The fuzzy dynamic programming problems

Abstract

In this work we consider the fuzzy dynamic programming problems. For this purpose, using the generalized Hukuhara differentiability for fuzzy functions, new concepts, for example: the fuzzy product, the fuzzy collocation and the Bellman’s principle, we have the neccesary and sufficient conditions.

Keywords

Fuzzy number fuzzy Hausdrorff metric space generalized Hukuhara differentiability the fuzzy dynamic programming problems Bellman’s principle

1 Introduction

In fuzzy metric Hausdorff space Eⁿ, there are several approaches to the study of fuzzy functions and fuzzy differential equations. One popular approach is based on H-differentiability. The first method was based on H-difference notation and was further investigated by the authors, for example: Allahviranloo [3 –7], Bede [9 –13], Kaleva [20], Lakshmikantham [23, 24], etc... The second approach is based on Zadeh’s extension principle [30] can be found in [32, 39], etc... Another approach can be found in [7 , 15] and it is based on strongly generalized differentiability of fuzzy functions [9, 37].

In this case the derivative exists and the solution of a fuzzy differential equation may have decreasing length of the support, but the uniqueness is lost. Therefore, our point is that the new concept generalized H - differentiability can be of great help in the dynamic study of relative problems of fuzzy functions, for example, the conditions of fuzzy optimality problems, for example: Najariyan [27, 28], Osuna - Gomez, Chalco-Cano,... [31], Ramik [35] and fuzzy differential equationswith fuzzy initial conditions, for example: Bede [9], Hoa, Phu [22], Tri [38], Vu [39].

In this work we consider the fuzzy dynamic programming problems, where the objectives are fuzzy functions, that are generalized H - differentiable. The paper is organized as follows: in Section 2, we recall some basic concepts and notations about fuzzy numbers, fuzzy function and some kinds of Hukuhara derivatives of fuzzy functions, which are useful in next sections. In Sections 3 we present the fuzzy dynamic programming problems. In Section 4 give illustration for example of the algorithm to find the necessary and sufficient conditions.

2 Preliminaries

We recall some notations and concepts presented in detail in recent series works of Professor Lakshmikantham V. et al …(see [3, 4]). Let $K_{C} (ℝ^{n})$ denote the collection of all nonempty, compact and convex subsets of $ℝ^{n} .$ Given A, B in $K_{C} (ℝ^{n}),$ the Hausdorff distance between A and B defined as

$\begin{matrix} d_{H} [A, B] & = & \max {sup_{a \in A} inf_{b \in B} ‖ a - b ‖ R^{n}, \\ sup_{b \in B} inf_{a \in A} ‖ a - b ‖ R^{n}}, \end{matrix}$ (2.1) where ∥ . ∥ _{R
ⁿ} denotes the Euclidean norm in $ℝ^{n}$ . It is known that $(K_{C} (ℝ^{n}), d_{H})$ is a complete metric space and if the space $K_{C} (ℝ^{n})$ is equipped with the natural algebraic operations of addition and nonegative scalar multiplication, then $K_{C} (ℝ^{n})$ becomes a semilinear metric space which can be embedded as a complete cone into a corresponding Banach space. Set $E^{n} = {ω : ℝ^{n} \to [0, 1] such that ω (z) satisfies (i) - (iv) stated below}$

ω is normal, that is, there exists an $z_{0} \in ℝ^{n}$ such that ω (z₀) =1;

ω is fuzzy convex, that is, for 0 ≤ λ ≤ 1 $ω (λ z_{1} + (1 - λ) z_{2}) \geq min {ω (z_{1}), ω (z_{2})};$

ω is upper semicontinuous;

$[ω]^{0} = cl {z \in ℝ^{n} : ω (z) > 0}$ is compact.

The element ω ∈ Eⁿ is called a fuzzy set. Whenn = 1, elements of E¹ are often called the fuzzy numbers.

The set $[ω]^{α} = {z \in ℝ^{n} : ω (z) \geq α, 0 < α \leq 1}$ is called the α-level set. For all 0 ⩽ α ⩽ β ⩽ 1 then we have [ω] ^β ⊂ [ω] ^α ⊂ [ω] ⁰.

For two fuzzy sets ω₁, ω₂, we denote ω₁ ⩽ ω₂ if and only if [ω₁] ^α ⊂ [ω₂] ^α. Let us denote $D_{0} [ω_{1}, ω_{2}] = sup_{α \in [0, 1]} {d_{H} ([ω_{1}]^{α}, [ω_{2}]^{α})}$ (2.2) the distance between ω₁ and ω₂ in Eⁿ, where d_H [[ω] ^α, [ω] ^α] is Hausdorff distance between two set [ω₁] ^α, [ω₂] ^α of $K_{C} (ℝ^{n})$ . Then (Eⁿ, D₀) is a complete space. Some properties of metric D₀ are as follows. $\begin{matrix} D_{0} [ω_{1} + ω_{3}, ω_{2} + ω_{3}] & = & D_{0} [ω_{1}, ω_{2}], \\ D_{0} [λ ω_{1}, λ ω_{2}] & = & | λ | D_{0} [ω_{1}, ω_{2}], \\ D_{0} [ω_{1}, ω_{2}] & \leq & D_{0} [ω_{1}, ω_{3}] + D_{0} [ω_{3}, ω_{2}], \end{matrix}$ for all ω₁, ω₂, ω₃ ∈ Eⁿ and $λ \in ℝ$ . Given an interval $[a, b] \subseteq ℝ_{+} .$

Let us denote θⁿ ∈ Eⁿ the zero element of Eⁿ as follows: $θ^{n} (z) = {\begin{matrix} 1, if z = \hat{0} \\ 0, if z \neq \hat{0}, \end{matrix}$ where $\hat{0}$ is the zero element of $ℝ^{n}$ .

Definition 2.1. Let x, y : [a, b] → E¹ be the fuzzy functions, that means $[x (t)]^{α} = [\underline{x} (t, α), \bar{x} (t, α)]$ and $[y (t)]^{α} = [\underline{y} (t, α), \bar{y} (t, α)], \forall α \in [0, 1]$ . We say that

scalar product λx (t) exists if $[λ x (t)]^{α} = λ k, k \in [x (t)]^{α}, \forall t \in [a, b], α \in [0, 1];$

fuzzy product z (t) = x (t) . y (t) exists if $\begin{matrix} \underline{z} (t, α) & = & min {\underline{x} (t, α) . \underline{y} (t, α), \underline{x} (t, α) . \bar{y} (t, α), \\ \bar{x} (t, α) . \underline{y} (t, α), \bar{x} (t, α) . \bar{y} (t, α)}; \\ \bar{z} (t, α) & = & max {\underline{x} (t, α) . \underline{y} (t, α), \underline{x} (t, α) . \bar{y} (t, α), \\ \bar{x} (t, α) . \underline{y} (t, α), \bar{x} (t, α) . \bar{y} (t, α)} . \end{matrix}$

Definition 2.2 Let x, y : [a, b] → E¹ be a fuzzy function, that means $[x (t)]^{α} = [\underline{x} (t, α), \bar{x} (t, α)]$ and $[y (t)]^{α} = [\underline{y} (t, α), \bar{y} (t, α)], \forall α \in [0, 1]$ . We say that x (t) ≺ y (t) if and only if satisfies one of the followings:

$\underline{x} (t, α) \geq \underline{y} (t, α), \bar{x} (t, α) ⩽ \bar{y} (t, α), \forall t \in [a, b], α \in [0, 1]$ ;

[x (t)] ^α⊆ [y (t)] ^α, ∀ t ∈ [a, b] , α ∈ [0, 1] ;

D₀ [x (t) , θ] ⩽ D₀ [y (t) , θ] , ∀ t ∈ [a, b] .

