Dynamic Systems and Applications 15 (2006) 147-154

BOUNDEDNESS PROPERTIES OF HYBRID FUZZY DIFFERENTIAL EQUATIONS

M. SAMBANDHAM

Department of Mathematics, Morehouse College, Atlanta, GA 30314

                              msamband@morehouse.edu

Dedicated to John Baxley on the occasion of his retirement from Wake Forest University.

ABSTRACT. By the application of the theory of fuzzy inequalities and utilizing the method of perturbing Lyapunov-like functions, we discuss the boundedness properties of hybrid fuzzy differential equations. AMS (MOS) Subject Classification. 34A19, 34D20, 34D99

1. INTRODUCTION

The study of differential equations through the interaction of both continuous valued and discrete valued variables leads to the study of hybrid systems. Hybrid systems contain mixture of logic, discrete value dynamics and continuous variable dynamics. The differential equations containing fuzzy valued functions and interaction with discrete time controller can be named as hybrid fuzzy differential equations. In this paper we develop nonuniform boundedness criteria for solutions of hybrid fuzzy differential equations under weaker assumptions using the idea of perturbation Lyapunov-like function [3].

In this section, we list some basic definitions of fuzzy valued functions needed for our discussion.

Let P_{k}(ℝ^{n}) denote the family of all nonempty compact, convex subsets of ℝ^{n}. If α, β ∈ ℝ and A, B ∈ P_{k}(ℝ^{n}), then

α(A + B) = αA + αB,  α(βA) = (αβ)A,  1A = A

and if α, β ≥ 0, then (α + β)A = αA + βA. Let I = [t_{0}, t_{0} + α], t_{0} ≥ 0 and α > 0 and denote by E^{n} = [u : ℝ^{n} → [0, 1] such that u satisfies (i) to (iv) mentioned below]:

(i) u is normal, that is, there exists an x_{0} ∈ ℝ^{n} such that u(x_{0}) = 1;

Received April 10, 2005                 1056-2176 $15.00 \begin{matrix}c \\ ◯ \end{matrix}Dynamic Publishers, Inc.

148 M. SAMBANDHAM

(ii) u is fuzzy convex, that is, for x, y ∈ ℝ^{n} and 0 ≤ λ ≤ 1,

u(λx + (1 − λ)y) ≥ min[u(x), u(y)];

(iii) u is upper semicontinuous; (iv) [u]^{0} = the closure of [x ∈ ℝ^{n} : u(x) > 0] is compact.

For 0 < α < 1, we denote [u]^{α} = [x ∈ ℝ^{n} : u(x) ≥ α]. Then form (i) to (iv), it follows that α-level sets [u]^{α} ∈ P_{k}(ℝ^{n}) for 0 ≤ α ≤ 1. For later purposes, we define ô ∈ E^{n} as ô(x) = 1 if x = 0 and ô(x) = 0 if x ≠ 0. Let d_{H}(A, B) be the Hausdorff distance between the sets A, B ∈ P_{k}(ℝ^{n}). Then we define

d[u, v] = sup_{0≤α≤1} d_{H}[[u]^{α}, [v]^{α}],

which defines a metric in E^{n} and (E^{n}, d) is a complete metric space. We list the following properties of d[u, v] (see [2])

d[u + w, v + w] = d[u, v] and d[u, v] = d[v, u],
d[λu, λv] = |λ| d[u, v],
d[u, v] ≤ d[u, w] + d[w, v],

for all u, v, w ∈ E^{n} and λ ∈ ℝ.

For x, y ∈ E^{n} if there exists a z ∈ E^{n} such that x = y + z, then z is called the H-difference of x and y and is denoted by x − y. A mapping F : I → E^{n} is differentiable at t = I if there exists a F′(t) ∈ E^{n} such that the limits

lim_{h→0^{+}} \frac{F (t + h) − F (t)}{h} and lim_{h→0^{+}} \frac{F (t) − F (t − h)}{h}

exist and are equal to F′(t). Here the limits are taken in the metric space (E^{n}, d).

