Intelligent Control and Automation, 2011, 2, 152159 doi:10.4236/ica.2011.22018 Published Online May 2011 (http://www.SciRP.org/journal/ica) Copyright © 2011 SciRes. ICA The TimeOptimal Problems for Controlled Fuzzy RSolutions Andrej V. Plotnikov, Tatyana A. Komleva, Irina V. Molchanyuk Odessa State Academy of Civil Engineering and Architecture, Odessa, Ukraine Email: aplotnikov@ukr.net, tkomleva@ukr.net, imolchanyuk@ukr.net Received April 21, 2011; revised May 12, 2011; accepted May 15, 2011 Abstract In the present paper, we show the some properties of the fuzzy Rsolution of the control linear fuzzy differ ential inclusions and research the timeoptimal problems for it. Keywords: Fuzzy Differential Inclusions, Control Problems, TimeOptimal Problems, Fuzzy RSolution 1. Introduction The first research of the differential equations with set valued righ thand side has b een fu lfilled b y A. March aud [1,2] and S. C. Zaremba [3]. In the early sixties, T. Wazewski [4,5], A. F. Filippov [6] had been obtained fundamental results about existence and properties of solutions of the differential equations with setvalued righthand side (differential inclusions). Connection de riving between differential inclusions and optimum con trol problems was one of the most important outcomes of these papers. These outcomes became impulse for de velopment of the theory of differential in clusions [79]. Considering of the differential inclusions required to study properties of setvalued maps, i.e. an elaboration the whole tool of mathematical analysis for setvalued maps [7,10,11]. In work [12] annotate of an Rsolution for differential inclusion is introduced as an absolutely continuous set valued maps. Various problems for the Rsolution theory were considered in [8,13]. The basic idea for a develop ment of an equation for Rsolutions (integral funnels) is contained in [14]. In the eighties the last century the control theory in the conditions of uncertainty began to be formed. The con trol differential equations with set of initial conditions [1517], control set differential equation s [1821] and the control differential inclusions [2132] are used in the given theory for exposition of dynamic processes. In recent years, the fuzzy set theory introduced by Zadeh [33] has emerged as an interesting and fascinating branch of pure and applied sciences. The applications of fuzzy set theory can be found in many branches of re gional, physical, mathematical, differential equations, and engineering sciences. Recently there have been new advances in the theory of fuzzy differential equations [3447] and inclusion s [4853] as well as in the theory of control fuzzy differential equations [5457] and inclu sions [5759]. In this article we consider the some properties of the fuzzy Rsolution of the control linear fuzzy differential inclusions and research the timeoptimal problems for it. 2. The Fundamental Definitions and Designations Let n comp Rconv Rn be a set of all nonempty (convex) compact subsets from the space , n R 0 ,min, rr r hABSABS BA be Hausdorff distance between sets and , B r SA is neighborhood of set . Let be the set of all n E :0 n uR,1 such that u satisfies the following conditions: 1) is normal, that is, there exists an u0n R such that 01ux ; 2) is fuzzy convex, that is, u 1min,uxy uxuy n for any , yR and 01 ; 3) is upper semicontinuou s, u 4) 0: n uclxRux0 n is compact. If , then is called a fuzz y number, and is said to be a fuzzy number space. For uEun E 01 , de note
