Nonlinear Dynamics 38: 323–337, 2004. C 2004 Kluwer Academic Publishers. Printed in the Netherlands. A General Formulation and Solution Scheme for Fractional Optimal Control Problems OM PRAKASH AGRAWAL Department of Mechanical Engineering, Southern Illinois University, Carbondale, IL 62901, U.S.A. (e-mail: [email protected]; fax: +1-618-453-7658) (Received: 16 November 2003; accepted: 9 April 2004) Abstract. Accurate modeling of many dynamic systems leads to a set of Fractional Differential Equations (FDEs). This paper presents a general formulation and a solution scheme for a class of Fractional Optimal Control Problems (FOCPs) for those systems. The fractional derivative is described in the Riemann–Liouville sense. The performance index of a FOCP is considered as a function of both the state and the control variables, and the dynamic constraints are expressed by a set of FDEs. The Calculus of Variations, the Lagrange multiplier, and the formula for fractional integration by parts are used to obtain Euler–Lagrange equations for the FOCP. The formulation presented and the resulting equations are very similar to those that appear in the classical optimal control theory. Thus, the present formulation essentially extends the classical control theory to fractional dynamic system. The formulation is used to derive the control equations for a quadratic linear fractional control problem. An approach similar to a variational virtual work coupled with the Lagrange multiplier technique is presented to find the approximate numerical solution of the resulting equations. Numerical solutions for two fractional systems, a time-invariant and a time-varying, are presented to demonstrate the feasibility of the method. It is shown that (1) the solutions converge as the number of approximating terms increase, and (2) the solutions approach to classical solutions as the order of the fractional derivatives approach to 1. The formulation presented is simple and can be extended to other FOCPs. It is hoped that the simplicity of this formulation will initiate a new interest in the area of optimal control of fractional systems. 1. Introduction This paper presents a new formulation and a new solution scheme for a class of fractional optimal control problems for fractional dynamic systems. A Fractional Dynamic System (FDS) is a system whose dynamics is described by Fractional Differential Equations (FDEs), and a Fractional Optimal Control Problem (FOCP) is an optimal control problem for a FDS. Considerable work has been done in the area of optimal control, and excellent textbooks exist on the subject (see, e.g. [1–3]). Because of its importance, there are several journals that publish articles of this field only. In addition, many other journals publish articles dealing with optimal control problems regularly. However, optimal control problems considered in these books and journal articles largely deal with systems dynamics whose behaviors are described by integral order differential equations. Recent investigations in engineering, science, and other fields have demonstrated that the dynamics of many systems are described more accurately using FDEs (see, e.g. [4–8] and the papers and references therein). As Miller and Ross  point out, there is hardly a field of science or engineering that has remained untouched by this field. However, very little work has been done in the area of FOCPs. As the demand for efficient, accurate, and high precision systems grows, the demand for optimal control theories, and the analytical and numerical schemes to solve the resulting equations will also grow. The formulation, numerical scheme, and numerical results for some FOCPs presented in this paper are attempts to fill this gap. Fractional derivatives, or more precisely derivatives of arbitrary orders, have played a significant role in engineering, science, and mathematics in recent years. Samko et al.  provide an encyclopedic 324 O. P. Agrawal treatment of this subject. Additional background, survey, and application of this field in science, engineering, and mathematics can be found in [11, 9, 12–14, 6, 7]. Only limited work has been done in the area of fractional control, or more specifically, in the area of Fractional Optimal Control (FOC). Early work in the area of FOC is documented in . Bode  suggested the idea of using FOC to ensure amplifier stability, but he did not implement the idea. Manabe  introduced fractional differentiation in feedback systems with saturation. Other works by Manabe in this field appear in . More recently, Skaar et al.  presented a root locus method to assess the stability of a fractionally controlled distributed viscoelastic structure whose constitutive law is modeled using fractional-order derivatives. They find stability criteria for the system which is similar to those for integral-order control system. Axtell and Bise  explored s-domain analysis