Research Article Dynamic Model of Urban Sports Infrastructure Supply and Demand Based on GDP Growth

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1 Research Journal of Appled Scences, Engneerng and Technology 7(1): 72-76, 2014 DOI: /rjaset ISSN: ; e-issn: Maxwell Scentfc Publcaton Corp. Subtted: January 26, 2013 Accepted: February 25, 2013 Publshed: January 01, 2014 Research Artcle Dynac Model of Urban Sports Infrastructure Supply and Deand Based on GDP Growth 1 Wenjng Zhang, 2 Keba L, 3 Peyan Fu and 4 Xao-Qn Xu 1 Departent of Physcal Educaton, Southeast Unversty, Nanjng , Chna 2 School of Econocs and Manageent, Nanjng Unversty of Inforaton Scence and Technology, Nanjng , Chna 3 Language Teachng and Research Group, Zhejang Pujang Mddle School, Jnhua , Chna 4 Departent of Physcal Educaton, Jangyn Polytechnc College, Jangyn , Chna Abstract: For dfferent cty, the change rate of the supply and deand of the sports nfrastructure s not the sae and s subject to regonal GDP growth constrants. Takng GDP growth as control varables, a dynac odel of ult-cty sports nfrastructure supply and deand syste was establshed. Accordng to Lyapunov stablty theory, the syste asyptotcally stable condton was obtaned. Usng the lnear atrx nequalty ethod, the paper gets a control ethod that ctes wth dfferent developent level can be unfed use for akng ther sports nfrastructure syste asyptotcally stable and supply-deand equlbru. The ethod can reduce the ctes sports nfrastructure constructon control coplexty. Keywords: Constructon control, dynac odel, GDP growth, sports nfrastructure, supply and deand INTRODUCTION At present, Chna's gross natonal product has rsen to the second place n the world. In order to ake econoc developental acheveent share wth the ctzens, the governent has stressed the sgnfcance and value of provng the socal and publc servces, ncludng culture and sports. For sports publc servce, sports nfrastructure s the ateral foundaton and basc guarantee. How to realze the balance of supply and deand of sports publc servce nfrastructure constructon for dfferent cty? How to coordnate sports nfrastructure constructon progress wth local GDP growth? Can we buld a unversal asyptotc stablty control schee sultaneously coordnatng ult cty s GDP growth and sports nfrastructure constructon progress? These probles are worthy of analyzng for cty anagers. For the ssues of sports publc servce nfrastructure and regonal econoc, current research anly focused on evaluaton and anageent. For exaple, through buldng the VEC odel of sports publc servce, Yao and Dng (2012) adopts the prary coponents extractng ethod to splfy the eprcal data n the feld of sports publc servce. Then analyzed relevant data both qualtatvely and quanttatvely to study the effect of the nput n the feld of sports publc servce. The research hopes to proote the role of publc sports organzatons, dstrbute publc sports nvestent ore ratonally, prove the level of sports publc servce effectvely and to ake the sports publc servce eet people s ncreasng requreents on t. Q et al. (2012) dealt wth how to reasonably arrange the start te of these actvtes whle eetng varous constrant condtons to obtan the entre project schedulng and to nze the total duraton of the projects occupancy for socal sports resources. Cobned wth the sulated annealng algorth, schedulng algorth for the socal sports project s desgned to study the proble of socal sports schedulng. On the bass of Porter's Daond Model, L (2012) establshed the evaluaton odel of Chnese sports ndustry copettveness. The copettveness odel of Chnese sports ndustry cobnes the characterstcs of sports ndustry and the stuaton n Chna. Eprcal result s consstent wth the actual stuaton of Chnese sports ndustry n Chna. Manly because soe evaluaton ndcators are not avalable or cannot be easured, the defcency of the research s that t s dffcult to select the evaluaton ndcators of operaton level n accordance wth the deal stuaton. Fang (2012) nvestgated the Multple Attrbute Decson Makng (MADM) probles for evaluatng the sports anageent syste wth nterval-valued ntutonstc fuzzy nforaton. Correspondng Author: Wenjng Zhang, Departent of Physcal Educaton, Southeast Unversty, Nanjng , Chna Ths work s lcensed under a Creatve Coons Attrbuton 4.0 Internatonal Lcense (URL: 72