A. V. PLOTNIKOV ET AL.153 : n uxRux . Then from (1)  (4), it follows that the level set for all 01 n uconvR . Let be the fuzzy mapping defined by 0x if and . 0x 01 : nn DE EDefine by the relation 0, 01 ,sup ,Duvh uv , where is the Hausdorff metric defined in h n comp R. Then is a metric in . Dn E Further we know that [60]: 1) is a complete metric space, , n ED Du w 2) for all , ,,v wDuv ,, n uvw E 3) ,Duv Duv , for all and ,n uv E R . Definition 1. [36] A mapping :0, n TE is measurable if for all 0,1 the setvalued map :0, n TconvR defined by tFt is Lebesgue measurable. Definition 2. [36] A mapping :0, n TE is said to be integrably bounded if there is an integrable fu nction such that ht tht for every 0 tFt. Definition 3. [36] The integral of a fuzzy mapping :0, n TE TT is defined levelwise by 00 dd tt Ftt . The set 0 d T tt of all 0 d T tt such that :0, n TR is a measurable selection for :0, n TconvR for all 0,1 . Definition 4. [36] A measurable and integrably bounded mapping :0, n TE is said to be integra ble over 0,T if 0 dn T ttE . Note that if :0, n TE is measurable and inte grably bounded, then is integrable. Further if :0, n TE is continuous, then it is integrable. Now we consider following control differential equa tions with the fuzzy parameter 0 ,,,, 0, ftxwv xx (1) where means d d t; tR is the time; n R k is the state; is the control; is the fuzzy parameter; m wRvV E k n :nm RRR m onvR R R . Let be the measurable setva lued map. : WR c Definition 5. The set of all measurable single valued branches of the setvalued map is the set of the admissible controls. LW W Further we consider following control fuzzy differen tial inclusions 0 ,,, 0, Ftxw xx n (2) where :nm RRR E is the fuzzy map such that ,, ,,, txwf txwV. Obviously, the control fuzzy differential inclusion (2) turns into the ordinary fuzzy differential in clusion 0 ,, 0, tx xx (3) if the control wLW is fixed and ,,txwt ,tx F . If righthand side of the fuzzy differential inclusion (3) satisfies some conditions (for example look [12]) then the fuzzy differential inclusions (3) has the fuzzy Rso lution. Let t denotes the fuzzy Rsolution of the differ ential inclusion (3), then , tw denotes the fuzzy Rsolution of the control differential inclusion (2) for the fixed wL W. Definition 6. The set ,:Tw wLWYT X be called the attainable set of the fuzzy system (2). 3. The Some Properties of the Fuzzy RSolution Further in the given paper, we consider following control linear fuzzy differential inclusio ns 0 ,, 0 , AtxGtwxx (4) where t is nn dimensional matrixvalued function; is the fuzzy map. :GRm R E n In this section, we consider the some properties of the fuzzy Rsolution of the control fuzzy differential inclu sion (4). Let the following condition is true. Condition A: A1. is measurable on 0,T; A2. The norm t of the matrix t is inte grable on 0,T; A3. The setvalued map 0 :, m WtT Rconv is measurable on 0,T; A4. The fuzzy map :0, mn GTR E satisfies the conditions 1) measurable in ; t 2) continuous in ; w A5. There exist 20,vLT and 20,lLT such that ,0, ,,hW tvtDGtwlt almost everywhere on 0,T and all wWt. A6. The set LW,():Qtwtw Gt is com pact and convex for almost every 0,tT, i.e. n Qt convE. Theorem 1. Let the condition A is true. Copyright © 2011 SciRes. ICA
