Transactions Template JOURNAL OF ENGINEERING RESEARCH AND TECHNOLOGY, VOLUME 1, ISSUE 3, SEPTEMBER 2014 79 Fuzzy optimal control of a poisoning-pest model by using 𝜶-cuts Mohamad Hadi Farahi 1, Mansooreh Keshtegar 2, Marzieh Najariyan 3 1Department of Applied Mathematics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad. Mashhad .Iran, Farahi@math.um.ac.ir 2 Department of Applied Mathematics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad. Mashhad .Iran, keshtegar_mino@hotmail.com 3 Department of Applied Mathematics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad. Mashhad .Iran marzieh.najariyan@gmail.com Abstract— In this article, a dynamical system represents the poisoning-pest model is considered. At First a mathematical model for the poisoning-pest model is simulated. Since there is no exact number of pests it is natural to consider the variables as fuzzy variables. Thus we need to consider a fuzzy dynamical system to the poisoning-pest model. To solve such a fuzzy optimal control system, using 𝛼-cut, and Zadeh’s extension principle, one can convert this system to a non-fuzzy optimal control system. The final optimal control problem is solved by discrelization method. Index Terms—Fuzzy optimal control, Poisoning-pest model, Zadeh’s extension principle, Fuzzy solution, Generalized differentiability I INTRODUCTION Hukuhara differentiability (H-differentiability) for fuzzy functions was originally introduced by Puri and Raelescu in [1]. After that Kelva [2] discussed the properties of differentiable fuzzy function using Hukuhara derivative. Fuzzy differential equations are studied in several papers [3, 4]. But Hukuhara derivative has a disadvantage : the fuzziness of solution increases when time goes on. Bede and Gal in [5] introduced the generalized differentiability. The presented differentiability has not this disadvantage. Apparently the disadvantage of generalized differentiability of a function compared to H- differentiability is that a fuzzy differential equation has no unique solution. Whenever differential equation and control functions are fuzzy in an optimal control problem, we are facing with a fuzzy optimal control problem. The so-called problem considered by many authors, for example: Diamond and Kandel in [4] showed the existence of the fuzzy optimal control for the system �̇̃�(𝑡) = 𝑎(𝑡)⨀�̃�(𝑡) ⊕ �̃�(𝑡), �̃�(0) = �̃�0. Najariyan and Farahi in [6, 7] found new techniques respectively for solving linear fuzzy controlled systems with fuzzy initial conditions and fuzzy optimal linear control systems with fuzzy coefficients by using 𝛼-cuts. One of the application of optimal control problems is the problem of controlling pests. Many attempts have been