An approximate method for solving fractional TBVP with state delay by Bernstein polynomials
- Elahe Safaie^{1}Email author and
- Mohammad Hadi Farahi^{1, 2}
https://doi.org/10.1186/s13662-016-1019-8
© Safaie and Farahi 2016
Received: 10 August 2016
Accepted: 2 November 2016
Published: 23 November 2016
Abstract
The current paper aims at investigating Fractional Hamiltonian Equations for a class of fractional optimal control problems with time delay. Furthermore, we introduce a method to solve the resulting two boundary values problem (TBVP) by extending Agrawal’s fractional variational method in (Nonlinear Dyn. 38:323-337, 2004) and using Bernstein polynomials (BPs). In this paper we use the Caputo fractional derivative of order α where \(0< \alpha<1\). Some numerical examples are included to demonstrate the validity of the present method.
Keywords
1 Introduction
Fractional calculus is a branch of mathematics that generalizes the derivative and the integral of a function to a noninteger order [2]. Fractional calculus has received considerable attention in recent years and there is hardly a field in science and engineering that has remained untouched by this field. It has been shown that materials with memory and hereditary effects and dynamical processes including gas diffusion and heat conduction can be more adequately modeled by fractional differential equations (FDEs) than by integer-order differential equations [3, 4].
In fractional calculus the Caputo and Riemann-Liouville are two main kinds of derivatives where each presents some advantages and disadvantages (see, e.g., [5]). The Caputo fractional derivative is commonly used in modeling physical phenomena but it is possible to assign a physical interpretation meaning for the Riemann-Liouville fractional derivative too. For instance, Heymans and Podlubny in [6] have illustrated some fractional differential equations with Riemann-Liouville fractional derivatives in the field of viscoelasticity.
A fractional optimal control problem (FOCP) is an optimal control problem in which the criterion and/or the differential equations governing the dynamics of the system contain at least one fractional derivative operator [7]. Most FOCPs do not have exact solutions, so in these cases approximation methods and numerical techniques must be used. Recently, several approximation methods to solve FOCPs have been introduced. Agrawal in [1] has introduced a general formula by making the TBVPs for some kinds of FOCP’s. Tricaud and Chen have solved fractional order optimal control problems by means of rational approximation [3]. Moreover, the effectiveness of using Legendre and Bernstein polynomials for approximating the solution of FOCPs has been demonstrated in [8–15]). Real life phenomena have been described more precisely with delay differential equations, so the delay fractional optimal control problem (DFOCP) has become the focus of many researchers in the last decade. Baleanu et al. in [16, 17] analyzed the fractional variational principles for some kinds of DFOCPs within the Riemann-Liouville and Caputo fractional derivatives, respectively, and made their corresponding Euler-Lagrange equations.
This paper is organized as follows. In Section 2 some preliminaries and definitions in the fractional calculus that used in this manuscript are reviewed. Section 3 gives a general introduction to Bernstein polynomials and their properties. In Section 4 the TBVP for a FOCP with time delay is analyzed and it is solved by using BPs. Section 5 contains some numerical examples. Conclusions are presented in Section 6.
2 Some preliminaries in fractional calculus
This section consists of some basic definitions and properties in fractional calculus [18, 19]. In the sequel, Γ represents the Gamma function.
Definition 2.1
Definition 2.2
Definition 2.3
Definition 2.4
Definition 2.5
Let \(\alpha>0\). Then \(_{0}I_{t}^{\alpha}(L^{p}([0,1]))\) denotes the space of all functions \(f(t)\), represented by \(_{0}I_{t}^{\alpha}\varphi\) where \(\varphi\in L^{p}([0,1])\).
Lemma 2.1
[18]
3 Bernstein polynomials (BPs) and their properties
Lemma 3.1
Lemma 3.2
[20]
Lemma 3.3
[21]
Lemma 3.4
[14]
For each given constant delay \(d > 0\), \(\Phi_{m}(t-d) = \Omega \Phi_{m}(t)\), where Ω is an \((m+1) \times(m+1)\) matrix in terms of d.
4 Fractional optimal control problems with state delay
In this section, first we state some lemmas to investigate the variational method for DFOCP (1)-(3) and make the corresponding TBVP, then two operational matrices to approximate the left-sided and right-sided Caputo fractional derivatives of \(\Phi_{m}(t)\) are introduced to numerically solve the TBVP.
