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85 | 85 | "Mathematically, you want to solve the following optimal control problem:\n", |
86 | 86 | "\n", |
87 | 87 | "$$\\begin{align*}\n", |
88 | | - "\\min_{u} \\quad & t_f \\\\\n", |
| 88 | + "\\min_{u,t_f} \\quad & t_f \\\\\n", |
89 | 89 | "\\mathrm{s.t.} \\quad & \\frac{dx}{dt} = v \\\\\n", |
90 | 90 | "& \\frac{dv}{dt} = u - R v^2 \\\\\n", |
91 | 91 | "& x(t=0) = 0, ~~ x(t=t_f) = L \\\\\n", |
92 | 92 | "& v(t=0) = 0, ~~ v(t=t_f) = 0 \\\\\n", |
93 | 93 | "& -3 \\leq u \\leq 1\n", |
94 | 94 | "\\end{align*}$$\n", |
95 | 95 | "\n", |
96 | | - "where $a$ is the acceleration/braking (your control variable) and $R$ is the drag coefficient (parameter)." |
| 96 | + "where $u$ is the acceleration/braking (your control variable) and $R$ is the drag coefficient (parameter)." |
97 | 97 | ] |
98 | 98 | }, |
99 | 99 | { |
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107 | 107 | "Let $t = \\tau \\cdot t_f$ where $\\tau \\in [0,1]$. Thus $dt = t_f d\\tau$. The optimal control problem becomes:\n", |
108 | 108 | "\n", |
109 | 109 | "$$\\begin{align*}\n", |
110 | | - "\\min_{u} \\quad & t_f \\\\\n", |
| 110 | + "\\min_{u,t_f} \\quad & t_f \\\\\n", |
111 | 111 | "\\mathrm{s.t.} \\quad & \\frac{dx}{d\\tau} = t_f v \\\\\n", |
112 | 112 | "& \\frac{dv}{d\\tau} = t_f (u - R v^2) \\\\\n", |
113 | 113 | "& x(\\tau = 0) = 0, ~~ x(\\tau = 1) = L \\\\\n", |
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960 | 960 | "provenance": [] |
961 | 961 | }, |
962 | 962 | "kernelspec": { |
963 | | - "display_name": "Python 3", |
| 963 | + "display_name": "summer2024", |
964 | 964 | "language": "python", |
965 | 965 | "name": "python3" |
966 | 966 | }, |
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