In Section 2 we list some known comparison theorems for hybrid fuzzy differential equations. We present our main results in Section 3. For related results in fuzzy differential equations and hybrid fuzzy differential equations, we refer to [1], [4]–[9].

2. COMPARISON THEOREM

Consider the hybrid fuzzy differential systems

u′(t) = f(t, u(t), λ_{k}(u_{k})), t ∈ [t_{k}, t_{k+1}],
(2.1)  u(t_{k}) = u_{k},  k = 0, 1, 2, …,

where

(i) 0 ≤ t_{0} < t_{1} < ⋯ < t_{k} < ⋯ and t_{k} → ∞ as t → ∞, (ii) f ∈ C[R_{+} × E^{n} × E^{n}, E^{n}], λ_{k} ∈ C[E^{n}, E^{n}], S(ρ) = [u ∈ E^{n} : d[u, ô] < ρ], and

ô(t) = 1 if t = t_{0} and zero elsewhere.

HYBRID FUZZY DIFFERENTIAL EQUATIONS 149

Let V ∈ C[R_{+} × E^{n}, R_{+}]. For t ∈ [t_{k}, t_{k+1}], u, u_{k} ∈ E^{n}, we define

(2.2)  D^{+}V (t, u) = lim_{h→0} sup \frac{1}{h} [V (t + h, u + hf(t, u, λ_{k}(u_{k}))) − V (t, u)] .

Then we state the following comparison theorem in terms of Lyapunov-like functions V. Theorem 2.1. Assume that

(i) V ∈ C[R_{+} × E^{n}, R_{+}], V (t, u) satisfy

|V (t, u) − V (t, v)| < Ld[u, v], L > 0.

for u, v ∈ s(ρ); (ii) D^{+}V (t, u) ≤ g(t, V (t, u), σ_{k}(V (t_{k}, u_{k}))), t ∈ (t_{k}, t_{k+1}], where u, u_{k} ∈ E^{n}, k =

0, 1, 2, …, σ_{k} ∈ C[R_{+}, R_{+}], g ∈ C[R_{+}^{3}, R], (iii) The maximal solution r(t) = r(t, t_{0}, ω_{0}) of the scalar hybrid differential equation

ω′ = g(t, ω, σ_{k}(ω_{k}),  t ∈ [t_{k}, t_{k+1}]
(2.3) ω(t_{k}) = ω_{k}, k = 0, 1, 2, …

exists on [t_{0}, ∞).

Then any solution u(t) = u(t, t_{0}, u_{0}) of (2.1) such that V (t_{0}, u_{0}) ≤ ω_{0} satisfies the estimate

(2.4) V (t, u(t)) ≤ r(t, t_{0}, ω_{0}), t ≥ t_{0}

We refer [9] for the proof of Theorem 2.1. Corollary 2.1. If g(t, ω, σ_{k}(ω_{k})) ≡ 0 is admissible in Theorem 2.1, then

V (t, u(t)) ≤ V (t_{0}, u_{0}), t ≥ t_{0}.

Corollary 2.2. Assume that f ∈ C[R_{+} × E^{n} × E^{n}, E^{n}] and

lim_{h→0^{+}} sup \frac{1}{h} [d[u + hf(t, u, λ_{k}(u_{k})), ô] − d[u, ô]] ≤ g(t, d[u, ô], σ_{k}(d[u_{k}, ô])),

where g ∈ C[R_{+}^{3}, R], σ_{k} ∈ C[R_{+}, R_{+}]. Then if d[u_{0}, ô] ≤ ω_{0}, we have

d[u(t), ô] ≤ r(t, t_{0}, ω_{0}), t ≥ t_{0},

where r(t, t_{0}, ω_{0}) is the maximal solution of (2.3).

3. CRITERIA FOR BOUNDEDNESS

We define boundedness concept in the following.

