arXiv:1403.5060v2 [math.OC] 23 Oct 2016 ae umte 6Nv21;rvsd1/ac/04 acce 17/March/2014; revised 26/Nov/2013; submitted Paper etrfrRsac n eeomn nMteaisadApplicatio and Mathematics in Development and Research for Center esatwt oedfiiin eddi h eul o oeo h s the 1. on more Definition For sequel. 15]. the 10, in 9, needed [7, definitions to some with start We Preliminaries w 2 illustrated, then and 3 Section in presented is method Our results. con we method. operator, fractiona Euler-like fractional original an the different the replacing apply after ordinary to solution here: of different the Then system is approximates a 12]. one to [2, solution, only reduced derivatives its 5 is integer 4, problem on the [1, depends o that conditions, reliable fractional so one, and the new to efficient operators, a difficult be fractional into are the to for proven problems approximation have fractional an which Such applied, often 14]. are [9, intege well only as not involves operators system control dynamic the fra where control, that fact the to due a interest [3]. of increasing phenomena integration an natural and received has differentiation It with [15]. deals calculus fractional The Introduction 1 2010: Keywords: Classification Subject Mathematics t ∈ ∗ eateto ahmtc,Uiest fAer,31–9 Aveir 3810–193 Aveiro, of University Mathematics, of Department h eti raie sflos nScin2w reyrcl h nec the recall briefly we 2 Section In follows. as organized is text The o that is area research active, very and interesting, particularly A hsi rpito ae hs nladdfiiefr ila will form definite and final whose paper a of preprint a is This [ ,b a, eilsrt h ffcieeso h rcdr iha exam an with procedure the finite of of effectiveness application the invo by that illustrate discretized, We one then new is a problem with a latter problem Our order derivatives. fractional fractional initial Caputo the and integer on depends ,w define we ], epeetamto oslefatoa pia oto prob control optimal fractional solve to method a present We rcinlcluu,fatoa pia oto,drc methods. direct control, optimal fractional calculus, fractional Let [email protected] iceeMto oSleFractional Solve to Method Discrete A x [ : iad Almeida Ricardo ,b a, pia oto Problems Control Optimal ] → R and > α Abstract eteodro h nerlo h eiaie For derivative. the or integral the of order the be 0 1 63,49M25. 26A33, tdfrpbiain20/March/2014. publication for pted efi .M Torres M. F. Delfim pa in ppear ieecs n ovdnumerically. solved and differences, [email protected] vsitgrdrvtvsol.The only. derivatives integer lves poc ossst approximate to consists pproach tmlcnrlpolmi rewritten is problem control ptimal ie h umne ucinland functional augmented the sider ple. olna Dynamics Nonlinear re eiaie u fractional but derivatives order r toa prtr r endby defined are operators ctional rbe 1] u approach Our [12]. problem l es hr h dynamic the where lems, a qain n,b finding by and, equations ial t neape nScin4. Section in example, an ith sn eesr optimality necessary using , ∗ v n ueia methods numerical and lve btay(oitgr order (noninteger) rbitrary betw ee h reader the refer we ubject ,1] yial,using Typically, 13]. 8, , h rcinloptimal fractional the f saycnet and concepts essary ,Portugal o, SN0924-090X. ISSN , s(CIDMA) ns 1. the left Riemann–Liouville fractional integral by

t α 1 α−1 aI x(t)= (t − τ) x(τ)dτ, t Γ(α) Za 2. the right Riemann–Liouville fractional integral by

b α 1 α−1 tI x(t)= (τ − t) x(τ)dτ, b Γ(α) Zt 3. the left Riemann–Liouville fractional derivative by

n t α 1 d n−α−1 aD x(t)= (t − τ) x(τ)dτ, t Γ(n − α) dtn Za 4. the right Riemann–Liouville fractional derivative by

n n b α (−1) d n−α−1 tD x(t)= (τ − t) x(τ)dτ, b Γ(n − α) dtn Zt 5. the left Caputo fractional derivative by 1 t C Dαx(t)= (t − τ)n−α−1x(n)(τ)dτ, a t Γ(n − α) Za 6. the right Caputo fractional derivative by

(−1)n b CDαx(t)= (τ − t)n−α−1x(n)(τ)dτ, t b Γ(n − α) Zt where n = [α] + 1 for the definitions of Riemann–Liouville fractional derivatives, and [α] + 1 if α∈ / N , n = 0 (1) α if α ∈ N ,  0 for the Caputo fractional derivatives. When α = n is an integer, the operators of Definition 1 reduce to standard ones:

t τ1 τn−1 n aIt x(t)= dτ1 dτ2 ... x(τn)dτn, a a a Z b Z b Zb n tIb x(t)= dτ1 dτ2 ... x(τn)dτn, Zt Zτ1 Zτn−1 n C n (n) aDt x(t)= a Dt x(t) = x (t), n C n n (n) tDb x(t)= t Db x(t) = (−1) x (t). There is an useful relation between Riemann–Liouville and Caputo fractional derivatives:

n−1 (k) C α α x (a) k−α D x(t)= aD x(t) − (t − a) . (2) a t t Γ(k − α + 1) k X=0 For numerical purposes, approximations are used to deal with these fractional operators. The Riemann–Liouville derivatives are expandable in a power series involving integer order derivatives only. If x is an analytic function, then (cf. [7])