Definition 2.3. Let x, y ∈ Eⁿ . If there exists z ∈ Eⁿ such that x = y + z then z is called a Hukuhara difference of x, y and it is denoted z = x ⊖ y .

Definition 2.4. (see [9]) Let x, y ∈ Eⁿ, the generalized Hukuhara difference of x and y is $x \underset{gH}{⊖} y = z \Leftrightarrow {\begin{matrix} (i) x = y + z, \\ or (ii) y = x + (- 1) z . \end{matrix}$ (2.3)

Definition 2.5. (see [24]) Let x : [a, b] → E¹ be a fuzzy function, that means $[x (t)]^{α} = [\underline{x} (t, α), \bar{x} (t, α)], \forall α \in [0, 1]$ . We say x is classical Hukuhara differentiable at t₀ ∈ (a, b) if there exists an element D_Hx (t₀) ∈ Eⁿ such that the limits

$\begin{matrix} lim_{h \to 0^{+}} h^{- 1} (x (t_{0} + h) ⊖ x (t_{0})) and \\ lim_{h \to 0^{+}} h^{- 1} (x (t_{0}) ⊖ x (t_{0} - h)) \end{matrix}$ (2.4) exist and equal to D_Hx (t₀) ∈ Eⁿ. Here the limits are taken in the metric space (Eⁿ, D₀), and at boundary points we consider only the one-side derivatives.

Definition 2.6. (see [22]) Let x : (a, b) → Eⁿ and t ∈ (a, b). We say that x is strongly generalized differentable at t, if there exists $D_{H}^{g} x (t) \in E^{n}$ , such that either (FH^g1) for all h > 0 sufficiently small, the generalized H-differences $x (t + h) \underset{gH}{⊖} x (t), x (t) \underset{gH}{⊖} x (t - h)$ exist and the limits (in the metric D₀) $\begin{matrix} lim_{h \to 0^{+}} D_{0} [\frac{x (t + h) ⊖_{gH} x (t)}{h}, D_{H}^{g} x (t)] \\ = lim_{h \to 0^{+}} D_{0} [\frac{x (t) ⊖_{gH} x (t - h)}{h}, D_{H}^{g} x (t)] = 0 \end{matrix}$ or (FH^g2) for all h > 0 sufficiently small, the generalized H-differences $x (t) \underset{gH}{⊖} x (t + h), x (t - h) \underset{gH}{⊖} x (t)$ exist and the limits $\begin{matrix} lim_{h \to 0^{+}} D_{0} [\frac{x (t) ⊖_{gH} x (t + h)}{- h}, D_{H}^{g} x (t)] \\ = lim_{h \to 0^{+}} D_{0} [\frac{x (t - h) ⊖_{gH} x (t)}{- h}, D_{H}^{g} x (t)] = 0 \end{matrix}$ or (FH^g3) for all h > 0 sufficiently small, the generalized H-differences $x (t + h) \underset{gH}{⊖} x (t), x (t - h) \underset{gH}{⊖} x (t)$ exist and the limits $\begin{matrix} lim_{h \to 0^{+}} D_{0} [\frac{x (t + h) ⊖_{gH} x (t)}{h}, D_{H}^{g} x (t)] \\ = lim_{h \to 0^{+}} D_{0} [\frac{x (t - h) ⊖_{gH} x (t)}{- h}, D_{H}^{g} x (t)] = 0 \end{matrix}$ or (FH^g4) for all h > 0 sufficiently small, the generalized H-differences $x (t) \underset{gH}{⊖} x (t + h), x (t) \underset{gH}{⊖} x (t - h)$ exist and the limits $\begin{matrix} lim_{h \to 0^{+}} D_{0} [\frac{x (t) ⊖_{gH} x (t + h)}{- h}, D_{H}^{g} x (t)] \\ = lim_{h \to 0^{+}} D_{0} [\frac{x (t) ⊖_{gH} x (t - h)}{h}, D_{H}^{g} x (t)] = 0 . \end{matrix}$

In this definition, case (FH^g1) corresponds to the classic H-derivative, so this differentiability concept is a generalization of the Hukuhara derivative. A function that is strongly generalized differentiable as in case (FH^g1) and (FH^g2), will be referred as (FH^g1) - differentiable or as(FH^g2) - differentiable, respectively. As for cases (FH^g3) and (FH^g4), it is known that (see [9]) a function may be differentiable as in (FH^g3) or (FH^g4) only on a discrete set of points (where differentiability switches between cases (FH^g1) and (FH^g2). In this paper we consider only the two first generalized H-differentiabilities of Definition 2.6. In the other cases, the derivative is trivial because it is reduced to a crisp element.

Definition 2.7. Let x : [a, b] → Eⁿ and f : [a, b] × Eⁿ → Eⁿ . We say that the fuzzy function f (t, x) is strongly generalized partial differentable at x if there exists $\frac{\partial f}{\partial x} \in E^{n}$ , such that for m ∈ Eⁿ we have: $\begin{matrix} lim_{s \to 0^{+}} D_{0} [\frac{f (t, x + sm) ⊖_{gH} f (t, x)}{s}, \frac{\partial f}{\partial x}] \\ = lim_{s \to 0^{+}} D_{0} [\frac{f (t, x - sm) ⊖_{gH} f (t, x)}{s}, \frac{\partial f}{\partial x}] = 0 . \end{matrix}$

3 Main results

3.1 The fuzzy dynamic programming problems

Definition 3.1. We consider the fuzzy dynamic programming problems: $(FDPP) {\begin{matrix} min_{u} I (u), \\ I (u) = \int_{0}^{T} φ (t, x (t), u (t)) dt + h (x (T), \\ D_{H}^{g} x (t) = f (t, x (t), u (t)), \\ x (0) = x_{0}, x (T) = x_{T} \in E^{1} . \end{matrix}$ (3.1) where $f, φ : ℝ \times E^{1} \times E^{1} \to E^{1}$ , the fuzzy states $x (t), D_{H}^{g} x (t) \in E^{1}$ and the admissible fuzzy control u (t) ∈ E¹ are fuzzy functions, t ∈ [0, T], the $I (u) = \int_{0}^{T} φ (t, x (t), u (t)) dt + h (x (T)) \to min$ means that there is exist I (u^*) such that I (u^*) ≺ I (u) for all admissible fuzzy control u (t) ∈ E¹ .

We have to solve this fuzzy dynamic programming problems (FDPP) (3.1), that means need to find the neccesary and sufficient conditions for FDPP. Solving this problems by Bellman’s principle, we have the Hamilton - Jacobi - Bellman (HJB) partial equation: $\frac{\partial V}{\partial t} + inf_{u \in E^{1}} [\frac{\partial V}{\partial x} f (t, x, u) + φ (t, x, u)] = θ$ (3.2) with the boundary conditions V (T, x (T)) = h (x (T)) , V (0, x (0)) = V (0, x₀) . The fuzzy product $\frac{\partial V}{\partial x} f (t, x, u)$ exists in force (ii) of Definition 2.1. Putting $W (t, x, u) = \frac{\partial V}{\partial t} + \frac{\partial V}{\partial x} f (t, x, u) + φ (t, x, u) .$ (3.3)

Lemma 3.1. (supplement lemma) Assume that V (t, x) is solution of HJB (3.2) which satisfy V (T, x (T)) = h (x (T)) , V (0, x (0)) = V (0, x₀) . If u (t) is any admissible fuzzy control, such that W (t, x, u) ≥0 then for FDPP (3.1) exists the estimate: $I (u) ≻ V (0, x_{0}) .$ (3.4)