of a fractional-order system. Bagley and Calico  presented a state space formulation to predict the effects of feedback to reduce motion of a fractionally damped system. These authors conclude that the fractional-order time derivative control improves the performance of a system exhibiting strong hereditary behavior. Makroglou et al.  presented a time domain method to assess the performance of the fractionally damped rotating beam and compared the results with a Kelvin–Voight constitutive model. Mbodje and Montseny  investigated the existence, uniqueness and asymptotic decay of the wave equation with fractional derivative feedback, and showed that the method developed can easily be adapted to a wide class of problems involving fractional derivative or integral operators of the time variable. Machado [22, 23] developed algorithms for fractional-order discrete-time controllers that are suited for z-transform analysis and discrete-time implementation and showed that classical P, I and D actions are special cases of the new fractional control scheme. Podlubny, Dorcak, and Kostial  compared the Letnikov–Riemann–Liouville and Caputo fractional derivatives from the point of view of their applications in a generalization of the PID-controller called the P I λ D µ controllers. Oustraloup and coworkers have applied fractional derivatives in system identification, robust control, and other fields (see, e.g. [25, 26] and references therein). Podlubny  presents a complete chapter on fractional-order systems and controllers and shows that PI-, PD-, and PID-controllers are particular cases of the P I λ D µ controller. Hotzel  investigated the stability conditions for fractional delay systems and proved that for stability, it is sufficient that the real parts of the transfer poles have a negative upper bound, and it is necessary that the real part of every transfer pole is negative. Hartley and Lorenzo  present a general fractional-order system and control theory that includes the time-varying initialization response. They also present the stability properties of fractional-order systems, and a fractional-order vector space representation, which is a generalization of the state space concept. Note that calculus of variations play a significant role in the field of classical optimal control. Given this fact, two recent papers by Riewe [29, 30] must be mentioned here. These papers are important because they use fractional calculus of variations to develop Larangian, Euler–Lagrange equations, and other concepts for mechanics of nonconservative systems. Agrawal  extends variational calculus to fractional variational problems. From the above and other literature in the field of fractional calculus it is clear that many of the ideas of the ordinary calculus can be extended to fractional calculus with only minor changes. In this paper, we present a new formulation and a new numerical scheme for a class of FOCPs, which can be considered as a direct extension of formulations and numerical schemes for classical optimal control problems. Our derivation uses Riewe’s results to develop the formulation. However, note that Riewe develops his formulation for nonconservative Mechanics in terms of the left Riemann–Liouville fractional derivatives. As a result, his formulation includes terms containing fractional power of (−1). In contrast, we use both the left and the right fractional derivatives to remove this problem. We further show that the numerical scheme to solve the fractional optimal control problem converges as the number A general formulation and solution scheme for fractional optimal control problems 325 of approximating terms is increased, and the solutions approach to classical optimal control solutions as the order of the fractional derivative approaches to 1. In contrast to numerical schemes for FOC, many numerical schemes have been developed to solve classical optimal control problems. Formulations and numerical schemes for classical optimal control can be found, among others, in [2, 3, 32], and the references listed there. The numerical scheme for fractional optimal control problems presented here follows the approach presented in Agrawal  for optimal control problems containing derivatives of integral orders only. Note that optimal control problems inherently lead to two-point boundary value problems. Lorenzo and Hartley  have discussed the problem of finding the correct form of the initial conditions in a more general setting. Deithelm, et al.  present a predictor-corrector type algorithm for fractional differential equations and cite several references for the same. Following these developments, a shooting method type algorithm may be possible for FOC. However, this will be considered in the future. 