 Intelligent Control and Automation, 2011, 2, 121-125 doi:10.4236/ica.2011.22014 Published Online May 2011 (http://www.SciRP.org/journal/ica) Copyright © 2011 SciRes. ICA A New Approach for a Class of Optimal Control Problems of Volterra Integral Equations Mohammad Hadi Noori Skandari1, Hamid Reza Erfanian2, Ali Vahidian Kamyad1 1Department of Applied Mathematics, School of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran 2Department of Mathematics & Statistics, University of Science and Culture, Tehran, Iran E-mail: hadinoori344@yahoo.com, Erfanian@usc.ac.ir, avkamyad@yahoo.com Received January 30, 2011; revised March 23, 2011; accep t e d M ay 5, 2011 Abstract In this paper, we propose a new approach for a class of optimal control problems governed by Volterra inte-gral equations which is based on linear combination property of intervals. We convert the nonlinear terms in constraints of problem to the corresponding linear terms. Discretization method is also applied to convert the new problems to the discrete-time problem. In addition, some numerical examples are presented to illustrate the effectiveness of the proposed approach. Keywords: Volterra Integral Equations, Optimal Control, Linear Programming 1. Introduction Consider the following optimal control problem gov-erned by Volterra integral equation (OCV):  0minimize, ,dTGyTFtyt utt (1)  0subject to,,,d,0tytpt KtsysusstT (2) where and are the state and control func-tions respectively on . The integral Equation (2) is applied in a natural way in the study of economic prob-lems, population dynamics and etc., see for instance Hri-tonenko and Yatsenco , and Kamien and Schwartz . The OCV problem (1)-(2) has been studied by many au-thors, including Neustadt [3-5], Bakke , Carlson , Vinokurov , Medhim , Schmidt [10-13], Wolfans-dorf , Elnagar, Kazemi and Kim , Pan and Teo , Angell [17,18], Belbas [19,20], Carlier and Tah-raoui , and Burnap, and Kazemi . The method usually employed for OCV problem (1)-(2) is method of necessary conditions of the type of Pontryagin maximum principle. In the Recent works, Vega  gives the nec-essary condition for optimal terminal time of OCV prob-lem (1)-(2) and verifies the terminal time T and final state (.)y(.)u[0, ]TyT by conditions ,0TTy and ,TyT 0. Bonnens and Vega  discuss problem (1)-(2) with running state on the initial and final states. Also, Belbas  applied the ideas of dynamic programming to OCV problem (1)-(2). In this paper, we are interested to solving the follow-ing class of the OCV problem (1)-(2) which we called it COCV problem:  0minimize dTctyt t (3)  0subject to,d,d,0tytptfsus tsysstT (4) where function is a continuous function. A con-trolled Volterra integral equation similar to equation (4) is discussed in . We suppose that (.,.)fut U, [0, ]tT where U a compact and connected set. In addi-tion, we let the final state yT is a given known num-ber. Here, the linear combination property of intervals is used to convert nonlinear controlled Volterra integral Equation (4) to the equivalent linear equation. The new optimal control problem with this linear Volterra integral equation is transformed to a discrete-time problem that could be solved by linear programming methods. This paper organized as follows. Section 2, transforms the nonlinear function to a corresponding function that is linear respect to a new control function. Section 3, converts the new problem to the discrete-time problem via discretization. In Section 4, numerical examples are presented to illustrated effectivness of this approach. (.f,.) M. H. N. SKANDARI ET AL. 122 Finally, the conclusion of this paper is given in Section 5. 2. Metamorphosis of the COCV Problem In this section, COCV problem (1) is transformed to the new equivalent problem. First, we state and prove the following two theorems: Theorem 2.1: Let be a continuous function on where U is a compact and connected subset of , then for any arbitrary (but fixed) (.,.)f[0, ]TUm[0, ]sT the set ,:fsuu U is a closed interval in . Proof: Assume that [0, ]sT be given. Let u ,fsu . Obviously, (.) is a continuous function on U. Since, continuous function preserve compactness and connectedness, the set is compact and connected. Therefore, is a closed inter-val in.