Proof. Putting P (t) = V (t, x (t)) , where V (t, x) is solution of HJB (3.2) and x (t) - solution of ${\begin{matrix} D_{H}^{g} x (t) = f (t, x (t), u (t)) \\ x (0) = x_{0}, x (T) = x_{T} \in E^{1} . \end{matrix}$ In problem FDPP (3.1) if admissible fuzzy controls are $u \in U_{[0, T]} \subseteq E^{1},$ then for any interval (γ, β) ⊂ [0, T] , we have $D_{H}^{g} P (t) = \frac{\partial V}{\partial t} + \frac{\partial V}{\partial x} f (t, x, u) + φ (t, x, u) .$ We infer that $\begin{matrix} P (β) ⊖ P (γ) \\ = \int_{α}^{β} (\frac{\partial V}{\partial t} + \frac{\partial V}{\partial x} f (t, x, u) + φ (t, x, u)) dt . \end{matrix}$ From HJB (3.2), we have $φ (t, x, u) + \frac{\partial V}{\partial x} f (t, x, u) ≻ - \frac{\partial V}{\partial t} (t, x), \forall u \in U_{[0, T]} .$ and $\frac{\partial V}{\partial x} f (t, x, u) + \frac{\partial V}{\partial t} (t, x) ≻ - φ (t, x, u), \forall u \in U_{[0, T]}$ $P (β) ⊖ P (γ) ≻ - \int_{γ}^{β} φ (t, x, u) dt, \forall u \in U_{[0, T]} .$ So that $\begin{matrix} V (β, x (β)) ⊖ V (γ, x (γ)) \\ ≻ - \int_{γ}^{β} φ (t, x, u) dt, \forall u \in U_{[0, T]} \\ V (T, x (T)) ⊖ V (0, x (0)) \\ ≻ - \int_{γ}^{β} φ (t, x, u) dt, \forall u \in U_{[0, T]} \end{matrix}$ or $h (x (T)) + \int_{γ}^{T} φ (t, x, u) dt ≻ V (0, x_{0}), \forall u \in U_{[0, T]}$ and $\begin{matrix} I (u) & = & \int_{γ}^{T} φ (t, x, u) dt + h (x (T)) \\ ≻ & V (0, x_{0}), \forall u \in U_{[0, T]} . □ \end{matrix}$

3.2 The neccesary conditions of fuzzy dynamic programming problems

Theorem 3.1. (Necessary conditions) If FDPP (3.1) has solution, that means there exists optimal control u^* (t) such that $I (u^{*}) = min_{u} I (u)$ and V^* (t, x) is a solution of HJB- partial differential Equation (3.2) with V^* (T, x) = h (x (T)) then the necessary conditions for FDPP (3.1) are:

$\int_{0}^{T} W^{*} (t, x (t), u^{*} (t)) dt = θ,$ (3.5) where W^* (t, x, u^*) was denoted in (3.3) according to solution V^* (t, x).

Proof. Because W (t, x, u) was denoted in (3.3) then W (t, x, u) satisfies:

$W (t, x, u) = \frac{\partial V}{\partial t} + \frac{\partial V}{\partial x} f (t, x, u) + φ (t, x, u) ≻ θ .$ By Lemma 3.1, if u (t) is admissible control then for fuzzy dynamic programming problems FDPP (3.1) exists estimation

$\begin{matrix} \int_{0}^{T} W (t, x (t), u (t)) dt \\ = \int_{0}^{T} [\frac{\partial V}{\partial t} + \frac{\partial V}{\partial x} f (t, x, u) + φ (t, x, u)] dt \\ = \int_{0}^{T} φ (t, x (t), u (t)) dxdt + V (T, x (T)) ⊖ V (0, x_{0}) \\ = \int_{0}^{T} φ (t, x (t), u (t)) dxdt + h (x (T) ⊖ V (0, x_{0}) \\ = I (u) ⊖ V (0, x_{0}) ≻ θ . \end{matrix}$ Because FDPP (3.1) satisfies (3.4) and has optimal control u^* (t) for all t ∈ [0, T] , such that $I (u^{*}) = min_{u} I (u)$ then we have:

$\int_{0}^{T} W^{*} (t, x (t), u^{*} (t)) dt = I (u^{*}) ⊖ V^{*} (0, x_{0}) = θ .$ □

3.3 The sufficient conditions of fuzzy dynamic programming problems

Theorem 3.2. (Sufficient conditions) Assume that u- any admissible control for FDPP (3.1) and V (t, x) solution of partial differential equation HJB (3.2), then the sufficient conditions of fuzzy dynamic programming problems FDPP (3.1) are: $I (u^{*}) = V^{*} (0, x_{0}),$ (3.6) and u^* (t) will be optimization control for FDPP (3.1).

Proof. Because W (t, x, u) was denoted in (3.3), it satisfies: $W (t, x, u) = \frac{\partial V}{\partial t} + \frac{\partial V}{\partial x} f (t, x, u) + φ (t, x, u) ≻ θ .$ For any admissible control $u \in U_{[0, T]} \subseteq E^{1}$ and V (t, x) - solution of HJB (3.2), that satisfy V (T, x (T)) = h (x (T)), then we have $I (u) = \int_{0}^{T} φ (t, x (t), u (t)) dxdt + h (x (T) .$ We infer that $\begin{matrix} I (u) ⊖ V (0, x_{0}) \\ = \int_{0}^{T} φ (t, x (t), u (t)) dxdt + V (T, x (T)) ⊖ V (0, x_{0}) \\ = \int_{0}^{T} [\frac{\partial V}{\partial t} + \frac{\partial V}{\partial x} f (t, x, u) + φ (t, x, u)] dt \\ = \int_{0}^{T} W (t, x (t), u (t)) dt ≻ θ . \end{matrix}$ For any control u^*, we have solution V^* (t, x) of HJB (3.2) so implies that: $I (u^{*}) ⊖ V^{*} (0, x_{0}) = \int_{0}^{T} W^{*} (t, x (t), u^{*} (t)) dt = θ$ then we have: $I (u^{*}) = V^{*} (0, x_{0})$ and u^* (t) will be optimization control for FDPP(3.1). □

4 Illustrations

Example 4.1. We consider the fuzzy dynamic programming problems: $(FDPP) {\begin{matrix} min_{u (t) \in E^{1}} I (u) \\ D_{H}^{g} x (t) = x (t) + u (t), t \in [0, T] \\ x (0) = x_{0}, x (T) = x_{T}, \\ I (u) = \frac{1}{2} x^{2} (T) + \int_{0}^{T} \frac{r}{2} u^{2} (t) dt, \end{matrix}$ (4.1) where α ∈ [0, 1] , x₀, x_T, x (t) , u (t) ∈ E¹, r > 0, and [x₀] ^α = [-1 + α, 1 - α] .

