Optimal Control Summary

Author

Affiliation

Enrico Bertolazzi

University of Trento, Department of Industrial Engineering

SUMMARY

The OCP problem

Minimize

\large \underbrace{ \underbrace{ \colorbox{LightYellow}{$\Phi({\color{blue}\bm{x}}(a),{\color{blue}\bm{x}}(b),{\color{DeepPink}\bm{p}})$} }_{\textbf{\color{blue}Mayer}} + \int_a^b \underbrace{ \colorbox{Azure}{$L({\color{blue}\bm{x}},{\color{DarkGreen}\bm{u}},{\color{DeepPink}\bm{p}},t)$} }_{\textbf{\color{blue}Lagrange}} \mathrm{d}t }_{\textbf{\color{blue}Bolza}}

Differential constraints

\large {\color{blue}\bm{x}}^\prime = \bm{f}({\color{blue}\bm{x}},{\color{DarkGreen}\bm{u}},{\color{DeepPink}\bm{p}},t)

Boundary conditions

\large \bm{b}({\color{blue}\bm{x}}(a),{\color{blue}\bm{x}}(b),{\color{DeepPink}\bm{p}}) = \bm{0}

Integral constraints

\large \bm{\Psi}\big({\color{blue}\bm{x}}(a),{\color{blue}\bm{x}}(b),{\color{DeepPink}\bm{p}}\big)+ \int_a^b \bm{g}({\color{blue}\bm{x}},{\color{DarkGreen}\bm{u}},{\color{DeepPink}\bm{p}},t)\,\mathrm{d}t = 0

Controls bounds

\large {\color{DarkGreen}\bm{u}}(t)\in\mathcal{U}

Tranformation

Minimize

\Phi({\color{blue}\bm{x}}(a),{\color{blue}\bm{x}}(b),{\color{DeepPink}\bm{p}}) + \int_a^b L({\color{blue}\bm{x}},{\color{DarkGreen}\bm{u}},{\color{DeepPink}\bm{p}},t)\,\mathrm{d}t

Differential constraints

\begin{aligned} {\color{blue}\bm{x}}^\prime &= \bm{f}({\color{blue}\bm{x}},{\color{DarkGreen}\bm{u}},{\color{DeepPink}\bm{p}},t) \\[1em] \bm{z}^\prime &= \bm{g}({\color{blue}\bm{x}},{\color{DarkGreen}\bm{u}},{\color{DeepPink}\bm{p}},t) \end{aligned}

Boundary conditions

\begin{aligned} \bm{b}({\color{blue}\bm{x}}(a),{\color{blue}\bm{x}}(b),{\color{DeepPink}\bm{p}}) &= \bm{0} \\[1em] \bm{z}(a) &= \bm{0} \\[1em] \bm{\Psi}\big({\color{blue}\bm{x}}(a),{\color{blue}\bm{x}}(b),{\color{DeepPink}\bm{p}}\big)+\bm{z}(b) &= \bm{0} \end{aligned}

Controls bounds

\large {\color{DarkGreen}\bm{u}}(t)\in\mathcal{U}

Functional

B\big({\color{blue}\bm{x}}(a),{\color{blue}\bm{x}}(b),{\color{DeepPink}\bm{p}},{\color{DodgerBlue}\bm{\omega}},{\color{SaddleBrown}\bm{\eta}}\big) +{\color{SaddleBrown}\bm{\eta}}\cdot\bm{z}(b)+ \int_a^b\Big( H\big({\color{blue}\bm{x}},{\color{OrangeRed}\bm{\lambda}},{\color{LimeGreen}\bm{\mu}},{\color{DarkGreen}\bm{u}},{\color{DeepPink}\bm{p}},t\big) -{\color{OrangeRed}\bm{\lambda}}\cdot{\color{blue}\bm{x}}^\prime -{\color{LimeGreen}\bm{\mu}}\cdot\bm{z}^\prime \Big)\mathrm{d}t

where \begin{aligned} H\big({\color{blue}\bm{x}},{\color{OrangeRed}\bm{\lambda}},{\color{LimeGreen}\bm{\mu}},{\color{DarkGreen}\bm{u}},t\big) &= L({\color{blue}\bm{x}},{\color{DarkGreen}\bm{u}},t) +{\color{OrangeRed}\bm{\lambda}}\cdot\bm{f}({\color{blue}\bm{x}},{\color{DarkGreen}\bm{u}},t) +{\color{LimeGreen}\bm{\mu}}\cdot\bm{g}({\color{blue}\bm{x}},{\color{DarkGreen}\bm{u}},t) \\[1em] B\big({\color{blue}\bm{x}}_a,{\color{blue}\bm{x}}_b,{\color{DeepPink}\bm{p}},{\color{DodgerBlue}\bm{\omega}},{\color{SaddleBrown}\bm{\eta}}\big) &=\Phi\big({\color{blue}\bm{x}}_a,{\color{blue}\bm{x}}_b,{\color{DeepPink}\bm{p}}\big) +{\color{DodgerBlue}\bm{\omega}}\cdot\bm{b}\big({\color{blue}\bm{x}}_a,{\color{blue}\bm{x}}_b,{\color{DeepPink}\bm{p}}\big) +{\color{SaddleBrown}\bm{\eta}}\cdot \bm{\Psi}\big({\color{blue}\bm{x}}(a),{\color{blue}\bm{x}}(b),{\color{DeepPink}\bm{p}}\big) \end{aligned}

variation and integration by part of -{\color{OrangeRed}\bm{\lambda}}\cdot\delta{\color{blue}\bm{x}}^\prime and -{\color{LimeGreen}\bm{\mu}}\cdot\delta\bm{z}^\prime