Definition 3.1. The solutions of (2.1) are said to be

(B1) equi-bounded, if for any α > 0 and t_{0} ∈ R_{+} there exists a β = β(t_{0}, α) > 0 such

that d[u_{0}, ô] < α implies d[u(t), ô] < β, t ≥ t_{0} (B2) uniformly bounded, if β in (B1) does not depend on t_{0}; (B3) quasi-equi-ultimately bounded for a bounded B, if there exists a B > 0 and a

T = T (t_{0}, α) > 0 such that

d[u_{0}, ô] < α implies d[u(t), ô] < B, t ≥ t_{0} + 1

(B4) quasi-uniform ultimately bounded, if T in (B3) is independent of t_{0}; (B5) equi-ultimately bounded, if (B1) and (B3) holds simultaneously; (B6) uniformly ultimately bounded if (B2) and (B4) holds simultaneously.

Using the comparison results, in the following we discuss boundedness results by hybrid fuzzy differential equations.

Theorem 3.1. Assume that

(i) V ∈ C[R_{+} × E^{n}, R_{+}], V (t, u) satisfies

|V (t, u) − V (t, v)| ≤ Ld[u, v], L > 0,

and for (t, u) ∈ R_{+} × E^{n}, D^{+}V (t, u) ≤ 0; (ii) b(d[u, ô]) ≤ V (t, u) ≤ a(t, d[u_{0}, ô]) for (t, u) ∈ R_{+} × E^{n}, a(t, ·), b ∈ κ, where

κ : [σ ∈ C[R_{+}, R_{+}) : σ(ω) is increasing and σ(ω) → ∞ as ω → ∞].

Then (B1) holds.

Proof. Let α > 0 and t_{0} ∈ R_{+} be given. Choose β = β(t_{0}, α) such that

(3.1)  a(t_{0}, α) < b(β).

With this β, (B1) holds. If this is not true, there would exist a solution u(t) = u(t, t_{0}, u_{0}) of (2.1) and a t_{1} > t_{0} such that

d[u(t_{1}), ô] = β and d[u(t), ô] < β, t_{0} ≤ t ≤ t_{1}.

From the assumption (i) and Corollary 2.1, we obtain

(3.2)  V (t, u(t)) ≤ V (t_{0}, u_{0}),  t_{0} ≤ t ≤ t_{1}.

From assumption (ii), (3.1) and (3.2) we get

b(β) = b(d[u(t_{1}), ô]) ≤ V (t_{1}, u(t_{1})) ≤ V (t_{0}, u_{0})
≤ a(t_{0}, d[u_{0}, ô]) < a(t_{0}, α) < b(β).

This contradiction proves (B1). This completes the proof.

In the next theorem we discuss the uniform boundedness of (2.1).

Theorem 3.2. Assume that

(i) V ∈ C[R_{+} × S^{c}(ρ), R_{+}], where ρ may be large,

|V (t, u) − V (t, v) ≤ Ld[u, v],

and D^{+}V (t, u) ≤ 0 for (t, u) ∈ R_{+} × S^{c}(ρ); (ii) b(d[u, ô]) ≤ V (t, u) ≤ a[d[u, ô]), a, b ∈ κ, where a, b are defined only on [ρ, ∞).

Then (B2) is true.

Proof. The proof is similar to Theorem 3.1 except that β is independent of t_{0}, α > ρ, and

(3.3)  a(α) < b(β).

The next theorem discusses the conditions for equiultimate boundedness, that is, (B5).

Theorem 3.3. Let all the assumptions of Theorem 3.1 hold except that we define the estimate on D^{+}V (t, u) as

(3.4)  D^{+}V (t, u) ≤ −ηV (t, u),  η > 0,  (t, u) ∈ R_{+} × E^{n}

and suppose that the condition (ii) holds for d[u, ô] ≥ B. Then (B5) holds.

Proof. By Theorem 3.1, (B1) is satisfied. Hence

d[u_{0}, ô] < α implies d[u(t), ô] < β, t ≥ t_{0}.

From (3.4) we obtain the estimate

(3.5)  V (t, u(t)) ≤ V (t_{0}, u_{0})e^{−η(t−t_{0})},  t ≥ t_{0}.

Let T = \frac{1}{η} ln \frac{a(t_{0},α)}{b(B)} and suppose that for t ≥ t_{0} + T, d[u(t), ô] ≥ B. Then from (3.5) we get

b(B) ≤ b(d[u(t), ô]) ≤ V (t, u(t)) ≤ a(t_{0}, α)e^{−ηT} < b(B).