Building the Hamilton - Jacobi - Bellman (HJB) partial equation for FDPP (4.2): ${\begin{matrix} \frac{\partial V}{\partial t} (t, x) + inf_{u \in E^{1}} (\frac{r}{2} u^{2} (t) + (x + u) \frac{\partial V}{\partial x} (t, x)) = θ, \\ V (T, x) = \frac{1}{2} x^{2} (T) . \end{matrix}$ (4.2) Putting $W (t, x, u) = \frac{\partial V}{\partial t} (t, x) + (x + u) \frac{\partial V}{\partial x} (t, x) + \frac{r}{2} u^{2} (t),$ where u (t) admissible controls for (4.1) such that ${\begin{matrix} \frac{\partial W}{\partial u} (u) = θ \\ \frac{\partial^{2} W}{\partial u^{2}} (u) > θ \end{matrix} \Leftrightarrow {\begin{matrix} ru + \frac{\partial V}{\partial x} = θ \\ r > 0 \end{matrix} \Leftrightarrow {\begin{matrix} u = - \frac{1}{r} \frac{\partial V}{\partial x} \\ r > 0, \end{matrix}$ (4.3) we have $\begin{matrix} \frac{r}{2} (- \frac{1}{r} \frac{\partial V}{\partial x})^{2} + \frac{\partial V}{\partial x} . (x - \frac{1}{r} \frac{\partial V}{\partial x}) = - \frac{\partial V}{\partial t} (t, x) \\ \Leftrightarrow \frac{1}{2 r} (\frac{\partial V}{\partial x})^{2} + x \frac{\partial V}{\partial x} - \frac{1}{r} (\frac{\partial V}{\partial x})^{2} = - \frac{\partial V}{\partial t} (t, x) \\ \Leftrightarrow - \frac{1}{2 r} (\frac{\partial V}{\partial x})^{2} + x \frac{\partial V}{\partial x} = - \frac{\partial V}{\partial t} (t, x) . \end{matrix}$ Solving the HJB (4.2) we have V (t, x) : ${\begin{matrix} - \frac{\partial V}{\partial t} = x \frac{\partial V}{\partial x} - \frac{1}{2 r} (\frac{\partial V}{\partial x})^{2} \\ V (T, x) = \frac{1}{2} x^{2} (T) . \end{matrix}$ V (t, x) will be under type $V (t, x) = \frac{1}{2} S (t) x^{2},$ we have $\frac{\partial V}{\partial t} = \frac{1}{2} x^{2} (t) \frac{dS (t)}{dt} + x \frac{dx (t)}{dt} S (t)$ and because S (T) =1 then S (t) satisfies ${\begin{matrix} - (\frac{1}{2} x^{2} (t) \frac{dS (t)}{dt} + x \frac{dx (t)}{dt} S (t)) \\ = x^{2} S (t) - \frac{1}{2 r} . S^{2} (t) . x^{2} (t) \\ S (T) = 1 . \end{matrix}$ We have $\begin{matrix} {\begin{matrix} - (\frac{1}{2} x^{2} \frac{dS (t)}{dt} + x \frac{dx (t)}{dt} S (t)) \\ = x^{2} S (t) - \frac{1}{2 r} . S^{2} (t) . x^{2} (t) \\ S (T) = 1 \end{matrix} \\ \Leftrightarrow {\begin{matrix} - (\frac{1}{2} x^{2} \frac{dS (t)}{dt} + x (x + u) S (t)) \\ = x^{2} S (t) - \frac{1}{2 r} . S^{2} (t) . x^{2} (t) \\ S (T) = 1 \end{matrix} \\ \Leftrightarrow {\begin{matrix} - (\frac{1}{2} x^{2} \frac{dS (t)}{dt} + xuS (t) + x^{2} S (t)) \\ = x^{2} S (t) - \frac{1}{2 r} . S^{2} (t) . x^{2} (t) \\ S (T) = 1 \end{matrix} \\ \Leftrightarrow {\begin{matrix} \frac{1}{2} x^{2} \frac{dS (t)}{dt} + xuS (t) = \frac{1}{2 r} . S^{2} (t) . x^{2} (t) - 2 x^{2} S (t) \\ S (T) = 1 \end{matrix} \\ \Leftrightarrow {\begin{matrix} x^{2} \frac{dS (t)}{dt} - \frac{1}{r} S^{2} (t) . x^{2} (t) + 2 x (u + 2 x) S (t) = θ \\ S (T) = 1 . \end{matrix} \end{matrix}$ Replacing $u (t) = \frac{1}{r} \frac{\partial V}{\partial x}$ from (4.3) into this Cauchy problem, we have: ${\begin{matrix} x^{2} \frac{dS (t)}{dt} - \frac{1}{r} S^{2} (t) . x^{2} (t) \\ + 2 x (- \frac{1}{r} \frac{\partial V}{\partial x} + 2 x) S (t) = θ \\ S (T) = 1 \end{matrix}$ and replacing else $\frac{\partial V}{\partial x} = xS (t)$ into this Cauchy problem, we get: $\begin{matrix} {\begin{matrix} x^{2} \frac{dS (t)}{dt} - \frac{1}{r} S^{2} (t) . x^{2} (t) \\ + 2 x (- \frac{1}{r} xS (t) + 2 x) S (t) = θ \\ S (T) = 1 \end{matrix} \\ \Leftrightarrow {\begin{matrix} x^{2} \frac{dS (t)}{dt} = \frac{1}{r} S^{2} (t) . x^{2} (t) \\ - 2 x (- \frac{1}{r} xS (t) + 2 x) S (t) \\ S (T) = 1 \end{matrix} \\ \Leftrightarrow {\begin{matrix} x^{2} \frac{dS (t)}{dt} = \frac{3}{r} S^{2} (t) . x^{2} (t) - 4 x^{2} S (t) \\ S (T) = 1 \end{matrix} \\ \Leftrightarrow {\begin{matrix} \frac{dS (t)}{dt} = \frac{3}{r} S^{2} (t) - 4 S (t), x \neq θ \\ S (T) = 1 . \end{matrix} \end{matrix}$ This Cauchy problem of Riccati equation has one solution: $S (t) = \frac{4}{\frac{3}{r} - (\frac{3}{r} - 4) e^{4 (t - T)}}$ that implies $\begin{matrix} V (t, x) = \frac{1}{2} S (t) x^{2} & = & \frac{1}{2} \frac{4}{\frac{3}{r} - (\frac{3}{r} - 4) e^{4 (t - T)}} x^{2} \\ = & \frac{2 x^{2}}{\frac{3}{r} - (\frac{3}{r} - 4) e^{4 (t - T)}}, \end{matrix}$

$\begin{matrix} u (t) = - \frac{1}{r} \frac{\partial V}{\partial x} & = & - \frac{1}{r} \frac{4 x}{\frac{3}{r} - (\frac{3}{r} - 4) e^{4 (t - T)}} \\ = & \frac{- 4 x}{3 - (3 - 4 r) e^{4 (t - T)}} . \end{matrix}$ Finally, we have fuzzy control u (t) as feedback u (t) = u (x (t)) then need to find the optimal trajectory: $\begin{matrix} D_{H}^{g} x (t) = x (t) + u (t) \\ \Leftrightarrow D_{H}^{g} x (t) = x (t) + \frac{- 4 x (t)}{3 - (3 - 4 r) e^{4 (t - T)}} \\ \Leftrightarrow D_{H}^{g} x (t) = (1 - \frac{4}{3 - (3 - 4 r) e^{4 (t - T)}}) x (t) \\ D_{H}^{g} x (t) = \frac{1 + (3 - 4 r) e^{4 (t - T)}}{(3 - 4 r) e^{4 (t - T)} - 3} x (t) \end{matrix}$ (4.4) with fuzzy solution $x (t) = C e^{- \int p (t) d t},$ where $p (t) = \frac{1 + (3 - 4 r) e^{4 (t - T)}}{(3 - 4 r) e^{4 (t - T)} - 3} .$ We have the neccesary conditions of FDPP (4.1): $\begin{matrix} W^{*} (t, x^{*}, u^{*}) \\ = \frac{\partial V^{*}}{\partial t} (t, x) + (x^{*} + u^{*}) \frac{\partial V^{*}}{\partial x} (t, x) + \frac{r}{2} u^{* 2} (t) = θ \end{matrix}$ and the sufficient conditions of FDPP (4.1): $\begin{matrix} I (u^{*}) = V^{*} (0, x_{0}) = \frac{2 x_{0}^{2}}{\frac{3}{r} - (\frac{3}{r} - 4) e^{- 4 T}}, \\ u^{*} = u (x^{*}) = \frac{- 4 x_{0}}{3 - (3 - 4 r) e^{- 4 T}}, \end{matrix}$ that are in the real forms: $\begin{matrix} [I (u^{*})]^{α} & = & [\frac{2}{\frac{3}{r} - (\frac{3}{r} - 4) e^{- 4 T}} x_{0}^{2}]^{α} \\ = & \frac{2 [- 1 + 2 α - α^{2}, 1 - 2 α + α^{2}]}{\frac{3}{r} - (\frac{3}{r} - 4) e^{- 4 T}}, \end{matrix}$ ${[u^{*}]}^{α} = [\frac{- 4 x_{0}}{3 - (3 - 4 r) e^{- 4 T}}]^{α} = \frac{- 4 [- 1 + α, 1 - α]}{3 - (3 - 4 r) e^{- 4 T}},$ with product of fuzzy functions $[x_{0}^{2}]^{α} = [- 1 + 2 α - α^{2}, 1 - 2 α + α^{2}], \forall α \in [0, 1] .$