made in this area (see [8, 9]). Often times we are not dealing with an exact number of pests when we want to control pests. In such cases, researchers have used of the theory of fuzzy (see [10]). This article is based on minimizing the number of pests. Because the exact number of pests is not known for us so we associate with a optimal fuzzy control. This paper is organized as follows: In Section 2 we present basic definitions and theorems of fuzzy numbers and operations of fuzzy numbers. Also in this section we have discussed the definition of Zadeh’s extension principle and generalized differentiability. In Section 3 we define optimal fuzzy control of a poisoning-pest model problem. In Section 4, we applied the technique to a real poisoning-pest model. Finally, Section 5 will give a conclusion briefly. 2 Basic concepts Let Ω be a set in ℝ, then a fuzzy subset �̃� of Ω is defined by its membership function, �̃�(𝑡), which produces values in [0,1] for all 𝑡 in Ω. So, �̃�(𝑡): Ω → [0,1]. A fuzzy number is a convex, normalized fuzzy set of the real line ℝ whose membership function is piecewise continuous and we show it as ℱ(Ω). A triangular fuzzy number �̃� is defined by three numbers 𝑎 < 𝑏 < 𝑐 where the base of the triangle is the interval [𝑎, 𝑐] and its vertex is at Mohamad Hadi Farahi, Mansooreh Keshtegar, Marzieh Najariyan / Fuzzy optimal control of a poisoning-pest model by using 𝜶-cuts (2014) 80 𝑡 = 𝑏. Triangular fuzzy numbers will be written as �̃� = (𝑎, 𝑏, 𝑐) (see[11]). If �̃� is a fuzzy number then an 𝛼-cut of �̃�, written �̃�𝛼 is defined as: 𝜇𝛼 = [𝜇]𝛼 = { {𝑥 ∈ Ω|𝜇(𝑥) ≥ 𝛼} , 0 < 𝛼 ≤ 1 {𝑥 ∈ Ω|𝜇(𝑥) > 0} , 𝛼 = 0, where �̅� denotes the closure of 𝐴 ⊂ Ω and �̃�0 is the support of �̃�, (see[12]). In this paper, we show the lower bound of �̃�𝛼 , as �̃�𝛼 and the upper bound of it as �̅̃�𝛼 . Definition 1 (Zadeh’s extension principle). Let Z be a cartesian product of universes, that is Z = Z1 × Z2 ×. . .× Zr and μ̃1, μ̃2, . . . , μ̃r be r fuzzy sets in Z1, Z2, . . . , Zr respectively and Y is a given space. Each function f: Z → Y induces corresponding function f̃ = ℱ(Z1) × ℱ(Z2) ×. . .× ℱ(Zr) → ℱ(Y) (i.e., f̃ is a function mapping fuzzy sets in Z to fuzzy sets in Y) defined for each fuzzy set μ̃ ∈ Z by f̃(μ̃1, μ̃2, . . . , μ̃r)(y) = { sup (Z1,Z2 ,...,Zr )=f −1(y) min{μ̃1(z1), μ̃2(z2), . . . , μ̃r(zr)}, f −1(y) ≠ ϕ, 0 f −1(y) = ϕ, where 𝑓 −1 is the inverse of 𝑓. The function 𝑓 is said to be obtained from 𝑓 by the extension principle. An important result of Zadeh’s extension principle is the characterization of the image levels of a fuzzy set through 𝑓, where 𝑓 is a continuous function. Therefore if 𝑓: ℝ × ℝ → ℝ is a continuous function then according to Zadeh’s extension principle one