2 Wu (2012) proposed a new coprehensve evaluaton odel wth cobnatonal weght based on ant colony optzaton. The use of evaluaton odel to evaluate the culture of people sports can analyze the coprehensve constructon of sports culture, and the developent of all factors of the sports culture, whch s helpful to grasp the overall and partal stuaton and proote the developent of people sports culture. Through a seres of research, Zheng and Wang (2008) and Zheng et al. (2008) proposed "the coprehensve benefts-net nvestent" natonal econoc evaluaton ethod and constructed the natonal econoc coprehensve beneft ndex syste n the fraework of "the coprehensve benefts - net nvestent", provdes a bass for large publc sports facltes scentfc and reasonable nvestent. Accordng to publc sports constructon stuaton, probles and nfluence factors n Shandong, Chna, Wang (2010) put forward the project anageent ode of governent nvestent n publc sports constructon. Wang et al. (2009) ade functon fttng wth publc sports nvestent and the GDP annual value n Yunnan Provnce, Chna, got the relatonshp between publc sports nvestent and econoc developent and predcted the probles about publc sports nvestent n the next few years. Exstng research has ade soe acheveents for evaluaton and anageent n sports publc servce nfrastructure and regonal econoc relatons. Whle for a specfc regon, the control balance dynac quanttatve relatonshp between GDP growth and sports nfrastructure constructon, less nvolved n the prevous study. Ths study akes research on the ult cty sultaneous stablzaton, supply and deand equlbru control ethod for GDP growth and sports nfrastructure constructon by usng econoc control theory, n order to reduce the sports nfrastructure constructon control coplexty. MODEL For the ctes wth dfferent developent level of econoy and socety, the supply and deand change rates of the sports nfrastructure are not the sae. Suppose there are ctes wth dfferent developent level, the state equaton of sports nfrastructure deand functon D (t) of cty s: D& ( t) = α D( t) + β R( t), = Res. J. Appl. Sc. Eng. Technol., 7(1): 72-76, 2014 (1) where, t s the te unt, R(t) s regonal GDP growth rate, α, β are the syste paraeters. The change rate of the urban resdents sports nfrastructure deand (t) s affected by two factors. One s the current level of deand, ths s the state varable. 73 The other s the regonal GDP growth rate R(t), ths s the control varable. Because of dfferent values of α, β, the deand of urban sports nfrastructure can show n the for of growth curve odel (Jan and Hu, 2009). The pact of sports nfrastructure deand degree D (t) on (t) s: n the hgh-speed econoc growth cty, the resdents' sports nfrastructure deand wll be rapd growth, α s a desrable larger value. In the stable developed cty, ost of resdents deand already has been et, α s a saller value. In the less-developed cty, resdents' sports nfrastructure deand growth s relatvely weak, α s a desrable saller value. The pact of sports nfrastructure deand degree R (t) on (t) s: Due to the econoc and socal developent level s dfferent, the sae GDP growth, for the ctes wth dfferent developent level, sports nfrastructure deand rate wll be dfferently affected. The developed ctes are ore wllng to spendng econoc growth for sports nfrastructure constructon and prootng the deand growth, so β s larger. In contrast, underdeveloped ctes are ore wllng to sparng econoc growth n expendtures for other econoc constructon, so β s saller. By eans of dfferent values of β, regonal GDP growth rate R (t) can regulate the change rate of the sports nfrastructure deand. Accordng to producton theory of the econocs (Takayaa, 2001), fxed captal s the portant factors of producton, the sports nfrastructure supply functon S(t) of cty can be expressed as: S( t) = θ K ( t), = (2) where, K (t) s fxed captal stock of sports nfrastructure supply agency, θ s the output captal rato. To dfferent developent level cty, the output captal rato s not the sae. Relatvely speakng, anufacturng level and captal utlzaton rate n the econoc developed cty are hgher than that n the underdeveloped cty. By the nvestent theory of the econocs (Takayaa, 2001), the relaton aong sports nfrastructure fxed captal stock changes, fxed captal nvestent and fxed captal deprecaton can be used dynac equaton to express as: K& ( t) = I( t) δ K( t), = (3) where, I(t) s fxed captal nvestent of sports nfrastructure supply, δ s fxed captal deprecaton rate. Sports nfrastructure nvestent s also constraned by the GDP growth n the regon. Supposng nvestent s a certan percentage of the regonal GDP growth, we have: I( t) = γ R( t), = (4)

3 Res. J. Appl. Sc. Eng. Technol., 7(1): 72-76, 2014 where, γ s the proporton coeffcent of nvestent to the regonal GDP growth. The econoc and socal dfferent stuaton wll have dfferent pact on nvestent rato coeffcent. The developed ctes can spend ore proporton of econoc growth on sports nfrastructure nvestent, so γ value s larger. In contrast, underdeveloped ctes are wllng to spend ore proporton of econoc growth on other econoc constructon expendtures, so γ s saller. Next we calculate the state equaton of sports nfrastructure supply functon. Frst dfferentate both sdes of the forula (2) to te t, then substtute forula (3) and forula (4), we get: (5) Forula (5) s the te varyng state equaton of the sports nfrastructure supply functon S(t) of cty. Jonng the forula (1) and forula (5) n a atrx for, we can get sports nfrastructure dynac supply and deand odel of cty as follows: (6) Constructon