Example 4.2. We consider the other fuzzy dynamic programming problems: $(FDPP) {\begin{matrix} min_{u (t) \in E^{1}} I (u), \\ D_{H}^{g} x (t) = Cg (t) + u (t), t \in [0, T], \\ x (0) = x_{0} = C, x (T) = x_{T}, \\ I (u) = \frac{1}{2} x^{2} (T) + \int_{0}^{T} \frac{r}{2} [x^{2} (t) + u^{2} (t)] dt, \end{matrix}$ (4.5) where [C] ^α = [0, 1 - α] , α ∈ [0, 1) , x (t) , u (t) ∈ E¹, g (t) ∈ C ([0, T] , R⁺) , g (0) =1, r > 0 .

Building the Hamilton - Jacobi - Bellman (HJB) partial equation for FDPP (4.5): ${\begin{matrix} \frac{\partial V}{\partial t} (t, x) \\ + inf_{u \in E^{1}} (\frac{r}{2} [x^{2} (t) + u^{2} (t)] + [Cg (t) + u] \frac{\partial V}{\partial x} (t, x)) = θ, \\ V (T, x) = \frac{1}{2} x^{2} (T) . \end{matrix}$ (4.6) Putting $\begin{matrix} W (t, x, u) & = & \frac{\partial V}{\partial t} (t, x) + [Cg (t) + u] \frac{\partial V}{\partial x} (t, x) \\ + \frac{r}{2} [x^{2} (t) + u^{2} (t)], \end{matrix}$ where u (t) admissible controls for (4.1) such that ${\begin{matrix} \frac{\partial W}{\partial u} (u) = θ \\ \frac{\partial^{2} W}{\partial u^{2}} (u) > θ \end{matrix} \Leftrightarrow {\begin{matrix} ru + \frac{\partial V}{\partial x} = θ \\ r > 0 \end{matrix} \Leftrightarrow {\begin{matrix} u = - \frac{1}{r} \frac{\partial V}{\partial x} \\ r > 0 . \end{matrix}$ (4.7) We have $\begin{matrix} \frac{r}{2} [x^{2} (t) + (- \frac{1}{r} \frac{\partial V}{\partial x})^{2}] \\ + \frac{\partial V}{\partial x} . (Cg (t) - \frac{1}{r} \frac{\partial V}{\partial x}) = - \frac{\partial V}{\partial t} \\ \Leftrightarrow \frac{1}{2 r} (\frac{\partial V}{\partial x})^{2} + Cg (t) \frac{\partial V}{\partial x} - \frac{1}{r} (\frac{\partial V}{\partial x})^{2} \\ = - \frac{\partial V}{\partial t} (t, x) - \frac{r}{2} x^{2} (t) \\ \Leftrightarrow - \frac{1}{2 r} (\frac{\partial V}{\partial x})^{2} + Cg (t) \frac{\partial V}{\partial x} \\ = - \frac{\partial V}{\partial t} (t, x) - \frac{r}{2} x^{2} (t) . \end{matrix}$ Solving the HJB (4.6) we have V (t, x) : ${\begin{matrix} - \frac{\partial V}{\partial t} = Cg (t) \frac{\partial V}{\partial x} - \frac{1}{2 r} (\frac{\partial V}{\partial x})^{2} + \frac{r}{2} x^{2} (t) \\ V (T, x) = \frac{1}{2} x^{2} (T) . \end{matrix}$ This solutions V (t, x) will be under type $V (t, x) = \frac{1}{2} S (t) x^{2},$ it implies that $\frac{\partial V}{\partial t} = \frac{1}{2} x^{2} (t) \frac{dS (t)}{dt} + x \frac{dx (t)}{dt} S (t)$ and because S (T) =1 then S (t) satisfies $\begin{matrix} {\begin{matrix} - (\frac{1}{2} x^{2} (t) \frac{dS (t)}{dt} + \frac{dx (t)}{dt} S (t) x (t)) \\ = Cg (t) S (t) x (t) - \frac{1}{2 r} S^{2} (t) x^{2} (t) + \frac{r}{2} x^{2} (t) \\ S (T) = 1 . \end{matrix} \\ \Leftrightarrow {\begin{matrix} - (\frac{1}{2} x^{2} (t) \frac{dS (t)}{dt} + (Cg (t) + u (t)) S (t) x (t)) \\ = Cg (t) S (t) x (t) - \frac{1}{2 r} S^{2} (t) x^{2} (t) + \frac{r}{2} x^{2} (t) \\ S (T) = 1 \end{matrix} \\ \Leftrightarrow {\begin{matrix} - (\frac{1}{2} x^{2} (t) \frac{dS (t)}{dt} + x (t) u (t) S (t)) \\ = - \frac{1}{2 r} S^{2} (t) x^{2} (t) + 2 Cg (t) S (t) x (t) + \frac{r}{2} x^{2} (t) \\ S (T) = 1 \end{matrix} \\ \Leftrightarrow {\begin{matrix} \frac{1}{2} x^{2} (t) \frac{dS (t)}{dt} + x (t) u (t) S (t) \\ = \frac{1}{2 r} S^{2} (t) x^{2} (t) - 2 Cg (t) S (t) x (t) - \frac{r}{2} x^{2} (t) \\ S (T) = 1 \end{matrix} \\ \Leftrightarrow {\begin{matrix} x^{2} (t) \frac{dS (t)}{dt} + 2 x (t) u (t) S (t) \\ = \frac{1}{r} S^{2} (t) x^{2} (t) - 4 Cg (t) S (t) x (t) - {rx}^{2} (t) \\ S (T) = 1 . \end{matrix} \end{matrix}$ Replacing $u (t) = - \frac{1}{r} \frac{\partial V}{\partial x}$ into this system, we have: ${\begin{matrix} x^{2} (t) \frac{dS (t)}{dt} - 2 x (t) \frac{1}{r} \frac{\partial V}{\partial x} S (t) \\ = \frac{1}{r} S^{2} (t) x^{2} (t) - 4 Cg (t) S (t) x (t) - {rx}^{2} (t) \\ S (T) = 1 \end{matrix}$ and replacing else $\frac{\partial V}{\partial x} = x (t) S (t)$ into this system, we have: $\begin{matrix} {\begin{matrix} x^{2} (t) \frac{dS (t)}{dt} - \frac{2}{r} x^{2} (t) S^{2} (t) \\ = \frac{1}{r} S^{2} (t) x^{2} (t) - 4 Cg (t) S (t) x (t) - {rx}^{2} (t) \\ S (T) = 1 \end{matrix} \\ \Leftrightarrow {\begin{matrix} x^{2} (t) \frac{dS (t)}{dt} \\ = \frac{3}{r} S^{2} (t) x^{2} (t) - 4 Cg (t) S (t) x (t) - {rx}^{2} (t) \\ S (T) = 1 \end{matrix} \\ \Leftrightarrow {\begin{matrix} \frac{dS (t)}{dt} = \frac{3}{r} S^{2} (t) - \frac{4 Cg (t) S (t)}{x (t)} - r, x \neq θ \\ S (T) = 1 . \end{matrix} \end{matrix}$ We consider the following differential equation: $S^{'} + \frac{4 Cg}{x} S (t) = \frac{3}{r} S^{2} (t) - r,$ (4.8) and the Bernoulli differential equation: $S^{'} + \frac{4 Cg}{x} S (t) = \frac{3}{r} S^{2} (t) .