can extend 𝑓 to 𝑓: ℱ(ℝ) × ℱ(ℝ) → ℱ(ℝ) by the equation 𝑓(�̃�, 𝜈)(𝑧) = 𝑠𝑢𝑝𝑧=𝑓(𝑠,𝑡)𝑚𝑖𝑛(�̃�(𝑠), 𝜈(𝑡)). (1) It is well known that 𝑓𝛼 (�̃�, 𝜈) = 𝑓(�̃�𝛼 , 𝜈𝛼 ), 𝛼 ∈ [0,1], �̃� ∈ ℱ(ℝ), 𝜈 ∈ ℱ(ℝ). (2) Using Zadeh’s extension principle the operations of addition, ⊕ , multiplication, ⊗, and scalar multiplication, ⨀, on the ℱ(ℝ) are defined respectively by (�̃� ⊕ 𝜈)(𝑠) = 𝑠𝑢𝑝𝑡∈ℝ𝑚𝑖𝑛{�̃�(𝑡), 𝜈(𝑠 − 𝑡)}, (𝜇 ⊗ 𝜈)(𝑠) = 𝑠𝑢𝑝𝑡∈ℝ𝑚𝑖𝑛{�̃�(𝑡), 𝜈(𝑠/𝑡)}, and (𝜆⨀𝜇)(𝑠) = { 𝜇( 𝑠 𝜆 ), 𝜆 ≠ 0, 𝜒{0}, 𝜆 = 0 where 𝜒{0} is the characteristic function of 0. It is clear that the following properties are true for all 𝛼-cuts [𝜇 ⊕ 𝜈]𝛼 = 𝜇𝛼 + 𝜈𝛼 , [𝜆 ⊙ 𝜇]𝛼 = 𝜆𝜇𝛼 , 𝛼 ∈ [0,1], and [𝜇 ⊗ 𝜈]𝛼 = [𝑚𝑖𝑛{𝜇𝛼 𝜈𝛼 , 𝜇𝛼 �̅�𝛼 , �̅�𝛼 𝜈𝛼 , �̅�𝛼 �̅�𝛼 }, 𝑚𝑎𝑥{𝜇𝛼 𝜈𝛼 , 𝜇𝛼 �̅�𝛼 , �̅�𝛼 𝜈𝛼 , �̅�𝛼 �̅�𝛼 }]. According to the definition of operations of addition, scaler multiplication, the operation subtraction, ⊖, is similarly defined . Definition 2 [5] Let ũ, υ̃ ∈ ℱ(ℝ) . If there exists w̃ ∈ ℱ(ℝ) such that ũ = υ̃ ⊕ w̃, then w̃ is called the Hukuhara- difference of ũ and υ̃ and it is denoted by ũ ⊖H υ̃. Definition 3 [5]. Let x̃: T ∈ ℝ → ℱ(ℝ) and t0 ∈ T. We say that x̃ is differentiable at t0 if : (I) there exists an element �̇̃�(𝑡0) ∈ ℱ(ℝ) such that, for all ℎ > 0 sufficiently near to 0, there are �̃�(𝑡0 + ℎ) ⊖𝐻 𝑥(𝑡0), 𝑥(𝑡0) ⊝𝐻 �̃�(𝑡0 − ℎ) and the limits 𝑙𝑖𝑚ℎ→0+ �̃�(𝑡0 + ℎ) ⊖𝐻 �̃�(𝑡0) ℎ = 𝑙𝑖𝑚ℎ→0+ �̃�(𝑡0) ⊖𝐻 �̃�(𝑡0 − ℎ) ℎ = �̇̃�(𝑡0), or (II) there is an element �̇̃�(𝑡0) ∈ ℱ(ℝ) such that, for all ℎ < 0 sufficiently near to 0, there are �̃�(𝑡0 + ℎ) ⊖𝐻 �̃�(𝑡0), �̃�(𝑡0) ⊝𝐻 �̃�(𝑡0 − ℎ) and the limits 𝑙𝑖𝑚ℎ→0− �̃�(𝑡0 + ℎ) ⊖𝐻 �̃�(𝑡0) ℎ = 𝑙𝑖𝑚ℎ→0− �̃�(𝑡0) ⊝𝐻 �̃�(𝑡0 − ℎ) ℎ = �̇̃�(𝑡0). Theorem 1 [12]. Let �̃�: 𝑇 → ℱ(ℝ) be a function and denote �̃�𝛼 (𝑡) = [𝑥𝛼 (𝑡), �̅�𝛼 (𝑡)] for each 𝛼 ∈ [0,1] . Then: (i) If �̃� is differentiable in the first form (I), then 𝑥𝛼 and �̅�𝛼 are differentiable functions and �̇̃�𝛼 (𝑡) = [𝑥𝛼 (𝑡), �̇̅�𝛼 (𝑡)]. (ii) If �̃� is differentiable in the second form (II), then 𝑥𝛼 and �̅�𝛼 are differentiable functions and �̇̃�𝛼 (𝑡) = [�̇̅�𝛼 (𝑡), 𝑥𝛼 (𝑡)]. Now we consider the fuzzy initial value problem �̇̃�(𝑡) = 𝑓(𝑡, �̃�(𝑡)), �̃�(0) = �̃�0 (3) where 𝑓: [0, 𝑇] × ℱ(ℝ) → ℱ(ℝ) is obtained by Zadeh’s extension principle from a continuous function 𝑓: [0, 𝑇] × ℝ → ℝ, Note that 𝑓 is continuous because 𝑓 is continuous (see [13]), and by (2) we have 𝑓𝛼 (𝑡, �̃�) = 𝑓(𝑡, �̃�𝛼 ) where 𝑓(𝑡, 𝐴) = {𝑓(𝑡, 𝑎)|𝑎 ∈ 𝐴}. Associated with (3) we can consider the following crisp differential equation �̇� = 𝑓(𝑡, 𝑥(𝑡)), 𝑥(0) = 𝑥0 (4) where �̇�(𝑡) is the derivative of a crisp function 𝑥: [0, 𝑇] → ℝ . For more details see [14]. Theorem 2 Let �̃� ∈ ℱ(ℝ). Suppose that 𝑓 is a continuous function and for each 𝑥0 ∈ ℝ there exists a unique solution 𝑥(. , 𝑥0) for (4) and that 𝑥(𝑡, . ) is