This contradiction proves (B5). This completes the proof.

In the next theorem we provide nonuniform boundedness property utilizing the method of Lyapunov functions.

Theorem 3.4. Assume that (i) ρ > 0, V_{1} ∈ C[R_{+} × S(ρ), R_{+}], V_{1} is bounded for (t, u) ∈ R_{+} × ∂S(ρ) and

|V_{1}(t, u) − V_{1}(t, v)| < L_{1}d[u, v], L_{1} > 0,
D^{+}V_{1}(t, u) = lim_{h→0^{+}} sup \frac{1}{h} [V_{1}(t + h, u + hf(t, u, λ_{k}(u_{k}))) − V_{1}(t, u)]
≤ g_{1}(t, V_{1}(t, u), σ_{k}(V_{1}(t_{k}, u_{k}))),
(t, u) ∈ R_{+} × ∂S(ρ), g_{1} ∈ C[R_{+}^{3}, R];

(ii) V_{2} ∈ C[R_{+} × S^{c}(ρ), R_{+}],

b(d[u, ô]) ≤ V_{2}(t, u) ≤ a(d[u, ô]), a(·), b(·) ∈ κ,
D^{+}V_{1} + D^{+}V_{2} ≤ g_{2}(t, V_{1}(t, u) + V_{2}(t, u), σ_{k}(V_{1}, V_{2})),

for (t, u) ∈ R_{+} × S^{c}(ρ), g_{2} ∈ C[R_{+}^{3}, R]; (iii) the scalar hybrid differential equation

(3.6)  ω′ = g_{1}(t, ω, σ_{k}(ω_{k})),  ω(t_{0}) = ω_{0} ≥ 0

is equibounded and

(3.7)  v′ = g_{2}(t, v, σ_{k}(v_{k})),  V (t_{0}) = v_{0} ≥ 0

is uniformly bounded.

Then the system (2.1) is equibounded.

Proof. Let B_{1} > ρ and t_{0} ∈ R_{+} be given. Let α_{1} = α_{1}(t_{0}, B_{1}) = max(α_{0}, α^{∗}), where α^{∗} ≥ V_{1}(t, u) for (t, u) ∈ R_{+}×∂S(ρ) and α_{0} = max[V_{1}(t_{0}, u_{0}); u_{0} ∈ closure of {S(B_{1})∩ S^{c}(ρ)}]. Since equation (3.6) is equibounded, given α_{1} > 0 and t_{0} ∈ R_{+}, there exists a β_{0} = β_{0}(t_{0}, α_{1}) such that

(3.8)  ω(t, t_{0}, ω_{0}) < β_{0},  t ≥ t_{0},

provided ω_{0} < α_{1}, where ω(t, t_{0}, ω_{0}) is any solution of (3.6). Let α_{2} = α(B_{1}) + β_{0}, then uniform boundedness of (3.7) yields that

(3.9)  V (t, t_{0}, v_{0}) < β_{1}(α_{2}),  t ≥ t_{0},

provided v_{0} < α_{2}, where v(t, t_{0}, v_{0}) is any solution of (3.7). Choose B_{2} satisfying

(3.10)  b(B_{2}) > β_{1}(α_{2}).

We now claim that u_{0} ∈ S(B_{1}) implies that u(t, t_{0}, u_{0}) ∈ S(B_{2}) for t ≥ t_{0}, where u(t, t_{0}, u_{0}) is any solution of (2.1)

If it is not true, there exists a solution u(t, t_{0}, u_{0}) of (2.1) with u_{0} ∈ S(B_{1}), such that for some t^{∗} > t_{0}, d[u(t^{∗}, t_{0}, u_{0}), ô] = B_{2}. Since B_{1} > ρ, there are two possibilities to consider:

(i) u(t, t_{0}, u_{0}) ∈ S^{c}(ρ) for t ∈ [t_{0}, t^{∗}];

HYBRID FUZZY DIFFERENTIAL EQUATIONS 153

(ii) there exists a t_{3} > t_{0} such that

u(t_{3}, t_{0}, u_{0}) ∈ ∂S(ρ) and u(t, t_{0}, u_{0}) ∈ S^{c}(ρ)

for t ∈ [t_{3}, t^{∗}].

