑅^𝑛)) be a family of all nonempty (convex) compact subsets from the space 𝑅^𝑛 with the Hausdorff metric: ℎ(𝐴,𝐵)=min_𝑟≥0𝑆_𝑟(𝐴)⊃𝐵,𝑆_𝑟(𝐵)⊃𝐴, (1) where 𝐴,𝐵∈comp(𝑅^𝑛) and 𝑆_𝑟(𝐴) is 𝑟-neighborhood of set 𝐴.

Let 𝐸^𝑛 be a family of all 𝑢∶𝑅^𝑛→[0,1] such that 𝑢 satisfies the following conditions:(1)𝑢is normal; that is, there exists an 𝑥₀∈𝑅^𝑛 such that 𝑢(𝑥₀)=1;(2)𝑢 is fuzzy convex; that is, 𝑢𝜆𝑥+(1−𝜆)𝑦≥min𝑢(𝑥),𝑢𝑦, (2) for any 𝑥,𝑦∈𝑅^𝑛 and 0≤𝜆≤1;(3)𝑢 is upper semicontinuous; that is, for any 𝑥₀∈𝑅^𝑛 and 𝜀>0 exists 𝛿(𝑥₀,𝜀)>0 such that 𝑢(𝑥)<𝑢(𝑥₀)+𝜀 whenever ‖𝑥−𝑥₀‖<𝛿(𝑥₀,𝜀), 𝑥∈𝑅^𝑛;(4)the closure of the set {𝑥∈𝑅^𝑛∶𝑢(𝑥)>0} is compact.

If 𝑢∈𝐸^𝑛, then 𝑢 is called a fuzzy number and 𝐸^𝑛 is said to be a fuzzy number space.

Definition 1. The set {𝑥∈𝑅^𝑛∶𝑢(𝑥)≥𝛼} is called the 𝛼-level [𝑢]^𝛼 of a fuzzy number 𝑢∈𝐸^𝑛, for 0<𝛼≤1. The closure of the set {𝑥∈𝑅^𝑛∶𝑢(𝑥)>0} is called the 0-level [𝑢]⁰ of a fuzzy number 𝑢∈𝐸^𝑛.
It is clear that the set [𝑢]^𝛼∈conv(𝑅^𝑛), for all 0≤𝛼≤1.

Theorem 2 (see [29] (stacking theorem)). If 𝑢∈𝐸^𝑛, then (1)[𝑢]^𝛼∈𝑐𝑜𝑛𝑣(𝑅^𝑛), for all 𝛼∈[0,1];(2)[𝑢]^𝛼2⊂[𝑢]^𝛼1, for all 0≤𝛼₁≤𝛼₂≤1;(3)if {𝛼_𝑘} is a nondecreasing sequence converging to 𝛼>0, then [𝑢]^𝛼=⋂_𝑘≥1[𝑢]^𝛼𝑘.Conversely, if {𝐴_𝛼∶𝛼∈[0,1]} is the family of subsets of 𝑅^𝑛 satisfying conditions (1)–(3), then there exists 𝑢∈𝐸^𝑛 such that [𝑢]^𝛼=𝐴_𝛼 for 0<𝛼≤1 and [𝑢]⁰=⋃_0<𝛼≤1𝐴_𝛼⊂𝐴₀.

Let 𝜃 be the fuzzy number defined by 𝜃(𝑥)=0, if 𝑥≠0 and 𝜃(0)=1.

Define 𝐷∶𝐸^𝑛×𝐸^𝑛→[0,∞) by the relation 𝐷(𝑢,𝑣)=sup_0≤𝛼≤1ℎ[𝑢]^𝛼,[𝑣]^𝛼. (3) Then, 𝐷 is a metric in 𝐸^𝑛. Further, we know that [30](i)(𝐸^𝑛,𝐷) is a complete metric space;(ii)𝐷(𝑢+𝑤,𝑣+𝑤)=𝐷(𝑢,𝑣), for all 𝑢,𝑣,𝑤∈𝐸^𝑛;(iii)𝐷(𝜆𝑢,𝜆𝑣)=|𝜆|𝐷(𝑢,𝑣), for all 𝑢,𝑣∈𝐸^𝑛 and 𝜆∈𝑅.

3. Fuzzy Differential Inclusion: 𝑅-Solution

Consider the fuzzy differential inclusion 𝑥̇∈𝐹(𝑡,𝑥),𝑥(0)∈𝑋₀, (4) where 𝑥∈𝑅^𝑛, 𝑡∈[0,𝑇]⊂𝑅₊, 𝐹∶[0,𝑇]×𝑅^𝑛→𝐸^𝑛, 𝑋₀∈𝐸^𝑛.