$ (4.9) We divide (4.9) by S² : $S^{'} S^{- 2} + \frac{4 Cg}{x} S^{- 1} = \frac{3}{r} .$ (4.10) Putting Z = S^-1, then Z′ = - S′S^-2 and the Equation (4.10) is: $\begin{matrix} - Z^{'} + \frac{4 Cg}{x} Z = \frac{3}{r} \\ \Leftrightarrow Z^{'} - \frac{4 Cg}{x} Z = - \frac{3}{r} . \end{matrix}$ Putting $p (t) = - 4 C \int_{0}^{t} \frac{g (γ) d γ}{x (γ)},$ the above equation has a solution: $\begin{matrix} Z = e^{- p (t)} [- \int \frac{3}{r} e^{p (t)} dt + K], K : const, \\ \Rightarrow S = \frac{e^{p (t)}}{K - \int \frac{3}{r} e^{p (t)} dt} . \end{matrix}$ Consider K = K (t) , we have $\begin{matrix} S^{'} & = & \frac{p^{'} e^{p (t)} (K - \int \frac{3}{r} e^{p (t)} dt) - e^{p (t)} (K^{'} - \frac{3}{r} e^{p (t)})}{(K - \int \frac{3}{r} e^{p (t)} dt)^{2}} \\ = & \frac{e^{p (t)} [p^{'} (t) (K - \int \frac{3}{r} e^{p (t)} dt) - (K^{'} - \frac{3}{r} e^{p (t)})]}{(K - \int \frac{3}{r} e^{p (t)} dt)^{2}} . \end{matrix}$ From (4.8), we have $\begin{matrix} \frac{e^{p (t)} [p^{'} (t) (K - \int \frac{3}{r} e^{p (t)} dt) - (K^{'} - \frac{3}{r} e^{p (t)})]}{(K - \int \frac{3}{r} e^{p (t)} dt)^{2}} \\ + \frac{4 Cg}{x} \frac{e^{p (t)}}{K - \int \frac{3}{r} e^{p (t)} dt} = \frac{3}{r} S^{2} - r \\ \Rightarrow e^{p} [p^{'} (t) (K - \int \frac{3}{r} e^{p (t)} dt) - (K^{'} - \frac{3}{r} e^{p (t)})] \\ + \frac{4 Cg}{x} e^{p (t)} (K - \int \frac{3}{r} e^{p (t)} dt) \\ = (\frac{3}{r} S^{2} - r) (K - \int \frac{3}{r} e^{p (t)} dt)^{2} \\ \Rightarrow e^{p} (t) (K - \int \frac{3}{r} e^{p (t)} dt) (p^{'} + \frac{4 Cg}{x}) \\ - (K^{'} - \frac{3}{r} e^{p (t)}) e^{p} (t) \\ = (\frac{3}{r} S^{2} - r) (K - \int \frac{3}{r} e^{p (t)} dt)^{2} . \end{matrix}$ Since $p^{'} (t) = - \frac{4 Cg (t)}{x (t)},$ we have $\begin{matrix} - (K^{'} - \frac{3}{r} e^{p (t)}) e^{p} (t) = (\frac{3}{r} S^{2} - r) (K - \int \frac{3}{r} e^{p (t)} dt)^{2} \\ \Leftrightarrow (K^{'} - \frac{3}{r} e^{p (t)}) e^{p} = (r - \frac{3}{r} S^{2}) (K - \int \frac{3}{r} e^{p (t)} dt)^{2} \\ \Leftrightarrow {\begin{matrix} (K^{'} - \frac{3}{r} e^{p (t)}) = θ \\ (r - \frac{3}{r} S^{2}) (K - \int \frac{3}{r} e^{p (t)} dt)^{2} = θ \end{matrix} \\ \Leftrightarrow {\begin{matrix} K^{'} = \frac{3}{r} e^{p (t)} \\ r = \frac{3}{r} S^{2} or K - \int \frac{3}{r} e^{p (t)} dt = 0 \end{matrix} \\ \Leftrightarrow {\begin{matrix} K = \int \frac{3}{r} e^{p (t)} dt + C_{1}, C_{1} : const, \\ S^{2} = \frac{r^{2}}{3} : \\ const, unreasonable (because S (T) = 1, r > 0) . \end{matrix} \\ \Leftrightarrow K = \int \frac{3}{r} e^{p (t)} dt + C_{1} . \end{matrix}$ The Equation (4.8) has one solution $S (t) = \frac{e^{p (t)}}{\int \frac{3}{r} e^{p (t)} dt + C_{1} - \int \frac{3}{r} e^{p (t)} dt} = \frac{e^{p (t)}}{C_{1}},$ where $p (t) = - 4 C \int_{0}^{t} \frac{g (γ) d γ}{x (γ)}, C_{1} is a const,$ and $S (T) = 1, V (T, x (T)) = \frac{x^{2} (T)}{2} .$ That implies $e^{- 4 C \int_{0}^{T} \frac{g (γ) d γ}{x (γ)}} = C_{1}$ $\begin{matrix} V (t, x) & = & \frac{1}{2} S (t) x^{2} (t) = \frac{x^{2} e^{p}}{2 C_{1}} = \frac{x^{2} e^{- 4 C \int_{0}^{t} \frac{g (γ) d γ}{x (γ)}}}{2 C_{1}}, \\ u (t) & = & - \frac{1}{r} \frac{\partial V}{\partial x} \\ = & - \frac{x (t)}{r C_{1}} e^{- 4 C \int_{0}^{t} \frac{g (γ) d γ}{x (γ)}} [1 + 2 Cx (t) \int_{0}^{t} \frac{g (γ) d γ}{x^{2} (γ)}] . \end{matrix}$ We have the neccesary conditions of FDPP (4.5): $\begin{matrix} W^{*} (t, x^{*}, u^{*}) \\ = \frac{\partial V^{*}}{\partial t} (t, x) + [Cg (t) + u^{*}] \frac{\partial V^{*}}{\partial x} (t, x) \\ + \frac{r}{2} [x^{* 2} (t) + u^{* 2} (t)] = θ, \end{matrix}$ and the sufficient conditions of FDPP (4.5): $I (u^{*}) = V^{*} (0, x_{0}) = \frac{x_{0}^{2}}{2 C_{1}},$ $u^{*} = u (x^{*}) = u (x_{0}) = - \frac{x_{0}}{{rC}_{1}}$ that are in the real forms: $[I (u^{*})]^{α} = [\frac{x_{0}^{2}}{2 C_{1}}]^{α} = \frac{[0, 1 - 2 α + α^{2}]}{2 C_{1}},$ ${[u^{*}]}^{α} = [- \frac{x_{0}}{{rC}_{1}}]^{α} = - \frac{[0, 1 - α]}{{rC}_{1}},$ with product of fuzzy functions $[x_{0}^{2}]^{α} = [C^{2}]^{α} = [0, 1 - 2 α + α^{2}],$ where [C] ^α = [0, 1 - α] , ∀ α ∈ [0, 1) , r > 0 and condition: $e^{- 4 C \int_{0}^{T} \frac{g (γ) d γ}{x (γ)}} = C_{1} .$

Example 4.3. We consider the other fuzzy dynamic programming problems: $(FDPP) {\begin{matrix} min_{u (t) \in E^{1}} I (u), \\ D_{H}^{g} x (t) = x (t) u (t), t \in [0, T], \\ x (0) = x_{0}, x (T) = x_{T} \in E^{1}, \\ I (u) = \frac{1}{2} x^{2} (T) + \int_{0}^{T} \frac{r}{2} u^{2} (t) dt, \end{matrix}$ (4.11) where [x₀] ^α = [1 + α, 3 - α] , α ∈ [0, 1] , x (t) , u (t) ∈ E¹, r > 0 .