2. Euler–Lagrange equations for FOCPs In this section, we first define a fractional derivative, and then formulate a FOCP and find the necessary conditions for optimality. Several definitions of a fractional derivative have been proposed. These definitions include Riemann– Liouville, Grunwald–Letnikov, Weyl, Caputo, Marchaud, and Riesz fractional derivatives [11, 9, 6, 35]. Here, we formulate the problem in terms of the Left and the Right Riemann–Liouville fractional derivatives, which are defined as , The Left Riemann-Liouville Fractional Derivative α a Dt 1 f (t) = (n − α) d dt n t (t − τ )n−α−1 f (τ )dτ, (1) a and The Right Riemann-Liouville Fractional Derivative α t Db 1 d n b f (t) = (t − τ )n−α−1 f (τ )dτ, − (n − α) dt t (2) where α is the order of the derivative such that n − 1 ≤ α < n. These derivatives will be denoted as the LRLFD and RRLFD, respectively. Note that in literature the Riemann–Liouville fractional derivative generally means the LRLFD. Using the above definitions, the FOCP under consideration can be defined as follows. Find the optimal control u(t) for a FDS that minimizes the performance index J (u) = 1 F(x, u, t)dt (3) 0 subject to the system dynamic constraints α 0 Dt x = G(x, u, t), (4) 326 O. P. Agrawal and the initial condition x(0) = x0 , (5) where x(t) is the state variable, t represents the time, and F and G are two arbitrary functions. Note that Equation (3) may also include some additional terms containing state variables at the end point. This term in not considered here for simplicity. When α = 1, the above problem reduces to a standard optimal control problem. Here the limits of integration have been taken as 0 and 1. Furthermore, we consider 0 < α < 1. These are not the limitations of the approach. Any limits can be considered and the derivative can be of any order. However, these conditions are considered for simplicity. To find the optimal control we follow the traditional approach and define a modified performance index as 1 J̄ (u) = 0 [F(x, u, t) + λ(G(x, u, t) − 0 Dtα x)]dt, (6) where λ is the Lagrange multiplier also known as a costate or an adjoint variable. Taking variation of Equation (6), we obtain 1 δ J̄ (u) = 0 α ∂F ∂G ∂F ∂G α dt, (7) δx + δu + δλ(G(x, u, t) − 0 Dt x) + λ δx + δu − δ 0 Dt x ∂x ∂u ∂x ∂u where δx, δu, and δλ, are the variation of x, u, and λ consistent with the specified terminal condition. Riewe  has demonstrated that for ν > 0, the following identity is satisfied b a d ν f (t) g(t)dt = (−1)−ν d(t − a)ν b f (t) a d ν g(t) dt d(t − b)ν (8) provided that d k f (t)/dt k = 0 or d k g(t)/dt k = 0 at t = a and t = b for k = 0 to n − 1, where d ν f (t) = a Dtν f (t) d(t − a)ν (9) is the LRLFD of f (t) of order ν, and n is the smallest integer greater than ν. In terms of our notations, Equation (8) is written as b a (a Dtα f (t))g(t)dt = a b f (t)(t Dbα g(t))dt. (10) Equation (10) – called the formula for fractional integration by parts – can also be found in , which lists additional requirements that functions f (t) and g(t) must satisfy. Using Equation (10), the last integral in Equation (7) can be written as 0 1 λδ(0 Dtα x)dt = 0 1 δx(t D1α λ)dt (11) provided δx(0) = 0 or λ(0) = 0, and δx(1) = 0 or λ(1) = 0. Because x(0) is specified, we have δx(0) = 0, and since x(1) is not specified, we require λ(1) to be zero. With these assumptions, the A general formulation and solution scheme for fractional optimal control problems 327 identity in Equation (11) is satisfied. Note that we have assumed that the order of variation and the fractional derivative can be interchanged. Using Equations (7) and (11), we obtain 1 δ J̄ (u) = 0 δλ(G(x, u, t) − 0 Dtα x) + δx ∂F ∂F ∂G ∂G +λ − t D1α λ + δu +λ dt. (12) ∂x ∂x ∂u ∂u Minimization of J̄ (u) (and hence minimization of J (u)) requires that the coefficients of δλ, δx, and δu in Equation (12) be zero. This leads to α 0 Dt x = G(x, u, t), ∂F ∂G α +λ , t D1 λ = ∂x ∂x ∂F ∂G +λ = 0, ∂u ∂u (13) (14) (15) and x(0) = x0 and λ(1) = 0. (16) Equations (13) to (15) represent the Euler-Lagrange equations for the FOCP. These equations give the necessary conditions for the optimality of the FOCP considered here. They are very similar to the EulerLagrange equations for classical optimal control problems except that the resulting differential equations contain the left and the right fractional derivatives. Furthermore, the derivation of these equations is very similar to the derivation for an optimal control problem containing integral order derivatives. Determination of the optimal control for the fractional system requires solution of Equations (13) to (16). Observe that Equation (13) contains LRLFD where as Equation (14) contains RRLFD. This clearly indicates that the solution of optimal control problems requires knowledge of not only forward derivatives but also backward derivatives to account for end conditions. In classical optimal control theories, this issue is either not discussed or they are not clearly stated. This is largely because the backward derivative of order 1 turns out to be the