∞ k−α α α (t − a) (k) aD x(t)= x (t), (3) t k Γ(k +1 − α) k X=0  

2 where α (−1)k−1αΓ(k − α) = . k Γ(1 − α)Γ(k + 1)   The obvious disadvantage of using (3) in numerical computations is that in order to have a small error, one has to sum a large number of terms and thus the function has to possess higher-order derivatives, which is not suitable for optimal control. To address this problem, a second approach was carried out in [2, 12], where a good approximation is obtained without the requirement of such higher-order smoothness on the admissible functions. The method can be explained, for left-derivatives, in the following way. Let α ∈ (0, 1) and x ∈ C2[a,b]. Then, ∞ α −α 1−α 1−p−α aDt x(t)= A(α)(t − a) x(t)+ B(α)(t − a) x˙(t) − C(α, p)(t − a) Vp(t), p=2 X where Vp(t) is the solution of the system

p−2 V˙p(t)=(1 − p)(t − a) x(t), (Vp(a)=0, for p =2, 3,..., and A, B and C are given by ∞ 1 Γ(p − 1+ α) A(α)= 1+ , Γ(1 − α) Γ(α)(p − 1)! " p=2 # X∞ 1 Γ(p − 1+ α) B(α)= 1+ , Γ(2 − α) Γ(α − 1)p! " p=1 # X 1 Γ(p − 1+ α) C(α, p)= . Γ(2 − α)Γ(α − 1) (p − 1)! Using (2), a similar formula can be deduced for the Caputo fractional derivative. When we consider finite sums only, that is, when we use the approximation

K α −α 1−α 1−p−α aDt x(t) ≈ A(α, K)(t − a) x(t)+ B(α, K)(t − a) x˙(t) − C(α, p)(t − a) Vp(t), (4) p=2 X where K ≥ 2 and K 1 Γ(p − 1+ α) A(α, K)= 1+ , Γ(1 − α) Γ(α)(p − 1)! " p=2 # X K 1 Γ(p − 1+ α) B(α, K)= 1+ , Γ(2 − α) Γ(α − 1)p! " p=1 # X the error is bounded by 2 exp((1 − α) +1 − α) 2−α |Etr(t)|≤ max |x¨(τ)| − (t − a) . τ∈[a,t] Γ(2 − α)(1 − α)K1 α See [12] for proofs and other details.

3 Problem Statement and the Approximation Method

The fractional optimal control problem that we consider here is the following one. Let α ∈ (0, 1) be the fixed fractional order, and consider two differentiable functions L and f with domain [a,b]×R2. Minimize the functional b J(x, u)= L(t, x(t),u(t)) dt Za

3 subject to the fractional dynamic constraint

C α Mx˙(t)+ N a Dt x(t)= f (t, x(t),u(t)) , t ∈ [a,b], and the initial condition x(a)= xa, where (M,N) 6= (0, 0) and xa is a fixed real number. Two situations are considered: x(b) fixed or free. Sufficient and necessary conditions to obtain solutions for this problem were studied in C α [14]. Here we proceed with a different approach. First, we replace the operator a Dt x(t) with the approximation given in (4). With relation (2) we get

Mx˙(t)+ N A(t − a)−αx(t)+ B(t − a)1−αx˙(t) " K −α xa(t − a) − C (t − a)1−p−αV (t) − = f (t, x(t),u(t)) , p p Γ(1 − a) p=2 # X where, for simplicity,

A = A(α, K) , B = A(α, K) and Cp = C(α, p).

Thus, one has

K −α −α 1−p−α Nxa(t − a) f(t, x(t),u(t)) − NA(t − a) x(t)+ NCp(t − a) Vp(t)+ Γ(1 − a) p=2 x˙(t)= X . M + NB(t − a)1−α Define the vector V (t) = (V2(t), V3(t),...,VK (t)) and the new function

K −α −α 1−p−α Nxa(t − a) f(t,x,u) − NA(t − a) x + NCp(t − a) Vp + Γ(1 − a) p=2 F (t, x, V,u)= X . M + NB(t − a)1−α We construct a new optimal control problem: minimize the functional

b J(x, V,u)= L(t, x(t),u(t)) dt (5) Za subject to the dynamic constraints

x˙(t)= F t, x(t), V (t),u(t) , (6)  ˙ p−2
Vp(t)=(1 − p)(t − a) x(t), p =2,...,K, and the initial conditions x(a)= x , a (7) (Vp(a)=0, p =2,...,K. To solve the problem (5)–(7), one can consider the Hamiltonian function

K p−2 H(t, x, V, λ, u)= L(t,x,u)+ λ1F (t, x, V,u)+ λp(1 − p)(t − a) x, p=2 X

4 where λ denotes the Lagrange multipliers,

λ(t) = (λ1(t), λ2(t),...,λK (t)).