Lemma 4.1
[18]
Lemma 4.2
[16]
Lemma 4.3
Proof
Equations (13) and (14) are proved in [16] when the fractional derivatives assumed to be Riemann-Liouville. Now, by assuming \(\varphi={^{c}_{t}D_{1}^{\alpha}} g\) and \(\psi={^{c}_{0}D_{t}^{\alpha}} f\) and applying Lemmas 4.1 and 4.2, considering \(f(0)=0\) and \(g(1)=0\), finding the results is straightforward. □
Remark
Theorem 4.1
Proof
According to Theorem 4.1 the necessary conditions for \((x^{*}(\cdot ),u^{*}(\cdot))\) being the optimal solution of (1)-(3) are to satisfy in (17)-(21). These conditions are also sufficient because of the convexity of the quadratic form of the objective function. To solve the system of equations (17)-(21), first from equation (17) one can conclude that \(u(t)=-R^{-1}(t)B^{T}(t)\lambda(t)\) (this is true since \(R(t)>0\)), then using the characteristic functions \(\chi_{[0,1-\eta]}(t)\) and \(\chi_{[1-\eta,1]}(t)\) we incorporate equations (19) and (20) and apply the Agrawal method in [23]. In this work we use Bernstein polynomials to approximate the solution of (18)-(21) in which \(u(t)\) is substituted by \(-R^{-1}(t)B^{T}(t)\lambda(t)\). Furthermore, to simplify the relations, in the sequel, we assume the matrix functions \(A(t)\), \(B(t)\), \(A_{d}(t)\), \(Q(t)\) and \(R(t)\) to be constant functions. Of course, when these functions are not constant the relations can be extended easily by approximating \(A(t)\), \(B(t)\), \(A_{d}(t)\), \(Q(t)\) and \(R(t)\) in terms of BPs.
In the above equations \(H=[H_{0}\ H_{1}\ \cdots\ H_{m}]\) is an \((m+1)\times (m+1)\) matrix where \(H_{i}\) (its \((i+1)\)th column of H) is the coefficients vector in approximating function \(h(t)=\int_{1-d} B_{i,m}(s)(s-t)^{\alpha-1}\,ds\) with BPs. Indeed \(_{\alpha}D^{1}\) and \(_{\alpha}D^{2}\) can be computed by substituting \(1-d\) and 1 in equation (33) instead of l (note that to achieve \(_{\alpha}D^{2}\) a suitable change of variable is needed to transform the time interval \([1-d,1]\) to \([0,1]\) before approximation). Also \(\Omega_{1}\), \(\Omega_{2}\) can be calculated by applying Lemma 3.4 to \(\Phi_{m}(t-\eta)\) and \(\Phi_{m}(t+\eta)\), respectively.
5 Numerical examples
In this section we give some numerical examples and apply the method presented in Section 4 for solving them. Our examples are solved using Matlab2011a on an Intel Core i5-430M processor with 4 GB of DDR3 Memory. These test problems demonstrate the validity and efficiency of this technique.
Example 1
The objective value and the end point of trajectory for \(\pmb{\alpha=1,0.9,0.8}\) in Example 1
α | Objective value | End points |
---|---|---|
1 | 1.9493 | 2.632,−7.3009 |
0.9 | 3.1472 | 2.4246,−8.7117 |
0.8 | 5.8783 | 1.8422,−10.5162 |
Example 2
The objective value and the end point of trajectory for \(\pmb{\alpha=1,0.9,0.8}\) in Example 2
α | Objective value | End point |
---|---|---|
1 | 1.0447 | 1.0772 |
0.9 | 1.0574 | 1.1032 |
0.8 | 1.0864 | 1.1240 |
Example 3
The objective value and the end point of trajectory for \(\pmb{\alpha=1,0.9,0.8}\) in Example 3
α | Objective value | End point |
---|---|---|
1 | 2.7384 | 0.7261 |
0.9 | 2.7504 | 0.6985 |
0.8 | 2.8108 | 0.5908 |
6 Conclusion
In this paper, we introduce the TBVP for the fractional optimal control problem with constant delay on trajectory. In order to solve the TBVP we have extended the method used in [1], then using Bernstein polynomials to approximate the solution. Since we use polynomials to approximate state and control, the approximating results are smooth and therefore no fitting curves are needed. We need to mention that in the case that the objective function is convex, the Hamiltonian condition would be necessary and sufficient. Thus by increasing the degree of Bernstein polynomials, the convergence should occur. Finally, some test problems are included to show the efficiency of this method.