negative of the forward derivative of order 1. For α = 1, 0 Dtα x and t D1α λ are written as d x/dt and −dλ/dt, and Equations (13) and (14) reduce to dx = G(x, u, t), dt and dλ ∂ F ∂G + +λ = 0, dt ∂x ∂x which are the same as those obtained using classical optimal control theories [2, 3]. As a special case, assume that the performance index is an integral of quadratic forms in the state and the control, 1 J (u) = 2 0 1 [q(t)x 2 (t) + r (t)u 2 ]dt, (17) 328 O. P. Agrawal where q(t) ≥ 0 and r (t) > 0, and the dynamics of the system is described by the following linear fractional differential equation, α 0 Dt x = a(t)x + b(t)u. (18) This linear system for α = 1 has been studied extensively, and formulations and solution schemes for this system are well documented in many textbooks and journal articles (see e.g. [2, 3)]. For 0 < α < 1, the Euler-Lagrange Equations (13) to (15) and (18) lead to Equation (18) and α t D1 λ = q(t)x + a(t)λ, (19) and r (t)u + b(t)λ = 0. (20) From Equations (18) and (19), we get α 0 Dt x = a(t)x − r −1 (t)b2 (t)λ. (21) The state x(t) and the costate λ(t) are obtained by solving the fractional differential equations (19) and (21) subject to the terminal conditions given by Equation (15). Once λ(t) is known, the control variable u(t) can be obtained using Equation (20). An approximate numerical method to find the state x(t) and the costate λ(t) is presented next. 3. Numerical Scheme to Solve the FOCPs Solution of a FOCP associated with a linear FDS with quadratic performance index requires solution of Equations (19) and (21) subject to the terminal conditions given by Equation (15). Equations (15), (19), and (21) provide a set of two point Fractional Boundary Value Problems (FBVP). Several methods have been presented to solve this class of problems for α = 1, for example see [2, 3, 32], and the references therein. Many of the techniques cited in  can be extended to the fractional optimal control problem formulated here. In this paper, we use the formulation of  for this task. To find an approximate solution of Equations (15), (19), and (21), assume that δx and δλ are arbitrary virtual variations of x and λ as defined above, except that they need not be consistent with the terminal conditions. Using an approach analogous to a variational work approach and the Lagrange multiplier technique, Equations (15), (19), and (21) can be restated as , a b [δx(0 Dtα x − a(t)x + r −1 (t)b2 (t)λ) + δλ(t D1α λ − q(t)x − a(t)λ)]dt +δ[µ1 (x(0) − x0 )] + δ[µ2 λ(1)] = 0 (22) where µ1 and µ2 are the Lagrange multipliers associated with the terminal conditions. The last two terms in Equation (22) ensure that the terminal conditions are satisfied. Equation (22) can also be obtained by eliminating u(t) from Equations (12) and (20), and adding the variations of the terminal conditions to the A general formulation and solution scheme for fractional optimal control problems 329 resulting equation. A more general formulation can be obtained by multiplying weighting coefficients to each variations. Details of this approach for α = 1 can be found in . Equation (22) is the desired variational formulation for numerical solution of the FOCP. For numerical solution, x, λ, δx, and δλ can be approximated using a set of basis functions, and Equation (22) can be used to convert the two point FBVP given by Equations (15), (19) and (21) to a set of algebraic equations. Solution of the algebraic equations will give the coefficients, which can then be substituted back into the approximating functions to find the desired solution. Since δx and δλ are arbitrary, different basis functions can be selected for x and λ, and δx, and δλ. Note that approximating functions for x and λ need not satisfy the terminal conditions a priori. The last two terms in Equation (22) enforce the terminal conditions. In this paper, we approximate x, λ, δx, and δλ as x(t) = m c j P j (t), (23) d j P j (t), (24) j=1 λ(t) = m j=1 δx(t) = m δc j P j (t), (25) j=1 and δλ(t) = m δd j P j (t), (26) j=1 where P j (t), j = 1, . . . , m, are the shifted Legendre polynomials which satisfy the following orthonormality conditions, 1 P j (t)Pk (t)dt = δ jk = 0 0 j = k 1 j =k , (27) c j and d j , j = 1, · · · , m, are polynomial coefficients, m is the number of polynomials selected, and δc j and δd j are the variations of the coefficients c j and d j . Here δ jk is the Kroneker delta function. The number of polynomials should be selected such that certain error is small in some sense. For example, as the number of polynomials changes, the change in the vale of the performance index or the change in the state variable measured in some norm space could be taken as a measure of the error. Note that it is not necessary to select orthonomal polynomials as the basis functions. Orthonormal polynomials are selected here because they lead to numerically stable sparse matrices, and in many cases the properties