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Theory and Modern Applications

Table 3 Results of simulation for \(n=5\)

From: Translation, solving scheme, and implementation of a periodic and optimal impulsive state control problem

Set

Non-optimal control

Optimal control

1

\(\tau_{1}=3.23\), \(\tau_{2}=1.98\), \(\tau_{3}=1.44 \)

\(\tau_{1}^{*}=2.99\), \(\tau_{2}^{*}=1.77\), \(\tau_{3}^{*}=1.26\)

\(\tau_{4}=1.13\), \(\tau_{5}=0.94 \)

\(\tau_{4}^{*}=0.98\), \(\tau _{5}^{*}=0.81\)

λ = 4.51, α = 0.8773, \(\varepsilon_{0}=0.1 \)

\(\lambda ^{*}=4.4\), \(\alpha^{*}=0.89\), \(\varepsilon^{*}=6.581e{-}5\)

\(J_{3}^{5}=18.2467\), \(x_{1}=0.49 \)

\(J_{3}^{5*}=14.1559\), \(x_{1}^{*}=0.6\)

2

\(\tau_{1}=2.35\), \(\tau_{2}=1.27\), \(\tau_{3}=0.88 \)

\(\tau_{1}^{*}=2.34\), \(\tau_{2}^{*}=1.26\), \(\tau_{3}^{*}=0.876\)

\(\tau_{4}=0.67\), \(\tau_{5}=0.54 \)

\(\tau_{4}^{*}=0.67\), \(\tau _{5}^{*}=0.54\)

λ = 4, α = 0.9028, \(\varepsilon_{0}=0.1 \)

\(\lambda ^{*}=3.99\), \(\alpha^{*}=0.9029\), \(\varepsilon^{*}=71e{-}5\)

\(J_{3}^{5}=13.5705\), \(x_{1}=1 \)

\(J_{3}^{5*}=10.6311\), \(x_{1}^{*}=4\)

3

\(\tau_{1}=4.77\), \(\tau_{2}=3.82\), \(\tau_{3}=3.2 \)

\(\tau_{1}^{*}=4.74\), \(\tau_{2}^{*}=3.80\), \(\tau_{3}^{*}=3.17\)

\(\tau_{4}=2.76\), \(\tau_{5}=2.44 \)

\(\tau_{4}^{*}=2.73\), \(\tau _{5}^{*}=2.40\)

λ = 4.8, α = 0.7724, \(\varepsilon_{0}=0.1 \)

\(\lambda ^{*}=4.79\), \(\alpha^{*}=0.7274\), \(\varepsilon^{*}=21e{-}4\)

\(J_{3}^{5}=24.2191\), \(x_{1}=0.2 \)

\(J_{3}^{5*}=21.30982\), \(x_{1}^{*}=0.21\)