continuous in ℝ for each 𝑡 ∈ [0, 𝑇]. Then: (i) If 𝑓 is nondecreasing with respect to the second argument, then the fuzzy solution of (3) and the solution of (4) via the derivative in the first from (I) are identical. (ii) If 𝑓 is nonincreasing with respect to the second argument, then the fuzzy solution of (3) and the solution of (4) via the derivative in the second from (II), if it exists, are identical. Proof 1 See [12] 3 Optimal control of the poisoning-pest model In this section we present a fuzzy optimal control for the poisoning-pest model. we are going to determine the sufficient amount of poison to kill the approximate number of pests. Suppose that �̃�(𝑡) is the pest density, �̃�(𝑡) is the speed of poison insufflation at time 𝑡 ≥ 0 , where �̃� and �̃� are fuzzy numbers. So we hope that in interval time [0, 𝑡𝑓 ], the pest density is reduced desirable. Consider the pest density at 𝑡 = 0 is �̃�1. We want to minimize the cost of the poison and the harm done to the crop. The fuzzy control Mohamad Hadi Farahi, Mansooreh Keshtegar, Marzieh Najariyan / Fuzzy optimal control of a poisoning-pest model by using 𝜶-cuts (2014) 81 model of the poisoning-pest is as follows: �̇̃�(𝑡) = �̃�(�̃�(𝑡)) ⊖ �̃�(𝑡) ⊗ �̃�(𝑡), (5) where the initial condition �̃�(0) = �̃�1, and the final condition is �̃�(𝑡𝑓 ) = �̃�𝑓 . The function �̃�(�̃�(𝑡)) can be written as 𝑟⨀(𝑥 ̃ ⊕ (𝑥 ̃ ⊗ 𝑥 ̃ )⨀ −1 𝑘 ),where 𝑟 is the growth rate of the pest density and 𝑘 is the maximum pest density of the environment (see [15] for more details). The objective function is as: min 𝐽(�̃�(𝑡), �̃�(𝑡)) = ∫ 𝑇 0 [�̃�(�̃�(𝑡)) ⊕ ℎ̃(�̃�(𝑡))]𝑑𝑡. (6) Suppose that the function �̃�(�̃�(𝑡)) denote the cost of pest harm and the function ℎ̃(�̃�(𝑡)) denotes the expense of the poison at the time 𝑡 ≥ 0. 4 Application We consider the following fuzzy optimal control problem (see[15]): �̇̃�(𝑡) = 𝑟⨀ (�̃� ⊕ (�̃� ⊗ �̃�)⨀ −1 𝑘 ) ⊝ �̃�(𝑡) ⊗ �̃�(𝑡) (7) with cost function: 𝐽(�̃�(𝑡), �̃�(𝑡)) = ∫ 𝑡𝑓 0 [�̃�(�̃�(𝑡)) + ℎ̃(�̃�(𝑡))]𝑑𝑡 , (8) where 𝑟 = 10 9 , 𝑘 = 20, 𝑡𝑓 = 10 ,�̃�(0) = 5̃ = (4,5,6), �̃�(𝑡𝑓 = 10) = 1̃ = (0,1,2), �̃�(�̃�(𝑡)) = 10⨀�̃�(𝑡), and ℎ̃(�̃�(𝑡)) = 2⨀�̃�(𝑡). So, the fuzzy optimal control problem can be written as: min 𝐽(�̃�(𝑡), �̃�(𝑡)) = ∫ 10 0 [10⨀�̃�(𝑡) ⊕ �̃�(𝑡)]𝑑𝑡 �̇̃�(𝑡) = 10 9 ⨀(�̃� ⊕ (�̃� ⊗ �̃�)⨀ 1 −20 ) ⊖ �̃�(𝑡) ⊗ �̃�(𝑡) �̃�(0) = (4,5,6), �̃�(𝑇 = 10) = (0,1,2) We solve this problem using 𝛼-cuts technique. We consider �̃�𝛼 = [𝑥𝛼 , �̅�𝛼 ] and �̃�𝛼 = [𝑢𝛼 , �̅�𝛼 ]. In the problem 𝑓(�̃�, �̃�) = 10 9 ⨀(�̃� ⊖ 1 20 ⨀(�̃� ⊗ �̃�)) ⊖ �̃�(𝑡) ⊗ �̃�(𝑡), is obtained by Zadeh’s extension principle from a continuous function 𝑓(𝑥, 𝑢) = 10 9 (𝑥(𝑡) − 1 20 𝑥 2) − 𝑢(𝑡)𝑥(𝑡) , Since 𝑢(. ) is a bounded function, it is not difficult to show that 𝑓(𝑥, 𝑢) is an increasing