A. V. PLOTNIKOV ET AL. 154 Then for every there exists the fuzzy Rsolution wLW , w such that 1) the fuzzy map , w has form 1 00 ,, td twt xtsGswss , where 0,tT; is Cauchy matrix of the differ ential equation t At nx EwtX ),( ; 1) for every ; Tt ,0 2) the fuzzy map , w is the absolutely continuous fuzzy map on 0,T. Proof. 1. Show that , tw is the fuzzy Rsolution of the fuzzy system (4). We have 1 00 1 00 1 00 , ,d (, d t t t ,d twtxtsGswss txtsGsws s txts Gswss for all 0,1 w, and . Since 0t wLW ,Xt is the Rsolution of the control differential inclusion 00 (,, Atx Gtwtxtx (see [30]), we obtain that , tw is fuzzy Rsolution of the control fuzzy differential inclusion (4). 2. By [36] and Condition A we have that ,n tw E for all and . 0t wLW ,Xtw 3. From [30] we have that is the abso lutely continuous setvalued map on 0,T for all 0, 1 , i.e. , tw is the absolutely continuous fuzzy map on 0,T. The theorem is proved. Theorem 2. Let the condition A is true. Then the attainable set is compact and convex. YT Proof. It is easy to check that 1 00 d T YTTxTsQs s and 1 00 d T YTTxTs Qss , where :0, n Q tTconvconvR for all 0, 1 T . From [20,21,30,57] we obtain that 1 0 dn TsQ ssconv conv R for all 0, 1 , i.e. . This ends the proof. YT convEn We obtained the basic properties of the fuzzy Rsolu tion of systems (4). Now, we consider the some fuzzy control p roblems. 4. TimeOptimal Problems Consider the following timeoptimal problem: it is nec essary to find the minimal time T and the control * wL W such that the fuzzy Rsolution of system (4) satisfies one of the conditions: * ,k XTw S, (5) * ,k Tw S, (6) * ,k Tw S, (7) where is the fuzzy terminal set. n kES It is obvious that optimum time and optimum controls for these problems will be different. Theorem 3. (necessary optimal condition for the timeoptimal problem (4), (5)). Let the condition A is true and the pair * ,Tw is optimality of the control problem (4), (5 ). Then there exists the vectorfunction , which is the solution of the system 1 , 0 T AtTS such that 1) 11 * ,,max ,, wWt CGtwtCGtw t almost everywhere on T,0 ; 2) , 11 * ,, , k CXTwT CST where 11 ,max, , n nn pP CPp pR n conv R. Proof. Let * w is the optimal control and * , w is the optimal fuzzy Rsolution of the problem (4), (5), i.e. 1) * ,; Tw YT 2) * ,. k XTw S From 1) and 2) we have 1 1 max ,, k XYT CXC S (8) for all 10S . Consequently 11 1 0 max min,,0. k S XYT pCXCS 1 0 From we have 1 * ,k XTw S 1 * 1 ,,,, k qTCXTwCS Copyright © 2011 SciRes. ICA
A. V. PLOTNIKOV ET AL.155 for all From 1 we have that the function 10S . Theoreman ,qT ve is contiR nuous on 10S . If ,0qT for all 10S then we ha 0 0 min 0 S qT qT . Hence there exists 1, T su equently 0 for all i.e. It contradicts that is optimal time. ch that0. Conswe have q 0 11 * ,, ,CX wCS k 10S , 11 * ,k XwS . T If, 0p 110 1 , ,, XY T k CX CXC S 1 max min,k SC S 1 and * , Tw X ce there exists , than we have a contradiction. Hen such that 10S * ,,w 1 1max ,, XYT CXT CX 11 * ,, , k CXTwCS . Consequently 0 CT 1 1* 1 1 0 ,d, max,d , T T wLW sGswss CT sGswss Then we have 1 1* 1 1 ,, max, , wWt CT sGsws CT sGsw for almost every 0, T. If 1 1 T T Tt tTt , than the theorem 3 is proved. Theorem 4. (necessary optimal condition for the tim . Let the condition A is tr system eoptimal problem (4), (6)) * ue and the pair ,Tw is optimality of the control problem (4), (6 ). Then there exist the vectorfunction , which is the solution of the 1 , 0 T AtTS d 0,1 such that 1) * () ,,max ,, wWt CGtwtCGtw t almo here on st everyw 0,T; all 2) for 0,1 * ,, , k CXTwT CST and * ,, , k CXTwT CST . This changes theorem is proved analogous theorem 5 with little of condition (8): for all 0, 1 max,, 0 k XYT CXCS for all 10S and there e 0,1 xist and n R such that max,, 0 k XYT CXCS . Theorem 5. (necessary optimal condition for the timeoptimal problem (4), (7)). Let the condition A is true and the pair * ,Tw is optimality of the control problem (4), (7 ). Then there exist the vectorfunction , which is the solution of the system , T At 10T S and 0,1 such that 1) * () ,,max ,, wWt CGtwtCGtw t almo here on st everyw 0,T; all 2) for 0,1 * ,, , k CXTwT CST and * ,, , k CXTwT CST . Also little ch this theorem is prov ed analogous theore m 5 with anges of condition (8): for all 0, 1 k XYT CXCS , max,, 0 for all 10S and there exist 0,1 and n R such that Copyright © 2011 SciRes. ICA