We interpret (4) as a family of differential inclusions (see [7, 9, 10]): 𝑥̇_𝛼∈𝐹𝑡,𝑥_𝛼^𝛼,𝑥_𝛼(0)∈𝑋₀^𝛼,𝛼∈[0,1]. (5)

An 𝛼-solution 𝑥_𝛼(⋅) of (4) is understood to be an absolutely continuous function 𝑥_𝛼∶[0,𝑇]→𝑅^𝑛 which satisfies (5) almost everywhere. Let 𝑋_𝛼 denote the 𝛼-solution set of (5) and let 𝑋_𝛼(𝑡)={𝑥_𝛼(𝑡)∶𝑥_𝛼(⋅)∈𝑋_𝛼}. Clearly, a family of subsets 𝑋_𝑡={𝑋_𝛼(𝑡)∶𝛼∈[0,1]} cannot satisfy the conditions of Theorem 2 (see [5, 6, 9]).

Therefore, we will consider an 𝑅-solution of fuzzy differential inclusion (4).

Definition 3. The upper semicontinuous fuzzy mapping 𝑋∶[0,𝑇]→𝐸^𝑛 which satisfies the system sup_[]𝛼∈0,1ℎ[𝑋(𝑡+𝜎)]^𝛼,_{𝑥∈[𝑋(𝑡)]𝛼}𝑥+_𝑡^𝑡+𝜎[𝐹(𝑠,𝑥)]^𝛼(𝜎)(0)d𝑠=𝑜,𝑋=𝑋₀ (6) is called the 𝑅-solution 𝑋(⋅) of differential inclusion (4), where lim_𝜎→0+(𝑜(𝜎)/𝜎)=0.

Theorem 4. Suppose that the following conditions hold:(1)fuzzy mapping 𝐹(⋅,𝑥) is measurable, for all 𝑥∈𝑅^𝑛;(2)there exists 𝜆>0 such that, for all 𝑥^′,𝑥^′′∈𝑅^𝑛, 𝐷𝐹𝑡,𝑥^′,𝐹𝑡,𝑥^′′≤𝜆𝑥^′−𝑥^′′, (7) for almost every 𝑡∈[0,𝑇];(3)there exists 𝛾>0 such that 𝐷(𝐹(𝑡,𝑥),0)≤𝛾, for almost every 𝑡∈[0,𝑇] and every 𝑥∈𝑅^𝑛;(4)for all 𝛽∈[0,1], 𝑥^′,𝑥^′′∈𝑅^𝑛 and almost every 𝑡∈[0,𝑇], 𝛽𝐹𝑡,𝑥^′+1−𝛽𝐹𝑡,𝑥^′′⊂Ft,𝛽𝑥^′+1−𝛽𝑥^′′. (8)Then, there exists a unique 𝑅-solution 𝑋(⋅) of fuzzy system (4) defined on the interval [0,𝜏]⊆[0,𝑇].

Proof. Let 𝑆_𝑟(𝑋₀)={𝑋∈𝐸^𝑛∶𝐷(𝑋,𝑋₀)≤𝑟} and 𝜏=min{𝑇,𝑟/𝛾}.
By [5, 6], it follows that a family of subsets 𝑋_𝑡={𝑋_𝛼(𝑡)∶𝛼∈[0,1]} satisfy the conditions of Theorem 2; that is, 𝑋_𝑡∈𝐸^𝑛, for every 𝑡∈[0,𝜏].
Divide the interval [0,𝜏] into partial intervals by the points 𝑡^𝑃_𝑘=𝑘𝜏2^−𝑝, 𝑘=0,…,𝑃, 𝑃=2^𝑝, 𝑝∈𝑁. We use Euler algorithm; let the mapping 𝑋^𝑃(⋅) be given by 𝑋^𝑃(𝑡)^𝛼=_{𝑥∈[𝑋(𝑡𝑃𝑘)]𝛼}𝑥+^𝑡_𝑡𝑃𝑘[𝐹(𝑠,𝑥)]^𝛼d𝑠, (9) where 𝑡∈[𝑡^𝑃_𝑘,𝑡^𝑃_𝑘+1], 𝑘∈{0,…,𝑃}, 𝑋(0)=𝑋₀, 𝛼∈[0,1].
By [7, 28], it follows that the sequence {[𝑋^𝑃(⋅)]^𝛼}^∞_𝑝=1 is equicontinuous and fundamental and its limit is a unique 𝑅-solution [𝑋(⋅)]^𝛼 of differential inclusion (5) and [𝑋(𝑡)]^𝛼=𝑋_𝛼(𝑡), for every 𝑡∈[0,𝜏] and 𝛼∈[0,1]. This concludes the proof.