Building the Hamilton - Jacobi - Bellman (HJB) partial equation for FDPP (4.11): ${\begin{matrix} \frac{\partial V}{\partial t} (t, x) + inf_{u \in E^{1}} (x (t) u \frac{\partial V}{\partial x} (t, x) + \frac{r}{2} u^{2} (t)) = θ, \\ V (T, x) = \frac{1}{2} x^{2} (T) . \end{matrix}$ (4.12) Putting $W (t, x, u) = \frac{\partial V}{\partial t} (t, x) + x (t) u \frac{\partial V}{\partial x} (t, x) + \frac{r}{2} u^{2} (t),$ where u (t) admissible controls for (4.11) such that ${\begin{matrix} \frac{\partial W}{\partial u} (u) = θ \\ \frac{\partial^{2} W}{\partial u^{2}} (u) > θ \end{matrix} \Leftrightarrow {\begin{matrix} ru + x \frac{\partial V}{\partial x} = θ \\ r > 0 \end{matrix} \Leftrightarrow {\begin{matrix} u = - \frac{x}{r} \frac{\partial V}{\partial x} \\ r > 0 . \end{matrix}$ (4.13) We have $\begin{matrix} \frac{r}{2} (- \frac{x}{r} \frac{\partial V}{\partial x})^{2} + \frac{\partial V}{\partial x} . (x . \frac{- x}{r} \frac{\partial V}{\partial x}) = - \frac{\partial V}{\partial t} (t, x) \\ \Leftrightarrow \frac{x^{2}}{2 r} (\frac{\partial V}{\partial x})^{2} - \frac{x^{2}}{r} (\frac{\partial V}{\partial x})^{2} = - \frac{\partial V}{\partial t} (t, x) \\ \Leftrightarrow \frac{- x^{2}}{2 r} (\frac{\partial V}{\partial x})^{2} = - \frac{\partial V}{\partial t} (t, x) . \end{matrix}$ Finally, we have V (t, x) : ${\begin{matrix} \frac{\partial V}{\partial t} = \frac{x^{2}}{2 r} (\frac{\partial V}{\partial x})^{2} \\ V (T, x) = \frac{1}{2} x^{2} (T) . \end{matrix}$ V (t, x) will be under type $V (t, x) = \frac{1}{2} S (t) x^{2},$ we have $\frac{\partial V}{\partial x} = xS (t)$ and $\frac{\partial V}{\partial t} = \frac{1}{2} x^{2} (t) \frac{dS (t)}{dt} + x \frac{dx (t)}{dt} S (t)$ then S (t) satisfies $\begin{matrix} {\begin{matrix} (\frac{1}{2} x^{2} (t) \frac{dS (t)}{dt} + x \frac{dx (t)}{dt} S (t)) = \frac{x^{4}}{2 r} . S^{2} (t) \\ S (T) = 1 . \end{matrix} \\ \Leftrightarrow {\begin{matrix} (\frac{1}{2} x^{2} (t) \frac{dS (t)}{dt} + x^{2} uS (t)) = \frac{x^{4}}{2 r} . S^{2} (t) \\ S (T) = 1 . \end{matrix} \end{matrix}$ Replacing $u (t) = \frac{- x}{r} \frac{\partial V}{\partial x}$ from (4.13) into Cauchy problem, we have: ${\begin{matrix} (\frac{1}{2} x^{2} (t) \frac{dS (t)}{dt} + x^{2} (- \frac{x}{r} \frac{\partial V}{\partial x}) S (t)) = \frac{x^{4}}{2 r} . S^{2} (t) \\ S (T) = 1 . \end{matrix}$ and replacing else $\frac{\partial V}{\partial x} = xS (t)$ into this Cauchy problem, we have: $\begin{matrix} {\begin{matrix} (\frac{1}{2} x^{2} (t) \frac{dS (t)}{dt} + x^{2} (- \frac{x}{r} xS (t)) S (t)) = \frac{x^{4}}{2 r} . S^{2} (t) \\ S (T) = 1 . \end{matrix} \\ \Leftrightarrow {\begin{matrix} (\frac{1}{2} x^{2} (t) \frac{dS (t)}{dt} - \frac{x^{4}}{r} S^{2} (t)) = \frac{x^{4}}{2 r} . S^{2} (t) \\ S (T) = 1 . \end{matrix} \\ \Leftrightarrow {\begin{matrix} \frac{dS (t)}{dt} = \frac{3 x^{2}}{r} S^{2} (t) \\ S (T) = 1 . \end{matrix} \end{matrix}$ This Cauchy problem of separated variable equations has one solution: $S (t) = \frac{1}{\frac{1}{S (0)} - (\frac{x^{3} (t)}{r} - \frac{x^{3} (0)}{r})},$ where $\begin{matrix} S (T) = 1 \\ \Rightarrow \frac{1}{\frac{1}{S (0)} - (\frac{x^{3} (T)}{r} - \frac{x^{3} (0)}{r})} = 1 \\ \Rightarrow S (0) = \frac{1}{1 - (\frac{x^{3} (T)}{r} - \frac{x^{3} (0)}{r})} . \end{matrix}$ That implies $\begin{matrix} V (t, x) & = & \frac{1}{2} S (t) x^{2} (t) = \frac{1}{2} \frac{x^{2} (t)}{\frac{1}{S (0)} - (\frac{x^{3} (t)}{r} - \frac{x^{3} (0)}{r})} \\ u (t) & = & - \frac{x}{r} \frac{\partial V}{\partial x} \\ = & - \frac{x (t)}{r} \frac{\frac{4 x}{S_{0}} - \frac{2 x^{4} (t)}{r} + \frac{2 x (t) x^{3} (0)}{r} - 6 x^{4} (t)}{4 {[\frac{1}{S_{0}} - \frac{x^{3} (t)}{r} + \frac{x^{3} (0)}{r}]}^{2}} . \end{matrix}$ We have the neccesary conditions of FDPP (4.11): $\begin{matrix} W^{*} (t, x^{*}, u^{*}) & = & \frac{\partial V^{*}}{\partial t} (t, x) + (x^{*} u^{*}) \frac{\partial V^{*}}{\partial x} (t, x) \\ + \frac{r}{2} u^{* 2} (t) = θ \end{matrix}$ and the sufficient conditions of FDPP (4.11): $\begin{matrix} I (u^{*}) = V^{*} (0, x_{0}) = \frac{S_{0} x^{2} (0)}{2}, \\ u^{*} = u (x^{*}) = u (x (0)) = S_{0} \frac{6 x^{5} (0) - 4 x^{2} (0)}{4 r}, \end{matrix}$ with product of fuzzy functions $\begin{matrix} [x_{0}^{2}]^{α} & = & [1 + 2 α + α^{2}, 9 - 6 α + α^{2}], \\ \forall α \in [0, 1], r > 0 . \end{matrix}$

5 Conclusion

In this work, the fuzzy dynamic programming problems (FDPP) in fuzzy metric space - the new object, that is interesting and important were investigated. To solve this problems for the fuzzy functions, we have used some new concepts, for example: the fuzzy product z (t) = x (t) · y (t), the fuzzy collocation x (t) ≺ y (t) and the Bellman’s principle for fuzzy optimization. The necessary and sufficient conditions can be obtained by the extension of Bellman’s principle. We give also some example of the algorithm to find the necessary and sufficient conditions for FDPP. The open problems are studying of the fuzzy dynamic programming problems in n- dimension fuzzy metric space Eⁿ.