A. V. PLOTNIKOV ET AL. 156 max,, 0 k XYT CXCS . Exam sider the followinl linear fple. Cong controuzzy differential inclusions , 00,xxwFx (9) where is the state; 1,1wW is the control; 1 E is the fuzzy set, where 0, 0,5 21,0,5 3,11 1,5 f f f f 1f 2 ,5 0, ff . Consider the following timeoptimal problem: it is necessary to find the minimal timeanthe control Wsuch that the fuzzy Rsionf system (9) satisfies of the condition (5), where te fuzzy terminal such, that T olut h d o * wL set 1 k SE 0, 1,75 47,1,752 1,2 3 4 x xx xx xx13,3 3,25 0, 3,25 . The control and time will be optimum pair foe given pr set *1wt r th ln 2T oblem. Fuzzy * , Tw Figure 1. inal set are shown in Obviously, this optimal pair satisfies to conditions of th and fuzzy termK S e theorem 3: 1) *,,wttCWt for almost every 0,ln2t ; 2) for almost every e consider the timeoptimal problem (9), (6hen 11 * ,, , k CXTwT CST , where 0,ln2t , 1t 1 * ,2XTw ., 12,3 K S If w) t the control and time *1wt13 ln 6 T will op timum pair. Fuzzy set be * , Tw and fuzzy terminal set S co are shown in Figure 2. This optimal pair satisfies to nditions of the theorem 4: 1) * wtCW tst every for almo,,t 13 0,ln 6 t ; 2) 0 * ,,CXTw T 0, k CS T , ln2,1X (−), K Figure 1. (+). 13 ln ,1 6 X (−), K Figure 2. (+). where for almost every 13 0,ln 6 t 1t , 0 *735 ,,, 4 0713 12 XTw , 4 . 4 K S If we consider a problem (9), (7) it is obvious that the olution does not exist. 5. Conclusions In the last decades, a nuber of works deted to prob le works fall into a subdivision of theory, namely, the theory of process tainty and fuzzy conditions. This is s m vo ms of optimal control of setvalued trajectories (fuzzy trajectories, trajectory bundles or an ensemble of trajec ories) appeared; these t the optimal control ontrol under uncerc conditioned by the fact that, in actual problems arising in economy and engineering in the course of construction of a mathematical model, it is practically impossible to exactly describe the behavior of an object. This is ex plained by the following fact. First, for some parameters of the object, it impossible to specify exact values and laws of their change, but it is possible to determine the domain of these changes. Second, for the sake of sim plicity of the mathematical model being constructed, the Copyright © 2011 SciRes. ICA
A. V. PLOTNIKOV ET AL.157 to describe behavior of objects. The rea so the necessary conditions of opti m . Marchaud, “Sur les Champs de Demicones et Equa erentilles du Premier Ordre,” Bulletin de tique de France, Vol. 63, 1934, pp , No. 4, 1967, pp. equations that describe the behavior of the object are simplified and one should estimate the consequences of such a simplification. Therefore, if is possible to divide the articles devoted to this direction into two types char acterized by the following distinctive features: 1) there exists an incomplete or fuzzy information on the initial data; 2) the equations describing the behavior of the object to be controlled are assumed to be inexact, for example, they can contain some parameters whose exact values and laws of variation are unknown but the domain of their values is fuzzy. 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