By the Pontryagin maximum principle [11], to solve the problem one should solve the following system of ODEs: ∂H =0, ∂u  ∂H  =x, ˙ ∂λ1   ∂H  = V˙p, p =2,...,K, ∂λ  p ∂H = −λ˙ 1,  ∂x   ∂H ˙  = −λp, p =2,...,K, ∂Vp   subject to the boundary conditions

x(a)= xa, V (a)=0, p =2,...,K,  p  λp(b)=0, p =1,...,K, if x(b) is free, or  x(a)= xa, Vp(a)=0, p =2,...,K,  otherwise. Instead of this indirect approach, we apply here a direct method, based on an Euler discretization, to obtain a finite-dimensional approximation of the continuous problem (5)–(7). We summarize briefly the method. Consider a finite grid

a = t0

xi+1 = xi + ∆tF (ti, xi, V i,ui) and p−2 Vp,i+1 = Vp,i + ∆t(1 − p)(ti − a) xi, p =2,...,K, where

xi = x(ti), ui = u(ti), V i = (V2(ti), V3(ti),...,VK (ti)) and Vp,i = Vp(ti).

The integral b L (t, x(t),u(t)) dt Za in replaced by the Riemann sum n−1 ∆t L (ti, xi,ui) . i=0 X The finite version of problem (5)–(7) is the following one: minimize

n−1 ∆t L (ti, xi,ui) i=0 X

5 subject to the dynamic constraints

xi+1 = xi + ∆tF ti, xi, V i,ui ,


p−2 Vp,i+1 = Vp,i + ∆t(1 − p)(ti − a) xi, p =2,...,K, and the initial conditions x0 = xa, (Vp,0 =0, p =2,...,K (x(b) fixed or free). In the next section we illustrate the method with an example.

4 An Illustrative Example

In this section we exemplify the procedure of Section 3 with a concrete example. To start, we recall the Caputo fractional derivative of a power function (cf. Property 2.16 of [7]). Let α > 0 be arbitrary, β>n with n given by (1), and define x(t) = (t − a)β−1. Then, Γ(β) C Dαx(t)= (t − a)β−α−1. a t Γ(β − α) Assume that we wish to minimize the functional 1 J(x, u)= (u2(t) − 4x(t))2 dt (8) Z0 subject to the dynamic constraint 2 x˙(t)+ C D0.5x(t)= u(t)+ t1.5, t ∈ [0, 1], (9) 0 t Γ(2.5) and the boundary conditions x(0) = 0 and x(1) = 1. (10) The solution is the pair (x(t), u(t)) = (t2, 2t). (11) Indeed, (11) satisfies both constraints (9) and (10) with J(x, u) = 0 while functional (8) is non- negative: J(x, u) ≥ 0 for all admissible pairs (x, u). After replacing the fractional derivative by the appropriate approximation, and fixing K ≥ 2, we get the following problem: minimize the functional 1 J(x, V,u)= (u2(t) − 4x(t))2 dt (12) Z0 subject to the dynamic constraints

2 1.5 −0.5 K 0.5−p u(t)+ Γ(2.5) t − At x(t)+ p=2 Cpt Vp(t) x˙(t)= 0.5 ,  1+ Bt P (13) p−2 V˙p(t)=(1 − p)t x(t), p =2,...,K, and the boundary conditions

x(0) = 0, Vp(0) = 0, p =2,...,K, (14) x(1) = 1.

We apply a direct method to the previous problem, also called a “first discretize then optimize method”. More precisely, the objective functional (12) and the system of ODEs (14) were dis- cretized by a simple Euler method with fixed step size, as explained in Section 3. Then we solved

6 the resulting nonlinear programming problem by using the AMPL modeling language for math- ematical programming [6] in connection with IPOPT of W¨achter and Biegler [16]. In Figure 1a we show the results for the state function x and in Figure 1b we show the results for the control function u, where we have used K = 3 and n = 100.

1 2

1.8

0.8 1.6

1.4

0.6 1.2

x(t) u(t) 1

0.4 0.8

0.6 0.2 0.4

0.2

0 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 t t (a) State function x(t). (b) Control function u(t).

Figure 1: Exact solution to (8)–(10) (continuous line) versus numerical approximation to solution of (12)–(14) with K = 3 and n = 100 (dot line).

Acknowledgments

This article was supported by Portuguese funds through the Center for Research and Development in Mathematics and Applications (CIDMA), and The Portuguese Foundation for Science and Technology (FCT), within project PEst-OE/MAT/UI4106/2014. Torres was also supported by the FCT project PTDC/EEI-AUT/1450/2012, co-financed by FEDER under POFC-QREN with COMPETE reference FCOMP-01-0124-FEDER-028894.

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