Also, we consider the differential inclusion 𝑦̇∈𝐺𝑡,𝑦,𝑦(0)∈𝑌₀, (10) where 𝑦∈𝑅^𝑛, 𝑡∈[0,𝑇]⊂𝑅₊, 𝐹∶[0,𝑇]×𝑅^𝑛→𝐸^𝑛, 𝑌₀∈𝐸^𝑛.

Lemma 5. Let 𝐹(𝑡,𝑥) and 𝐺(𝑡,𝑦) satisfy conditions (1)–(4) of Theorem 4 and there exist 𝜂>0 and 𝜇>0 such that 𝐷^𝑡2_𝑡1𝐹(𝑠,𝑥)𝑑𝑠,^𝑡2_𝑡1𝐺(𝑠,𝑥)𝑑𝑠<𝜂𝑡₂−𝑡₁,𝐷𝑋₀,𝑌₀<𝜇, (11) for every 𝑥∈𝑅^𝑛 and 𝑡₂>𝑡₁, 𝑡₁,𝑡₂∈[0,𝑇].
Then 𝐷(𝑋(𝑡),𝑌(𝑡))≤𝜇𝑒^𝜆𝑡+(𝜂/𝜆)(𝑒^𝜆𝑡−1), for every 𝑡∈[0,𝑇].

Proof. Divide the interval [0,𝑇] into partial intervals by the points 𝑡^𝑚_𝑘=𝑘Δ, Δ=(𝑇/𝑚), 𝑘=0,…,𝑚, 𝑚∈𝑁. By Definition 3, we have 𝐷(𝑋(𝑡),𝑌(𝑡))≤sup_[]𝛼∈0,1ℎ_{𝑥∈[𝑋(𝑡𝑘)]𝛼}𝑥+^𝑡_𝑡𝑘[𝐹(𝑠,𝑥)]^𝛼d𝑠_{𝑦∈[𝑌(𝑡𝑘)]𝛼},𝑦+^𝑡_𝑡𝑘𝐺𝑠,𝑦^𝛼(Δ)d𝑠+𝑜≤sup_𝛼∈[0,1]ℎ𝑋𝑡_𝑘^𝛼+^𝑡_𝑡𝑘𝐹𝑠,𝑋𝑡_𝑘^𝛼𝛼𝑌𝑡d𝑠,_𝑘𝛼+^𝑡_𝑡𝑘𝐺𝑠,𝑌𝑡_𝑘^𝛼𝛼(Δ)d𝑠+𝑜≤sup_[]𝛼∈0,1ℎ𝑋𝑡_𝑘^𝛼+^𝑡_𝑡𝑘𝐹𝑠,𝑋𝑡_𝑘^𝛼𝛼𝑋𝑡d𝑠,_𝑘𝛼+^𝑡_𝑡𝑘𝐹𝑠,𝑌𝑡_𝑘^𝛼𝛼d𝑠+sup_[]𝛼∈0,1ℎ𝑋𝑡_𝑘^𝛼+^𝑡_𝑡𝑘𝐹𝑠,𝑌𝑡_𝑘^𝛼𝛼𝑌𝑡d𝑠,_𝑘𝛼+^𝑡_𝑡𝑘𝐹𝑠,𝑌𝑡_𝑘^𝛼𝛼d𝑠+sup_[]𝛼∈0,1ℎ𝑌𝑡_𝑘^𝛼+^𝑡_𝑡𝑘𝐹𝑠,𝑌𝑡_𝑘^𝛼𝛼𝑌𝑡d𝑠,_𝑘𝛼+^𝑡_𝑡𝑘𝐺𝑠,𝑌𝑡_𝑘^𝛼𝛼(Δ)d𝑠+𝑜≤sup_[]𝛼∈0,1ℎ^𝑡_𝑡𝑘𝐹𝑠,𝑋𝑡_𝑘^𝛼𝛼d𝑠,_𝑡𝑡𝑘𝐹𝑠,𝑌𝑡_𝑘^𝛼𝛼d𝑠+sup_[]𝛼∈0,1ℎ𝑋𝑡_𝑘^𝛼,𝑌𝑡_𝑘^𝛼≤sup_[]𝛼∈0,1ℎ^𝑡_𝑡𝑘𝐹𝑠,𝑋𝑡_𝑘^𝛼𝛼d𝑠,_𝑡𝑡𝑘𝐺𝑠,𝑌𝑡_𝑘^𝛼𝛼d𝑠+𝑜(Δ)≤^𝑡_𝑡𝑘𝜆𝐷𝑋𝑡_𝑘,𝑌𝑡_𝑘𝑋𝑡d𝑠+𝐷_𝑘,𝑌𝑡_𝑘+𝜂𝑡−𝑡_𝑘(Δ)≤+𝑜𝑡−𝑡_𝑘𝜆+1𝐷𝑋𝑡_𝑘,𝑌𝑡_𝑘+𝜂𝑡−𝑡_𝑘+𝑜(Δ)≤𝜇𝑒^𝜆𝑡+𝜂𝜆𝑒^𝜆𝑡−1 (12) for every 𝑡∈[0,𝑇]. This concludes the proof.

Remark 6. If 𝑋₀=𝑌₀, then 𝐷(𝑋(𝑡),𝑌(𝑡))≤(𝜂/𝜆)(𝑒^𝜆𝑡−1), for every 𝑡∈[0,𝑇].

4. The Method of Partial Averaging

Now, consider the fuzzy differential inclusion with a small parameter 𝑥̇∈𝜀𝐹(𝑡,𝑥),𝑥(0)∈𝑋₀, (13) where 𝑋∈𝑅^𝑛, 𝑡∈𝑅₊, 𝐹∶𝑅₊×𝑅^𝑛→𝐸^𝑛, 𝑋₀∈𝐸^𝑛, and 𝜀>0 is a small parameter.

In this work, we associate the following partial averaged fuzzy differential inclusion with the inclusion (10): 𝑦̇∈𝜀𝐺𝑡,𝑦,𝑦(0)∈𝑋₀, (14) where 𝐺∶𝑅₊×𝑅^𝑛→𝐸^𝑛 such that lim_𝑇→∞𝐷1𝑇^𝑇₀𝐹()1𝑡,𝑥d𝑡,𝑇^𝑇₀𝐺(𝑡,𝑥)d𝑡=0. (15)

Theorem 7. Let in domain 𝑄={(𝑡,𝑥)∶𝑡≥0,𝑥∈𝐷∈𝑐𝑜𝑛𝑣(𝑅^𝑛)} the following conditions hold:(1)mappings 𝐹(⋅,𝑥), 𝐺(⋅,𝑥) are measurable on 𝑅₊;(2)mappings 𝐹(𝑡,⋅), 𝐺(𝑡,⋅) satisfy a Lipschitz condition 𝐷𝐹𝑡,𝑥^′,𝐹𝑡,𝑥^′′≤𝜆𝑥^′−𝑥^′′,𝐷𝐺𝑡,𝑥^′,𝐺𝑡,𝑥^′′≤𝜆𝑥^′−𝑥^′′, (16) with a Lipschitz constant 𝜆>0;(3)there exists 𝛾>0 such that 𝐷𝐹(𝑡,𝑥),0≤𝛾,𝐷𝐺(𝑡,𝑥),0≤𝛾; (17) for almost every 𝑡∈[0,𝑇] and every 𝑥∈𝑅^𝑛;(4)for all 𝛽∈[0,1], 𝑥^′,𝑥^′′∈𝑅^𝑛 and almost every 𝑡∈[0,𝑇], 𝛽𝐹𝑡,𝑥^′+1−𝛽𝐹𝑡,𝑥^′′⊂𝐹𝑡,𝛽𝑥^′+1−𝛽𝑥^′′,𝛽𝐺𝑡,𝑥^′+1−𝛽𝐺𝑡,𝑥^′′⊂𝐺𝑡,𝛽𝑥^′+1−𝛽𝑥^′′. (18)(5)limit (15) exists uniformly with respect to x in the domain 𝐺;(6)for any 𝑋₀([𝑋₀]⁰⊂𝐷^′⊂𝐷), 𝜀∈(0,𝜈], and 𝑡>0, the 𝑅-solution of the inclusion (10) together with a 𝜌-neighborhood belongs to the domain G; that is, [𝑋(𝑡)]⁰+𝑆_𝜌(0)⊂𝐷, for every 𝑡>0.
Then, for any 𝜂∈(0,𝜌] and L > 0, there exists 𝜀₀(𝜂,𝐿)>0 such that, for all 𝜀∈(0,𝜀₀] and 𝑡∈[0,𝐿𝜀⁻¹], the following inequality holds: 𝐷(𝑋(𝑡),𝑌(𝑡))<𝜂, (19) where 𝑋(⋅), 𝑌(⋅) are the 𝑅-solutions of initial and partial averaged inclusions.