control goal s that, for the ctes wth dfferent econoc and socal developent degree, to deterne a unfed control schee for the control varable R (t). Through the pleentaton of ths schee, can ake the sports nfrastructure dynac supply and deand systes of the ctes wth dfferent developent degree to be asyptotcally stable and fnally to acheve the balance of supply and deand. The advantage of ths pleentaton s that, an unfed constructon control schee can be used for a nuber of ctes wth dfferent developent level and can reduce the coplexty of the constructon control. CONSTRUCTION CONTROL SCHEME Forula (6) sports nfrastructure dynac supply and deand odel can be expressed as: where, S & ( t) = θ K & ( t) = θ I( t) δ K( t) = θγ R( t) δ S( t), = 1, L, D& ( t) α 0 D( t) β = + R( t), = S& ( t) 0 δ S( t) θγ x& ( t) = A x( t) + B u( t), = x( t) = D( t), S( t) T s 2-densonal state vector: (7) s a 1-densonal control nput: are respectvely 2 2 and 2 1 constant coeffcent atrx. Control schee s desgned to ake use of the state feedback control, that s to let: (8) where G s 1 2 densonal state feedback gan atrx. Substtutng forula (8) nto forula (7), we have: (9) By the Lyapunov stablty theory (Yu, 2011), the necessary and suffcent condton of the closed-loop syste forula (9) asyptotc stablty s to exst a syetrc postve defnte atrx P, akng: (10) Therefore, the stablzng controller desgn proble s reduce to deterne the atrx G and the syetrc postve defnte atrx P and to ake the atrx nequalty (10) hold. For the atrx varables G and P, forula (10) s not lnear atrx nequalty. In order to solved by lnear atrx nequalty ethod, we need try to transfor t nto lnear atrx nequalty. So we expand forula (10) as: (11) Left ultpled and rght ultpled P -1 on both sdes of forula (11), we have: As P s a syetrc postve defnte atrx, there s: Let: α 0 A 0 δ β B = θγ u( t) = Gx( t) x& ( t) = A + B G x( t), = T ( + ) + ( + ) < 0, = A B G P P A B G PA + A P+ G B P+ PB G< 0, = AP + P A + P G B + B GP < 0, = T T T ( P ) X = P 1 T Y = KP = P, forula (12) can be transfored nto: (12) u( t) = R( t) 74 A X + XA + Y B + B Y < 0, = (13)

4 Res. J. Appl. Sc. Eng. Technol., 7(1): 72-76, 2014 For atrx varable X and Y, forula (13) s a lnear atrx nequalty group contanng nequaltes. Due to the postve defnteness of the atrx P and that of the atrx X s equvalent, f the lnear atrx nequalty group: A X + XA + Y B + BY < 0, = X > 0 (14) exstng feasble soluton, then forula (8) stablzaton controller of forula (7) syste also has feasble soluton. If X and Y are feasble solutons of forula (14) lnear atrx nequalty group, then G= YX s a stablzng state feedback gan atrx of forula (7) syste and P= X s a Lyapunov atrx for forula (9) correspondng closed-loop syste. Now the forula (8) control schee can ake the ctes wth dfferent level of econoc and socal developent reach sports nfrastructure dynac syste asyptotcally stable and supply and deand balance. Exaple: Three ctes A, B and C are, respectvely the developed cty wth steady econoc growth, the edu-developed cty wth rapd econoc growth and the less developed cty wth slower econoc growth. To eet the publc sports servce needs of the cty resdents, the three ctes all plan the constructon of sports nfrastructure. The ctes sports nfrastructure dynac supply and deand syste odel paraeters are as follows: Cty A s: α = 0.05, β = 0.09, θ = 0.8, δ = 0.05, γ = A1 B1 A2 B A3 =, B 3 = Substtutng the syste paraeter nto forula (14), usng lnear atrx nequalty toolbox of MATLAB software (Xue and Chen, 2007), we can get: X, Y are feasble soluton of lnear atrx nequalty group forula (14). Substtutng X, Y nto: and: we get: X =, Y = ( ) P= X G= YX P G = ( ) The atrx P s the syetrc postve defnte Lyapunov atrx and the atrx G s the stablzng state feedback gan atrx. Substtutng G nto forula (8), we get: D( t) R( t) = ( ) S( t) Cty B s: α = 0.08, β = 0.05, θ = 0.07, δ = 0.04, γ = Cty C s: α = 0.02, β = 0.01, θ = 0.06, δ = 0.03, γ = To reduce the coplexty of the constructon control, the syste target s to deterne a unfed control schee for GDP growth n varous ctes and can ake the three ctes wth dfferent level of econoc and socal developent reach sports nfrastructure dynac syste asyptotcally stable and supply and deand balance. Here we have: = 3, 75 The above solved R (t) s the constructon control desgn schee. The coeffcent of stablzng state feedback gan atrx G s negatve, ndcatng that the control desgn schee s the negatve feedback control. When the value of state varables D (t) or S (t) s less than ts stable value, ts value should be regulated to ncrease. Whle when the value of state varables D (t) or S (t) s hgher than ts stable value, ts value should be regulated to decrease. By eans of an unfed negatve feedback control desgn schee, can ake the A, B and C three ctes wth dfferent level of econoc and socal developent reach sports nfrastructure dynac syste asyptotcally stable and supply and deand balance.