Acknowledgments

The authors are very grateful to the anoymous referees for their careful reading and many valuable remarks which improved the presentation of the paper.

References

Agarwal

R.P.

, Regan

D.O.

and Lakshmikantham

, A stacking theorem approach for fuzzy differential equations, J Nonlinear Anal 55 (2003), 299–312.

Agarwal

R.P.

, ÓRegan

and Lakshmikantham

, Viability theory and fuzzy differential equations, Fuzzys and Systems 151 (2005), 536–580.

Allahviranloo

, Kiani

N.A.

and Motamedi

, Solving fuzzy differential equations by differential transformation method, Information Sciences 179 (2009), 956–966.

Allahviranloo

, Amirteimoori

, Khezerloo

and Khezerloo

, A new method for solving fuzzy volterra integro –differential equations, Australian Journal of Basic and Applied Sciences 5(4) (2011), 154–164.

Allahviranloo

, Ghanbari

, Haghi

, Hosseinzadeh

and Nouraei

, A note on “Fuzzy linear systems, Journal of Fuzzy Sets and Systems 177 (2011), 87–92.

Allahviranloo

, Salahshour

and Abbasbandy

, Solving fuzzy fractional differential equations by fuzzy Laplace transforms, Communications in Nonlinear Science and Numerical Simulation 17 (2012), 1372–1381. doi: 10.1016/j.cnsns.2011.07.005

Allahviranloo

, Abbasbandy

, Sedaghatfar

and Darabi

, A new method for solving fuzzy integro-differential equation under generalized differentiability, J Neural Computing and Applications 21 (2012), 191–196. doi: 10.1007/s00521-011-0759-3

Barros

L.C.

, Bassanezi

R.C.

and Tonelli

P.A.

, Fuzzy modeling in population dynamics, J Ecological Modeling 128 (2000), 27–33.

Bede

and Gal

S.G.

, Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations, J Fuzzy Sets and Systems 151 (2005), 581–599.

10.

Bede

, Rudas

I.J.

and Bencsik

A.L.

, First order linear fuzzy differential equations under generalized differentiability, J Information Sciences 177 (2007), 1648–1662.

11.

Bede

and Rudas

I.J.

, Approximation properties of fuzzy transforms, J Fuzzy Sets and Systems 180 (2011), 1648–1662.

12.

Bede

, Mathematics of Fuzzy Sets and Fuzzy Logic, Springer Verlag Berlin Heidelberg, 2013, p. 276.

13.

Bede

and Stefanini

, Generalized differentiability of fuzzyvalued functions, Fuzzy Sets and Systems 230 (2013), 119–141.

14.

Bezdek

J.C.

, Partern Recognition with Fuzzy Objective Function Algoriths, Plenum Press, New York, 1981.

15.

Chalco-Cano

and Román-Flores

, On new solutions of fuzzy differential equations, Chaos Solitons Fractals 38 (2008), 112–119.

16.

Chang

S.S.L.

and Zadeh

, On fuzzy mapping and control, IEEE Transactions on System, Man and Cybernetics 2 (1972), 30–34.

17.

Dubois

and Prade

, Operation on fuzzy numbers, Int J Syst Sci 9 (1978), 613–626.

18.

Dubois

and Prade

, Towards fuzzy differential calculus, Fuzzy Sets and Systems 8 (1982), 225–233.

19.

Feng

and Hu

, On the quasi-controllability of continuoustime dynamic fuzzy control systems, Chaos, Solitons and Fractals 30 (2006), 177–188.

20.

Kaleva

, Fuzzy differential equations, Fuzzy Sets and Systems 24 (1987), 301–317.

21.

Khastan

, Nieto

J.J.

and RodrÃ–uez–LÃşez

, Variation of constant formula for first order fuzzy differential equations, Fuzzy Sets Syst 177 (2011), 20–33.

22.

Hoa

N.V.

and Phu

N.D.

, Fuzzy functional integro-differential equations under generalized H-differentiability, Journal of Intelligent and Fuzzy Systems 26 (2014), 2073–2085.

23.

Lakshmikantham

and Leela

, Fuzzy differential systems and the new concept of stability, J Nonlinear Dynamics and Systems Theory 1 (2001), 111–119.

24.

Lakshmikantham

and Mohapatra , Theory of fuzzy differential equations and inclusions Taylor Francis Inc, London, 2003.

25.

Lupulescu

, On a class of fuzzy functional differential equations, Fuzzy Sets and Systems 160 (2009), 1547–1562.

26.

Malinowski

M.T.

, Second type Hukuhara differentiable solutions to the delay set-valued differential equations, Applied Mathematics Letters 218 (2012), 9427–9437.

27.

Najariyan

and Farahi

M.H.

, Optimal control of fuzzy linear controlled systems with fuzzy initial conditions, Iranian Journal of Fuzzy Systems 10 (2013), 21–35.

28.

Najariyan

and Farahi

M.H.

, A new approach for the optimal fuzzy linear time invariant controlled system with fuzzy coefficients, Journal of Computational and Applied Mathematics 259 (2014), 682–694.

29.

Nieto

J.J.

, The Cauchy problem for continuous fuzzy differential equations, Fuzzy Sets and Systems 102 (1999), 259–262.

30.

Nguyen

H.T.

, A note on the extension principle for fuzzy sets, Journal Math Anal Aplications 64 (1978), 369–380.

31.

Osuna-Gomez

, Chalco-Cano

, Ruflian-Lizana

and Hernadez Jimenez

, Necessary and sufficient conditions for fuzzy optimality problems,20. In Press, Fuzzy Sets and Systems (2015), 20. In Press.

32.

Phu

N.D.

, An

T.V.

, Hoa

N.V.

and Hien

N.T.

, Interval-valued functional di_erential equations under dissipative conditions, J. Advances in Di_erence Equations, (2014), 2014, 198.

33.

Phu

N.D.

and Dung

L.Q.

, On the Stability and Contrllability of Fuzzy Set Control Di_erential Equations, International Journal of Reliability and Safety, Vol 5 Nos 3/4, 320-335, 2011.

34.

Puri

M.L.

and Ralescu

, Di_erential for fuzzy function, Journal of Mathematical Analysis and Applications 91 (1983), 552–558.

35.

Ramik

, Duality in fuzzy linear programming: Some new concepts and results, Journal Fuzzy Optimization and Dicision Making, Vol 4 (2005), pp. 25–39.

36.

Song

and Wu

, Existence and uniqueness of solutions to Cauchy problem of fuzzy di_erential equations, Fuzzy Sets and Systems 110 (2000), 55–67.

37.

Stefanini

, A generalization of Hukuhara di_erence and division for interval and fuzzy arithmetic, Fuzzy sets syst 161 (2010), 1564–1584.

38.

Tri

P.V.

, Hoa

N.V.

and Phu

N.D.

, Sheaf fuzzy problems for functional di_erential equations, J, Advanced in Di_erence Equations 2014 (2014), 156.

39.

, Hoa

N.V.

and Phu

N.D.

, The local existence of solutions for random fuzzy integro-diff_erential equations under generalized H-di_erentiability, Journal of Intelligent and Fuzzy Systems 26 (2014), 2701–2717.

40.

Zadeh

L.A.

, Fuzzy sets, Information and Control 8 (1965), 338–353.