□ :uu:uuUUFor any [0, ]sT, we may suppose the lower and upper bounds of interval ,:Fsuu U are gs and respectively. Thus we have: ws,,0,gsfsuwss T  (5) In other words  min, :,0,ugsfsuuUsT (6)  max, :,0,uwsf suuUsT (7) Theorem 2.2: Let functions (.)g and be de-fined by relations (6) and (7). Then they are uniformly continuous on . (.)w[0,]TProof: We will show that (.)g is a uniformly con-tinuous function. It is sufficient that we show that for any given 0, there exists 0 such that if 12sNs then 12gsgs where Nz is a (.,.fneighborhood of . Since, any continuous function on compact set is a uniformly continuous. The function on compact set is a uni-formly continuous, i.e. for any z)[0, ]TU0 there exists 0, such that if 12,,suNsu then 12,u f,sufs. Thus fsu12,,suf. In addition, by (5), 11 ,gsfsu and so 1gs 2,fsu. Now, by taking infimum on the right hand side of the last inequality gs1gs2gs21. By a simi-lar procedure, we havegs. Thus  12gs gs. The proof of uniformly continuity of is similar. □ (.)wBy linear combination property of intervals and rela-tion (5), we have for any [0, ]sT:  ,fsuswsgss gss ,[0,1], (8) Thus, we transform COCV problem (3)-(4) by relation (6) as the following continuous-time problem:  0dTminimizectytt (9)   0,d01,0,tsubject toytqthssdtsyssttT  where and  0dtqtptgs shtwtgt for any [0, ].tT Note that in the new problem (9), which is a optimal control of linear Volterra integral equation, (.) is the new control function. Next section, converts the problem (9) to the corresponding linear pro-gramming problem. 3. Discrete-Time Problem Now, discretization method enables us transforming con-tinuous problem (9) to the corresponding discrete form. Consider equidistance points 0120NssssT  of interval which defined as [0, ]TjTsjN, 0,1,,jNwhere N is a given big number. Also, we set jjts for 0,1,,jN. By trapezoidal approximation in nu-merical integration, problem (9) is converted to the fol-lowing discrete-time problem: 1001minimize 22Njj NNjTT Tcycyc yNNN (10) 100 010000subject to122 2,201,,0,1, 2,,jjjji jiijjijjjNTTTTdyhh dyhjjj jTqdyjyqy jN  where jjyyt, jjhhs, , ,ijijddtsjjccs, jjt and jjqqt for all . In problem (10), final state is ,0,1,2,,ij N that is a known number. By solving problem (10), which is a linear programming problem, we are able to obtain the optimal solution j and jy for all 0,1,2,,jN. Note that, for evaluat-ing optimal control variable , we must use the fol-lowing equation: (.u) ,.fsuhsgs (11) 4. Numerical Examples Here, we use our approach to obtain approximate optimal Copyright © 2011 SciRes. ICA M. H. N. SKANDARI ET AL.123 solution of the following two COCV problems by solv-ing linear programming (LP) problem (10) via simplex method  in MATLAB software. Example 4.1: Consider the following COCV problem: 10minimizecos 3πdtyt t (12) 0subject toπsin 3πsin 4 cos3πd00.5,0,1(1) 1.tyttus stsys sut ty    Here, π,sin4fsuu s, ,cos3 πdtst s cos 3πct t andfor all  sin 3πptt[0,1]t, and u. Thus by (6), (7) [0,1]s[0,0.5] [0,0.5]πmin sinsin,0,14ugsu sss [0,0.5]πmax sinsin,0,148uwsu sss . hence  0dsin3πcos1,0,1tqtptgs sttt   πsinsin ,0,18hswsgsss s  Assume that and 100N100jjs for all 0,1,j 2, ,100. The optimal solutions jy and j, 0,1,j of problem (12) is obtained by solving prob-lem (10) which is illustrated in Figures 1 and 2 respec-tively. Here, the value of optimal solution of objective function is –0.470. The corresponding Equation (11) for this example is 2, ,100 Figure 1. Optimal trajectory of Ex.12. (.)y Figure 2. Corresponding optimal control of Ex.12. (.)  πsin,0,1, 2,,1004jjjj jushsgs j   therefore  14sin,0,1, 2,,100πjjjjjuhsgssj  The optimal control *ju, of this example is showed in Figure 3. 0,1, 2,,100jExample 4.2: Consider the following COCV problem: 10minimize 0.5dtyt t (13)  03subject to3ln2d01,0,1(1) 1.ttsyttu ssteyssut ty  Here ,3lnfsuu ss32, ,)dtstets, ct 0.5 t and pt t for all , [0,1][0,1]st and [0,1]u. In this example for all [0,1]s [0,1]3min3ln23ln3 ,ugsu sss   [0,1]3max 3ln23ln(2),uwsu sss 3ln3ln2 .hsws gsss  Moreover for all [0,1]t   0d33ln3 33log33tqtptgs sttt t, Let 100N and 100jjt for all . 0,1, 2,,100jWe obtain the optimal solution jy and j, 0,1,j of problem (13) by solving problem (10) which is illustrated in Figures 4 and 5 respectively. In 2, ,100Copyright © 2011 SciRes. ICA M. H. N. SKANDARI ET AL. 124 Figure 3. Optimal control of Ex.12. (.)u Figure 4. Optimal trajectory of Ex.13. (.)y Figure 5. Corresponding optimal control of Ex.13. (.) addition, by (11) the corresponding for this ex-ample is (.)u 1/332,0,1,2,,100jj jhs gsjjues j The optimal control *ju, of prob-lem (10) is showed in Figure 6. Here, the value of opti-mal solution of objective function is 0.071. 0,1, 2,,100j Figure 6. Optimal control of Ex.13. (.)u 5. 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