5 Res. J. Appl. Sc. Eng. Technol., 7(1): 72-76, 2014 The control desgn schee can acheve syste goals and reduce constructon control coplexty. CONCLUSION Dfferent ctes wth dfferent developent level and dfferent paraeters of sports nfrastructure dynac supply and deand syste, results n that the change rates of cty sports nfrastructure supply and deand functon are not the sae. But for syste control s concerned, the ctes for the syste asyptotcally stable and equlbru of supply and deand have the sae requreents. Regonal GDP growth rate has portant nfluence on both the supply and deand of cty sports nfrastructure. Usng Regonal GDP growth rate as the syste control varables, applcaton of Lyapunov stablty theory, can establsh an unfed control desgn schee. Usng lnear atrx nequalty ethod, can acheve the target that sports nfrastructure dynac supply and deand syste of the ctes wth dfferent level of econoc and socal developent are asyptotcally stable and balanced supply and deand. ACKNOWLEDGMENT The authors thank the anonyous revewers for ther valuable rearks and coents. Ths study s supported by Huantes and Socal Scence Research of the Mnstry of Educaton Youth Project of Chna (Grant No. 10YJC880158); the Natonal Socal Scence Foundaton of Physcal Educaton Youth Project of Chna (Grant No. 12CTY017); Hgher School n Jangsu Provnce Departent of Educaton Phlosophy and Socal Scence Fund Gudance Project (Grant No. 2012SJD890014); the Prorty Acadec Progra Developent of Jangsu Hgher Educaton Insttutons (PAPD); the State Scholarshp Fund by the Chna Scholarshp Councl (CSC) (Fle No ), Jangsu Socal Scence Fund Project (Grant No. 12GLC010), Chna Postdoctoral Scence Foundaton Funded Projects (Grant No ). REFERENCES Fang, Q., Research on the coprehensve evaluaton of sports anageent syste wth nterval-valued ntutonstc fuzzy nforaton. Int. J. Adv. Coput. Technol., 4: Jan, M. and Y.L. Hu, Market Forecast and Manageent Decson Makng. 4th Edn., Chna Renn Unversty Press, Bejng. L, D.Y., Research on the copettveness of chnese sports ndustry based on the daond odel. Int. J. Adv. Coput. Technol., 4: Q, A.L., F.T. Wang, F. Xu and F. Zhen, A study of socal sports schedulng based on genetc algorth. Adv. Infor. Sc. Serv. Sc., 4: Takayaa, A., Analytcal Method n Econocs. Chna Renn Unversty Press, Bejng. Wang, H.Y., L.G. Zhang, B. Gong and Y.L. Zhao, Matheatcal odelng of relatonshp between Yunnan publc sports nput and the GDP. J. Wenshan Teach. Coll., 22: Wang, Y., Research on Shandong provnce governent nvestent n publc sports venues constructon project anageent ode. M.A. Thess, Shandong Unversty, Chna. Wu, Y., An proved coprehensve evaluaton odel based on ant colony optzaton for people sports culture. Adv. Infor. Sc. Serv. Sc., 4: Xue, D. and Y.Q. Chen, Control Matheatcal Proble Solvng wth MATLAB. Tsnghua Unversty Press, Bejng. Yao, J.W. and X.S. Dng, A VEC-odel-based sports publc servce analyss. Int. J. Dg. Cont. Tech. Appl., 6: Yu, L., Modern Control Theory. Tsnghua Unversty Press, Bejng. Zheng, M.Y. and Z.L. Wang, Large publc sports facltes natonal econoc evaluaton studes (I): The conceptual fraework and evaluaton ethod. J. Nanjng Inst. Phys. Educ. Soc. Sc. Edn., 22: 1-5. Zheng, M.Y., Z.L. Wang and Y. Zhang, Large publc sports facltes natonal econoc evaluaton studes (II): Econoc ratonalty coprehensve evaluaton ndcator syste. J. Nanjng Inst. Phys. Educ. Soc. Sc. Edn., 22: