# American Institute of Mathematical Sciences

March  2020, 10(1): 171-187. doi: 10.3934/mcrf.2019035

## A direct method based on the Clenshaw-Curtis formula for fractional optimal control problems

 Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Iran

Received  November 2017 Revised  February 2019 Published  March 2020 Early access  August 2019

In this paper, we present a new method based on the Clenshaw-Curtis formula to solve a class of fractional optimal control problems. First, we convert the fractional optimal control problem to an equivalent problem in the fractional calculus of variations. Then, by utilizing the Clenshaw-Curtis formula and the Chebyshev-Gauss-Lobatto points, we transform the problem to a discrete form. By this approach, we get a nonlinear programming problem by solving of which we can approximate the optimal solution of the main problem. We analyze the convergence of the obtained approximate solution and solve some numerical examples to show the efficiency of the method.

Citation: Mohammad Hadi Noori Skandari, Marzieh Habibli, Alireza Nazemi. A direct method based on the Clenshaw-Curtis formula for fractional optimal control problems. Mathematical Control & Related Fields, 2020, 10 (1) : 171-187. doi: 10.3934/mcrf.2019035
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##### References:
The exact and approximate optimal state for $N = 4$ in Example 6.1
The exact and approximate optimal control for $N = 4$ in Example 6.1
The absolute error of the approximate optimal state in Example 6.1
The absolute error of the approximate optimal control in Example 6.1
The exact and approximate optimal state for $N = 6$ in Example 6.2
The exact and approximate optimal control for $N = 6$ in Example 6.2
The absolute error of approximate optimal state in Example 6.2
The absolute error of the approximate control in Example 6.2
The exact and approximate optimal state for $N = 6$ in Example 6.3
The exact and approximate optimal control for $N = 6$ in Example 6.3
The absolute error of the approximate optimal state in Example 6.3
The absolute error of the approximate optimal control in Example 6.3
The maximum absolute error for $N = 4$ in Example 6.1
 $\alpha = 0.5$ $\alpha = 0.6$ $\alpha = 0.7$ $\underset{t}{\mathrm{Max}}|E_x(t)|$ $6.077585\times 10^{-5}$ $2.236463 \times 10^{-5}$ $1.662597\times 10^{-3}$ $\underset{t}{\mathrm{Max}}|E_u(t)|$ $2.031847\times 10^{-5}$ $6.593832 \times 10^{-5}$ $4.616561\times 10^{-4}$
 $\alpha = 0.5$ $\alpha = 0.6$ $\alpha = 0.7$ $\underset{t}{\mathrm{Max}}|E_x(t)|$ $6.077585\times 10^{-5}$ $2.236463 \times 10^{-5}$ $1.662597\times 10^{-3}$ $\underset{t}{\mathrm{Max}}|E_u(t)|$ $2.031847\times 10^{-5}$ $6.593832 \times 10^{-5}$ $4.616561\times 10^{-4}$
The optimal value of the objective functional for $N = 4$ in Example 6.name-style="western"
 $\alpha = 0.5$ $\alpha = 0.6$ $\alpha = 0.7$ $J^*$ $7.986571 \times 10^{-10}$ $1.200881\times 10^{-8}$ $6.080480\times 10^{-7}$
 $\alpha = 0.5$ $\alpha = 0.6$ $\alpha = 0.7$ $J^*$ $7.986571 \times 10^{-10}$ $1.200881\times 10^{-8}$ $6.080480\times 10^{-7}$
The maximum absolute error for $N = 6$ in Example 6.2
 $\alpha = 0.5$ $\alpha = 0.6$ $\alpha = 0.7$ $\underset{t}{\mathrm{Max}}|E_x(t)|$ $1.128957\times 10^{-3}$ $2.967422 \times 10^{-4}$ $1.393032\times 10^{-3}$ $\underset{t}{\mathrm{Max}}|E_u(t)|$ $1.508986\times 10^{-2}$ $5.269326 \times 10^{-3}$ $2.011737\times 10^{-3}$
 $\alpha = 0.5$ $\alpha = 0.6$ $\alpha = 0.7$ $\underset{t}{\mathrm{Max}}|E_x(t)|$ $1.128957\times 10^{-3}$ $2.967422 \times 10^{-4}$ $1.393032\times 10^{-3}$ $\underset{t}{\mathrm{Max}}|E_u(t)|$ $1.508986\times 10^{-2}$ $5.269326 \times 10^{-3}$ $2.011737\times 10^{-3}$
The optimal value of the objective function for $N = 6$ in Example 6.2
 $\alpha = 0.5$ $\alpha = 0.6$ $\alpha = 0.7$ $J^*$ $4.699837 \times 10^{-6}$ $5.516687\times 10^{-7}$ $8.165173\times 10^{-8}$
 $\alpha = 0.5$ $\alpha = 0.6$ $\alpha = 0.7$ $J^*$ $4.699837 \times 10^{-6}$ $5.516687\times 10^{-7}$ $8.165173\times 10^{-8}$
The maximum absolute error for $N = 6$ in Example 6.3
 $\alpha = 0.5$ $\alpha = 0.6$ $\alpha = 0.7$ $\underset{t}{\mathrm{Max}}|E_x(t)|$ $2.475186\times 10^{-3}$ $3.593088 \times 10^{-4}$ $1.005558\times 10^{-3}$ $\underset{t}{\mathrm{Max}}|E_u(t)|$ $5.873404\times 10^{-2}$ $5.118654 \times 10^{-3}$ $5.399343\times 10^{-3}$
 $\alpha = 0.5$ $\alpha = 0.6$ $\alpha = 0.7$ $\underset{t}{\mathrm{Max}}|E_x(t)|$ $2.475186\times 10^{-3}$ $3.593088 \times 10^{-4}$ $1.005558\times 10^{-3}$ $\underset{t}{\mathrm{Max}}|E_u(t)|$ $5.873404\times 10^{-2}$ $5.118654 \times 10^{-3}$ $5.399343\times 10^{-3}$
The optimal value of the objective functional for $N = 6$ in Example 6.3
 $\alpha = 0.5$ $\alpha = 0.6$ $\alpha = 0.7$ $J^*$ $1.080409 \times 10^{-7}$ $2.074337\times 10^{-9}$ $4.013053\times 10^{-9}$
 $\alpha = 0.5$ $\alpha = 0.6$ $\alpha = 0.7$ $J^*$ $1.080409 \times 10^{-7}$ $2.074337\times 10^{-9}$ $4.013053\times 10^{-9}$
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