of the polynomials can be used to generate the desired matrices efficiently. It is not necessary to select the shifted Legendre orthonormal polynomials only. Other orthonormal polynomials can also be selected for this task. However, this may require some modifications in the formulation so that one can take advantage of the properties of the orthonormal polynomials. This issue for α = 1 is discussed in . Substituting Equations (23) and (24) into Equation (17), the performance index may be written as J= m m 1 [F0x ( j, k)c j ck + F0u ( j, k)d j dk ], 2 j=1 k=1 (28) 330 O. P. Agrawal where F0x and F0u are defined as 1 F0x ( j, k) = q(t)P j (t)Pk (t)dt (29) r (t)P j (t)Pk (t)dt. (30) 0 and 1 F0u ( j, k) = 0 Substituting Equations (23) to (27) into Equation (22), and setting the coefficients of δµ1 , δµ2 , δc j and δd j , j = 1, . . . , m to zero, we obtain m [F1 ( j, k) − F2 ( j, k)]ck + k=1 − F3 ( j, k)dk + P j (0)µ1 = 0, j = 1, . . . , m, (31) k=1 m F0x ( j, k)ck + k=1 m m m [F4 ( j, k) − F2 ( j, k)]dk + P j (1)µ2 = 0, j = 1, . . . , m, (32) k=1 Pk (0)ck = x0 , (33) k=1 and m Pk (1)dk = 0, (34) k=1 where F1 ( j, k) through F4 ( j, k) are defined as 1 F1 ( j, k) = 0 0 P j (t)(0 Dtα Pk (t))dt, (35) a(t)P j (t)Pk (t)dt, (36) r −1 (t)b2 (t)P j (t)Pk (t)dt, (37) 1 F2 ( j, k) = 1 F3 ( j, k) = 0 and 1 F4 ( j, k) = 0 P j (t)(t D1α Pk (t))dt. (38) Note that Equations (35) and (38) contain the fractional derivatives of the basis functions. For Legendre functions, these derivatives could be obtained in close form. For other basis functions, numerical integration may be necessary. Equations (31) to (34) provide a set of (2m + 2) linear equations in (2m + 2) unknowns which can be solved using a standard subroutine. Once the unknowns c j and d j , j = 1, . . . , m are known, the state and the control variables are obtained using Equations (20), (23), and (24). An approach similar to A general formulation and solution scheme for fractional optimal control problems 331 the one presented here has been used in conjunction with Hamilton’s law of varying action to find the response of a dynamic system whose dynamics is described using integral derivative terms only . Thus, the present numerical scheme can be extended to fractional differential equations. This it will be considered in the future. Applications of the formulations presented in Sections 2 and 3 are presented next. 4. Numerical Examples To demonstrate the applicability of the formulation, and to validate the numerical scheme, we derive the differential equations and present numerical results for two FOCPs, one time invariant and the other time varying. We also demonstrate that the solutions converge as the number of basis polynomials selected is increased, and the solutions for the fractional control problems approach to the solutions for standard control problems as the order of the fractional derivative (i.e. α) approaches to 1. 4.1. TIME INVARIANT FOCP As a first example, consider the following time invariant FOCP: Find the control u(t) which minimizes the quadratic performance index 1 J (u) = 2 1 [x 2 (t) + u 2 ]dt (39) 0 subject to the system dynamics α 0 Dt x = −x + u. (40) and the initial condition x(0) = 1. (41) Note that in this example, q(t) = r (t) = −a(t) = b(t) = x0 = 1, (42) and Equations (19) and (20) are given as α t D1 λ = x − λ, (43) and u + λ = 0, (44) and thus the optimal control function u(t) is negative of the costate variable λ(t). Analytical and numerical results for this example for α = 1 can be found in [2, 3, 32] and references therein. Substituting Equation (42) into Equations (29), (30), (36), and (37), and using Equation (27), we obtain F0x ( j, k) = F0u ( j, k) = −F2 ( j, k) = F3 ( j, k) = δ jk . (45) 332 O. P. Agrawal Figure 1. Convergence of the state variable for the time-invariant system for α = 3/4 (: m = 4; ×: m = 6; o: m = 8; +: m = 10). Figure 2. Convergence of the control variable for the time-invariant system for α = 3/4 (: m = 4; ×: m = 6; o: m = 8; +: m = 10). The problem is solved for different values of m and α. Figures 1 and 2 show the state and the control variables, respectively, as a function of time for α = 3/4 for different values of m. From these figures it is clear that both the state and the control variables converge as the number of approximating polynomials is increased. Figures 3 and 4 show the state and the control variables, respectively, as a function of time for m = 10 for different values of α. These figures show that as α approaches close to 1, the numerical solutions for both the state and the control variables approach to the analytical solutions for α = 1 as expected. In Figures 3 and 4, only the numerical results for α = 1 are presented. This is because, for α = 1 the analytical and the numerical results overlap. Furthermore, for α = 1, comparison of analytical and numerical results appear in . Note that in these figures amplitudes of both x(t) and u(t) decrease as α is decreased. For α = 0, Equation (40) essentially represents a linear algebraic equation, and in that case, we obtain the trivial optimal solution as x(t) = u(t) = 0 for t > 0. A general formulation and solution scheme for fractional optimal control problems 333 Figure 3. State variable as a function of time for fractional derivative models of different order for the time-invariant system (: α=1/2; ×: α = 3/4; o: α = 7/8; +: α = 15/16; −: α = 1). Figure 4. Control variable as a function of time for fractional derivative models of different order for the time-invariant system (: α = 1/2; ×: α = 3/4; o: α = 7/8; +: α = 15/16; −: α = 1). 4.2. TIME VARYING FOCP As a second example consider a linear time varying system with the same performance index and the same initial condition as those considered in Section 4.1, except in this example, the system is subjected to the following dynamic constraint, α 0 Dt x = t x + u. (46) 334 O. P. Agrawal Figure 5. Convergence of the state variable for the time-varying system for α = 3/4 (: m = 4; ×: m = 6; o: m = 8; +: m = 10). For this case, q(t) = r (t) = b(t) = x0 = 1, α t D1 λ = x + tλ, a(t) = t, (47) (48) and u + λ = 0. (49) This problem for α = 1 has been considered by several investigators in the past (see e.g.  and the references therein). For this example also F0x ( j, k), F0u ( j, k), and F3 ( j, k) are given by Equation (45), and F2 ( j, k) is obtained by replacing a(t) in Equation (36) by t. Using the properties of the Legendre polynomials it can be shown that for this example F2 ( j, k) leads to a tri-diagonal matrix. Like the previous problem, this problem also is solved for different values of m and α. Figures 5 and 6 show the state and the control variables, respectively, as a function of time for α = 3/4 for different values of m. Figures 7 and 8 show the state and the control variables, respectively, as a function of time for m = 10 for different values of α. From these figures we see that in this example also both the state and the control variables converge as the number of approximating polynomials is increased (see Figures 5 and 6). It is clear from Figures 3, 4, 7, and 8 that as α decreases the solutions for x(t) and u(t) oscillate more. As a result for smaller α more number of polynomials are needed for the convergence of the solutions. As a final remark, we note that very little progress has been made in the field of FOCP. This is largely due to the fact that the underlying mathematics for fractional derivatives was not well developed. Recent development in the field of fractional derivatives has eliminated this barrier. From the formulation and the numerical examples presented above, it is clear that many of the concepts of classical control theory can be directly extended to FOCPs. Although only one class of FOCPs was considered here, the A general formulation and solution scheme for fractional optimal control problems 335 Figure 6. Convergence of the control variable for the time- varying system for α = 3/4 (: m = 4; ×: m = 6; o: m = 8; +: m = 10). Figure 7. State variable as a function of time for fractional derivative models of different order for the time-varying system (: α = 1/2; ×: α = 3/4; o: α = 7/8; +: α = 15/16; −: α = 1). formulation can be extended to many other FOCPs. It is hoped that this observation will initiate some interest in the areas of fractional variational calculus and fractional optimal control. 5. Conclusions A general formulation has been presented for a class of fractional optimal control problems. The formulation utilized the calculus of variations, the Lagrange multiplier technique, and the formula for fractional integration by parts to obtain the Euler–Lagrange equations for the fractional optimal control problems. The formulation presented and the resulting equations are very similar to those for classical 336 O. P. Agrawal Figure 8. Control variable as a function of time for fractional derivative models of different order for the time-varying system (: α = 1/2; ×: α = 3/4; o: α = 7/8; +: α = 15/16; −: α = 1). optimal control problems. The formulation is specialized for a system with quadratic performance index subject to a fractional system dynamic constraint. Two numerical examples, one time invariant and the other time varying are presented to show the applications of the formulations. Numerical results show that the approximate solutions converge as the number of approximating polynomials increase, and as α approaches close to 1, the numerical solutions for both the state and the control variables approach to the analytical solutions for α = 1. It is hoped that the simplicity of the formulation and the numerical scheme presented here will initiate new research in the areas of fractional variational calculus and fractional optimal control. 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