If (i) holds, we can find t_{1} > t_{0} such that

u(t_{1}, t_{0}, u_{0}) ∈ ∂S(B_{1}),
(3.11)  u(t^{∗}, t_{0}, u_{0}) ∈ ∂S(B_{2}), and
u(t, t_{0}, u_{0}) ∈ S^{c}(B_{1}),  t ∈ [t_{1}, t^{∗}].

Setting m(t) = V_{1}(t, u(t, t_{0}, u_{0})) + V_{2}(t, u(t, t_{0}, u_{0})) for t ∈ [t_{1}, t^{∗}], then using Theorem 2.1, we obtain the differential inequality

D^{+}m(t) ≤ g_{2}(t, m(t), σ_{k}(m_{k})), t ∈ [t_{1}, t^{∗}],

and therefore

m(t) ≤ ν_{2}(t, t_{1}, m(t)), t ∈ [t_{1}, t^{∗}],

where ν_{2}(t, t_{1}, v_{0}) is the maximal solution of (3.7) with ν_{2}(t_{1}, t_{1}, v_{0}) = v_{0}. Thus

V_{1}(t^{∗}, u(t^{∗}, t_{0}, u_{0})) + V_{2}(t^{∗}, u(t^{∗}, t_{0}, u_{0}))

≤ ν_{2}(t^{∗}, t_{1}, V_{1}(t_{1}, u(t_{1}, t_{0}, u_{0})) + V_{2}(t_{1}, u(t_{1}, t_{0}, u_{0})). (3.12)

Similarly, we have

(3.13)  V_{1}(t_{1}, u(t_{1}, t_{0}, u_{0})) ≤ ν_{1}(t_{1}, t_{0}, V_{1}(t_{0}, u_{0})),

where ν_{1}(t, t_{0}, u_{0}) is the maximal solution of (3.6). Setting ω_{0} = V_{1}(t_{0}, u_{0}) < α_{1}, then from (3.8)

V_{1}(t_{1}, u(t_{1}, t_{0}, u_{0})) ≤ ν_{1}(t_{1}, t_{0}, V_{1}(t_{0}, u_{0})) ≤ β_{0}.

From (3.1) and V_{2}(t_{1}, u(t_{1}, t_{0}, u_{0})) < a(B_{1}), we have

(3.14)  v_{0} = V_{1}(t_{1}, u(t_{1}, t_{0}, u_{0})) + V_{2}(t_{1}, u(t_{1}, t_{0}, u_{0})) ≤ β_{0} + a(B_{1}) = α_{2}.

From (3.9), (3.10), (3.11) and (3.14), we obtain

(3.15)  b(B_{2}) ≤ m(t^{∗}) ≤ ν(t^{∗}) ≤ β_{1}(α_{2}) < b(B_{2}),

which is a contradiction.

If case (ii) holds, we arrive at the inequality (3.12), where t_{1} > t_{3} satisfies (3.11). We now have the relation (3.13) as

V_{1}(t_{1}, u(t_{1}, t_{0}, u_{0})) ≤ ν_{1}(t_{1}, t_{3}, V_{1}(t_{3}, u(t_{3}, t_{0}, u_{0}))).

Since u(t_{3}, t_{0}, u_{0}) ∈ ∂S(ρ) and V_{1}(t_{3}, u(t_{3}, t_{0}, u_{0})) ≤ α^{∗} ≤ α_{1}, arguing as before we get the contradiction (3.15).

This proves that for any given B_{1} > ρ, t_{0} > 0, there exists a B_{2} such that u_{0} ∈ S(B_{1}) implies u(t, t_{0}, u_{0}) ∈ S(B_{2}), t ≥ t_{0}. For B_{1} < ρ, we set B_{2}(t_{0}, B_{1}) = B_{2}(t_{0}, ρ) and hence the proof is complete.

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