function with respect to 𝑥(𝑡) for all |𝑢(𝑡)| ≤ 1 in [0,10], so we must use the first form (I) derivative. Now �̇�𝛼 = 𝑓(𝑥𝛼 , 𝑢𝛼 , �̅�𝛼 ) and �̅̇�𝛼 = 𝑓(�̅�𝛼 , 𝑢𝛼 , �̅�𝛼 ). the objective function is the average of 10𝑥𝛼 + 2𝑢𝛼 and 10�̅�𝛼 + 2�̅�𝛼 . So one interfaces the following non-fuzzy optimal control problem: 𝑚𝑖𝑛𝐽 = 1 2 ∫ 10 0 (10𝑥𝛼 (𝑡) + 10�̅�𝛼 (𝑡) + 2𝑢𝛼 (𝑡) + 2�̅�𝛼 (𝑡)) 𝑑𝑡 (9) 𝑠𝑡: �̇�𝛼 (𝑡) = 10 9 (𝑥𝛼 (𝑡) − (𝑥𝛼 2 )(𝑡) 20 ) − �̅�𝛼 (𝑡)𝑥𝛼 (𝑡) (10) �̅̇�𝛼 (𝑡) = 10 9 (�̅�𝛼 (𝑡) − (�̅�𝛼) 2(𝑡) 20 ) − 𝑢𝛼 (𝑡)�̅�𝛼 (𝑡) (11) where the initial conditions are 𝑥𝛼 (0) = 5𝛼 + 4(1 − 𝛼), �̅�𝛼 (0) = 5𝛼 + 6(1 − 𝛼) and final conditions are 𝑥𝛼 (10) = 𝛼, �̅�𝛼 (10) = 𝛼 + 2(1 − 𝛼) for all 𝛼 ∈ [0,1]. Because 𝑢𝛼 ∈ [−1,1] and �̅�𝛼 ∈ [−1,1] ,so one can define function 𝑢𝛼 = 𝐴1sin(𝑡 × 𝜋 4 ), and �̅�𝛼 = 𝐴2sin(𝑡 × 𝜋 4 ) where 𝐴1 ∈ [−1,1], 𝐴2 ∈ [−1,1]. We solve this problem by discretization method (see for more details [16]), the solutions have obtained for 𝛼 = 0,0.25,0.5,0.75,1. The solutions of 𝑥𝛼 , �̅�𝛼 and 𝑢𝛼 , �̅�𝛼 are shown in Figure 1 and Figure 2 respectively. 5 CONCLUSION Optimal fuzzy control theory is applied to a poisoning-pest problem. By applying 𝛼-cuts, and using Zadeh’s extension principle the fuzzy optimal control of a poisoning-pest system, extended to a new form involve in lower and upper state and control. Based on diseretization method, the above metioned non- fuzzy optimal control problem is solved. ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their careful reading, constractive comments, and nice suggestions which have improved the paper very much. This research was supported by a grant from Ferdowsi University of Mashhad , No MA89194FAR Figure 1: The number of pests , 𝑥𝛼 , �̅�𝛼 Figure 2: The speed of poison, 𝑢𝛼 , �̅�𝛼 Mohamad Hadi Farahi, Mansooreh Keshtegar, Marzieh Najariyan / Fuzzy optimal control of a poisoning-pest model by using 𝜶-cuts (2014) 82 REFERENCES [1] Puri. M and Ralescu. D, Differential and fuzzy functions, J.Math.Anal.Appl, 91:552–558, 1983. [2] Kelva. O, Fuzzy differential equations, Fuzzy sets and Systems , 24:301–317, 1987. [3] Daimond. P, Brief note on the variation of constants formula for fuzzy differential equations, Fuzzy sets and Systems , 129:65–217, 2002. [4] Diamond. P and Kandel. P, Metric Space of Fuzzy Sets, Theory and Application , World scientific , Singapore , 1994. [5] Barnabas Bede and Sorin G. Gal, Generaizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations, Fuzzy Sets and Systems, 151:581–599, 2005. [6] Najarian. M and Farahi. M. H, Optimal control of fuzzy linear controlled system with fuzzy initial conditions, Iranian Journal of Fuzzy Systems , 3:21–35, 2013. [7] Najariyan. M and Farahi. M. H, A new approach for optimal fuzzy linear