Declarations
Acknowledgements
The authors like to express their sincere gratitude to referees and editor for their very constructive advices.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Agrawal, OP: A general formulation and solution scheme for fractional and optimal control problems. Nonlinear Dyn. 38, 323-337 (2004) MathSciNetView ArticleMATHGoogle Scholar
- Almeida, R, Torres, D: Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives. Commun. Nonlinear Sci. Numer. Simul. 16, 1490-1500 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Tricaud, C, Chen, YQ: An approximate method for numerically solving fractional order optimal control problems of general form. Comput. Math. Appl. 59, 1644-1655 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Zamani, M, Karimi, G, Sadati, N: FOPID controller design for robust performance using particle swarm optimization. Fract. Calc. Appl. Anal. 10, 169-188 (2007) MathSciNetMATHGoogle Scholar
- Miller, KS, Ross, B: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993) MATHGoogle Scholar
- Heymans, N, Podlubny, I: Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives. Rheol. Acta 45, 765-771 (2006) View ArticleGoogle Scholar
- Tricaud, C, Chen, YQ: Solution of fractional order optimal control problems using SVD-based rational approximations. In: American Control Conference Hyatt Regency Riverfront, St. Louis, MO, USA, June 10-12 (2009) Google Scholar
- Agrawal, OP: A formulation and a numerical scheme for fractional optimal control problems. J. Vib. Control 14, 1291-1299 (2008) MathSciNetView ArticleMATHGoogle Scholar
- Agrawal, OP: A quadratic numerical scheme for fractional optimal control problems. J. Dyn. Syst. Meas. Control 130, 011010 (2008) View ArticleGoogle Scholar
- Alipour, M, Rostamy, D, Baleanu, D: Solving multi-dimensional fractional optimal control problems with inequality constraint by Bernstein polynomials operational matrices. J. Vib. Control (2012). doi:10.1177/1077546312458308 MathSciNetGoogle Scholar
- Jafari, H, Yousefi, SA, Firoozjaee, MA, Momanic, S, Khalique, CM: Application of Legendre wavelets for solving fractional differential equations. Comput. Math. Appl. 62, 1038-1045 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Lotfi, A, Dehghan, M, Yousefi, SA: A numerical technique for solving fractional optimal control problems. Comput. Math. Appl. 62, 1055-1067 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Lotfi, A, Yousefi, SA, Dehghan, M: Numerical solution of a class of fractional optimal control problems via the Legendre orthonormal basis combined with the operational matrix and the Gauss quadrature rule. J. Comput. Appl. Math. 250, 143-160 (2013) MathSciNetView ArticleMATHGoogle Scholar
- Safaie, E, Farahi, MH, Farmani Ardehaie, M: An approximate method for numerically solving multi-dimensional delay fractional optimal control problems by Bernstein polynomials. Comput. Appl. Math. 34, 831-846 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Safaie, E, Farahi, MH: An approximation method for numerical solution of multi-dimensional feedback delay fractional optimal control problems by Bernstein polynomials. Iran. J. Numer. Anal. Optim. 4(1), 77-94 (2014) MATHGoogle Scholar
- Baleanu, D, Maaraba Abdeljawad, T, Jarad, F: Fractional variational principles with delay. J. Phys. A, Math. Theor. 41, 315403 (2008). doi:10.1088/1751-8113/41/31/315403 MathSciNetView ArticleMATHGoogle Scholar
- Jarad, F, Maaraba Abdeljawad, T, Baleanu, D: Fractional variational principles with delay within Caputo derivatives. Rep. Math. Phys. 65, 17-28 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Samko, SG, Kilbas, AA, Marichev, OI: Fractional Integrals and Derivatives: Theory and Applications. Gordon & Breach, Amsterdam (1993) MATHGoogle Scholar
- Diethelm, K: The Analysis of Fractional Differential Equations. Springer, Berlin (2010) View ArticleMATHGoogle Scholar
- Kreyszig, E: Introduction to Functional Analysis with Applications. Wiley, New York (1978) MATHGoogle Scholar
- Lorentz, GG: Zur theorie der polynome von S. Bernstein. Mat. Sb. 2, 543-556 (1937) MATHGoogle Scholar
- Pinch, ER: Optimal Control and Calculus of Variations. Oxford Science Publications, Oxford (1993) MATHGoogle Scholar
- Agrawal, OP: General formulation for the numerical solution of optimal control problems. Int. J. Control 50(2), 627-638 (1989) MathSciNetView ArticleMATHGoogle Scholar
- Wang, XT: Numerical solutions of optimal control for time delay systems by hybrid of block-pulse functions and Legendre polynomials. Appl. Math. Comput. 184, 849-856 (2007) MathSciNetMATHGoogle Scholar
- Ghomanjani, F, Farahi, MH, Gachpazan, M: Optimal control of time-varying linear delay systems based on the Bezier curves. Comput. Appl. Math. (2013). doi:10.1007/s40314-013-0089-4 MathSciNetMATHGoogle Scholar
- Haddadi, N, Ordokhani, Y, Razzaghi, M: Optimal control of delay systems by using a hybrid functions approximation. J. Optim. Theory Appl. 153, 338-356 (2012) MathSciNetView ArticleMATHGoogle Scholar
- Khellat, F: Optimal control of linear time-delayed systems by linear Legendre multiwavelets. J. Optim. Theory Appl. 143, 107-121 (2009) MathSciNetView ArticleMATHGoogle Scholar