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Theory and Modern Applications

Table 3 The maximum absolute errors \(E_{x}\) and \(E_{u}\) at \(\mathcal{N} = 5\) and various values of α, ρ for \(x(t)\) and \(u(t)\), respectively

From: Numerical solutions of fractional optimal control with Caputo–Katugampola derivative

Example 1

ρ

\(E_{x}\)

\(E_{u}\)

\(\mathcal{J}\)

α = 0.5

0.2

1.3154e − 4

9.1543e − 3

3.6957e − 8

0.5

2.0416e − 4

1.3077e − 2

9.6857e − 8

0.8

1.9006e − 4

1.1307e − 2

8.9359e − 8

1

1.5831e − 4

8.9932e − 3

6.3605e − 8

1.2

1.2060e − 4

6.5490e − 3

3.7228e − 8

1.5

6.5823e − 5

3.3397e − 3

1.0760e − 8

α = 0.9

0.2

2.1582e − 4

1.2958e − 2

9.2709e − 8

0.5

2.0571e − 4

1.0850e − 2

1.0133e − 7

0.8

8.7454e − 5

4.1592e − 3

2.1681e − 8

1

2.4125e − 5

1.0802e − 3

1.8261e − 9

1.2

6.6853e − 5

1.1549e − 3

8.3696e − 10

1.5

1.0467e − 4

2.8705e − 3

7.8248e − 9