time invariant controlled system with fuzzy coefficients , Journal of Computational and Applied Mathematics, 259:682–694, 2014. [8] Kar. T. K, Abhijit Ghorai and Soovoojeet Jana, Dynamics of pest and its predator model with disease in the pest and optimal use of pesticide, Journal of Theoretical Biology , 310: 187–197, 2012. [9] Ghosh. S and Bhattacharya. D. K, Optimization in microbial pest control: An integrated approach, Applied Mathematical Modelling, 34:1382–1395, 2010. [10] Scherm. H, Simulating uncertainty in climate–pest models with fuzzy numbers, Environmental Pollution , 108:373–379, 2000. [11] Buckley. James. J and Jawers. Leonard. J, Monte Carlo Methods in Fuzzy Optimization, Studies in Fuzziness and soft computing , 222. [12] Chalco-Cano. Y and Roman Flores, Comparation between some approaches to solve fuzzy differential equations, Fuzzy Sets and Systems, 160:1517–1527, 2009. [13] Roman Flores, Barros. L and Bassanezi. R, A note on the Zadeh’s extensions, Fuzzy sets on Banach spaces, Inform.Sci, 144:227–247, 2002. [14] Chalco-Cano. Y and Roman Flores, On the new solution of fuzzy differential equations, Chaos Solitons Fractals , 38:112–119, 2006. [15] Zhenkun Huang and Shuili Chen, Optimal fuzzy control of a poisoning-pest model, Applied Mathematics and Computation , 171: 730–737 ,2005. [16] Najariyan. M , Farahi. M. H, M. Alavian, Optimal con- trol of HIV infection by using fuzzy dynamical systems, The Journal of Mathematics and Computer Section , 2:639- 649,(2011). Mohammad Hadi Farahi is a full professor at the Department of Applied Mathematics, School of Mathematics, Ferdowsi University of Mashhad, Iran. He obtained his B.Sc, M.Sc and PhD, respective- ly from Ferdowsi University of Mashhad, Iran in 1972, Brunel University, UK in 1978, and Leeds University, UK in 1996. He has published more than 70 technical papers in international journals, and also five text books. His scientific interests include optimal control, optimization, sliding mode control, bio-mathematics, ODE 'S, PDE 'S and approximation theory. Mansooreh Keshtegar received her B.Sc and M.Sc , from Fer- dowsi University of Mashhad, Iran respectively in 2011, and 2013. Now she is doing some researchs in the area of control theo- ry. Marzieh Najariyan received her B.Sc. degree from Payam Noor University of Torbat-e-Heydarieh im 2005, M.Sc and PhD degrees from Department of Applied Mathematics, Ferdowsi University of Mashhad,Iran respectively in July 2008 and December 2013. She has published nine papers in international journals and also she has attended several national and international conferences with oral presentations. She is currently working on fuzzy differential equa- tion and fuzzy control theory.