Proof. Divide the interval [0,𝐿𝜀⁻¹] on the partial intervals by the points 𝑡_𝑘=(𝑘𝐿/𝜀𝑚), 𝑘∈{0,1,…,𝑚−1}. We denote fuzzy mappings 𝑋^𝑚(⋅) and 𝑌^𝑚(⋅) such that 𝑋^𝑚(𝑡)^𝛼=_{𝑥∈[𝑋(𝑡𝑘)]𝛼}𝑥+𝜀^𝑡_𝑡𝑘[𝐹(𝑠,𝑥)]^𝛼,𝑋d𝑠_𝑚(0)^𝛼=𝑋₀^𝛼,𝑌^𝑚(𝑡)^𝛼=_{𝑦∈[𝑌(𝑡𝑘)]𝛼}𝑦+𝜀^𝑡_𝑡𝑘𝐺𝑠,𝑦^𝛼,𝑌d𝑠_𝑚(0)^𝛼=𝑋₀^𝛼, (20)(21) for every 𝛼∈[0,1], 𝑡∈[𝑡_𝑘,𝑡_𝑘+1], 𝑘∈{0,1,…,𝑚−1}.
Then 𝐷𝑋^𝑚𝑡_𝑘,𝑋𝑡_𝑘≤sup_𝛼∈[0,1]ℎ_{𝑥∈[𝑋𝑚(𝑡𝑘−1)]𝛼}𝑥+𝜀^𝑡𝑘_𝑡𝑘−1[𝐹(𝑠,𝑥)]^𝛼d𝑠_{𝑥∈[𝑋(𝑡𝑘−1)]𝛼},𝑥+𝜀^𝑡𝑘_𝑡𝑘−1[𝐹(𝑠,𝑥)]^𝛼𝑡d𝑠+𝑜_𝑘−𝑡_𝑘−1≤1+𝜀𝑡_𝑘−𝑡_𝑘−1𝜆𝐷𝑋^𝑚𝑡_𝑘−1,𝑋𝑡_𝑘−1+𝑜𝑡_𝑘−𝑡_𝑘−1≤𝑜𝑡_𝑘−𝑡_𝑘−1𝑡_𝑘−𝑡_𝑘−1𝑒^𝜆𝐿−1. (22)
Also, we take 𝐷𝑌^𝑚𝑡_𝑘,𝑌𝑡_𝑘≤𝑜𝑡_𝑘−𝑡_𝑘−1𝑡_𝑘−𝑡_𝑘−1𝑒^𝜆𝐿−1. (23)
As for 𝑡∈[𝑡_𝑘,𝑡_𝑘+1], 𝐷𝑋^𝑚(𝑡),𝑋^𝑚𝑡_𝑘≤sup_[]𝛼∈0,1ℎ×_{𝑥∈[𝑋𝑚(𝑡𝑘)]𝛼}𝑥+𝜀^𝑡_𝑡𝑘[𝐹(𝑠,𝑥)]^𝛼d𝑠,𝑋^𝑚𝑡_𝑘^𝛼≤𝜀𝛾𝑡−𝑡_𝑘≤𝛾𝐿𝑚,𝐷𝑌^𝑚(𝑡),𝑌^𝑚𝑡_𝑘≤𝜀𝛾𝑡−𝑡_𝑘≤𝛾𝐿𝑚. (24)(25)
Using estimates (22)–(25), for any 𝜂>0, there exists 𝑚₀ such that, for 𝑚>𝑚₀, we have 𝐷𝑋^𝑚(𝑡)(𝑡)≤𝜂,𝑋4,𝐷𝑌^𝑚(𝑡)(𝑡)≤𝜂,𝑌4. (26)
Taking into account Lemma 5, for any 𝜈>0, there exists 𝜀₀>0 such that, for all 𝜀∈(0,𝜀₀], the following inequality holds: 𝐷𝑋^𝑚𝑡_𝑘+1,𝑌𝑡_𝑘+1≤𝜈𝜆𝑒^𝜆𝐿−1. (27)
By combining (26) and (27) and choosing 𝑚≥max{𝑚₀,8𝛾𝐿/𝜂} and 𝜈<(𝜂𝜆/4(𝑒^𝜆𝐿−1)), we obtain (19). The theorem is proved.

5. Conclusion

If 𝐹(⋅,𝑥) is continuous on [0,𝑇], then, instead of (5), it is possible to consider the following more simple equation: sup_[]𝛼∈0,1ℎ[𝑋(𝑡+𝜎)]^𝛼,_{𝑥∈[𝑋(𝑡)]𝛼}𝑥+𝜎[𝐹(𝑡,𝑥)]^𝛼=𝑜(𝜎),𝑋(0)=𝑋₀, (28) and, similarly, we can prove all the results received earlier.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.