\begin{aligned} \int_a^b -\delta{\color{OrangeRed}\bm{\lambda}}\cdot{\color{blue}\bm{x}}^\prime -{\color{OrangeRed}\bm{\lambda}}\cdot\delta{\color{blue}\bm{x}}^\prime\,\mathrm{d}t & = \int_a^b -\delta{\color{OrangeRed}\bm{\lambda}}\cdot{\color{blue}\bm{x}}^\prime -\left[ \dfrac{\mathrm{d}}{\mathrm{d}t} \left( {\color{OrangeRed}\bm{\lambda}}\cdot\delta{\color{blue}\bm{x}} \right) -{\color{OrangeRed}\bm{\lambda}}^\prime\cdot\delta{\color{blue}\bm{x}} \right]\,\mathrm{d}t \\ & = \int_a^b \Big( -\delta{\color{OrangeRed}\bm{\lambda}}\cdot{\color{blue}\bm{x}}^\prime +{\color{OrangeRed}\bm{\lambda}}^\prime\cdot\delta{\color{blue}\bm{x}} \Big)\mathrm{d}t +{\color{OrangeRed}\bm{\lambda}}(a)\delta{\color{blue}\bm{x}}(a) -{\color{OrangeRed}\bm{\lambda}}(b)\delta{\color{blue}\bm{x}}(b) \\[1em] \int_a^b -\delta{\color{LimeGreen}\bm{\mu}}\cdot\bm{z}^\prime -{\color{LimeGreen}\bm{\mu}}\cdot\delta\bm{z}^\prime\,\mathrm{d}t &= \int_a^b -\delta{\color{LimeGreen}\bm{\mu}}\cdot\bm{z}^\prime -\left[ \dfrac{\mathrm{d}}{\mathrm{d}t} \left( {\color{LimeGreen}\bm{\mu}}\cdot\delta\bm{z} \right) - {\color{LimeGreen}\bm{\mu}}^\prime\cdot\delta\bm{z} \right]\mathrm{d}t \\ & = \int_a^b \Big(-\delta{\color{LimeGreen}\bm{\mu}}\cdot\bm{z}^\prime +{\color{LimeGreen}\bm{\mu}}^\prime\cdot\delta\bm{z} \Big)\mathrm{d}t +\underbrace{\cancel{{\color{LimeGreen}\bm{\mu}}(a)\delta\bm{z}(a)}}_{\delta\bm{z}(a)=\bm{0}} -{\color{LimeGreen}\bm{\mu}}(b)\delta\bm{z}(b) \end{aligned}

after variations:

Variation of \delta{\color{blue}\bm{x}}(t), \delta{\color{blue}\bm{x}}(a) e \delta{\color{blue}\bm{x}}(b)

\begin{aligned} {\color{OrangeRed}\bm{\lambda}}^\prime + \dfrac{\partial H}{\partial{\color{blue}\bm{x}}}({\color{blue}\bm{x}},{\color{OrangeRed}\bm{\lambda}},{\color{LimeGreen}\bm{\mu}},{\color{DarkGreen}\bm{u}},t) &= \bm{0} \\[1em] \dfrac{\partial B}{\partial{\color{blue}\bm{x}}_a}\big({\color{blue}\bm{x}}(a),{\color{blue}\bm{x}}(b),{\color{DeepPink}\bm{p}},{\color{DodgerBlue}\bm{\omega}},{\color{SaddleBrown}\bm{\eta}}\big) +{\color{OrangeRed}\bm{\lambda}}(a) &=\bm{0} \\[1em] \dfrac{\partial B}{\partial{\color{blue}\bm{x}}_b}\big({\color{blue}\bm{x}}(a),{\color{blue}\bm{x}}(b),{\color{DeepPink}\bm{p}},{\color{DodgerBlue}\bm{\omega}},{\color{SaddleBrown}\bm{\eta}}\big) -{\color{OrangeRed}\bm{\lambda}}(b) &=\bm{0} \end{aligned}

Variation of \delta\bm{z}(t) e \delta\bm{z}(b)

\left. \begin{aligned} {\color{LimeGreen}\bm{\mu}}^\prime &= \bm{0} \\[1em] {\color{SaddleBrown}\bm{\eta}}-{\color{LimeGreen}\bm{\mu}}(b) &=\bm{0} \end{aligned} \right\} \quad\implies\quad {\color{LimeGreen}\bm{\mu}}(t)={\color{SaddleBrown}\bm{\eta}}

Variation of \delta{\color{DeepPink}\bm{p}}

\dfrac{\partial B}{\partial{\color{DeepPink}\bm{p}}}\big({\color{blue}\bm{x}}(a),{\color{blue}\bm{x}}(b),{\color{DeepPink}\bm{p}},{\color{DodgerBlue}\bm{\omega}},{\color{SaddleBrown}\bm{\eta}}\big) + \int_a^b \dfrac{\partial H}{\partial{\color{DeepPink}\bm{p}}}\big({\color{blue}\bm{x}},{\color{OrangeRed}\bm{\lambda}},{\color{LimeGreen}\bm{\mu}},{\color{DarkGreen}\bm{u}},t\big) \,\mathrm{d}t = \bm{0}

Variation of \delta{\color{OrangeRed}\bm{\lambda}}(t) e \delta{\color{LimeGreen}\bm{\mu}}(t)

\begin{aligned} {\color{blue}\bm{x}}^\prime &= \dfrac{\partial H}{\partial{\color{OrangeRed}\bm{\lambda}}}({\color{blue}\bm{x}},{\color{OrangeRed}\bm{\lambda}},{\color{LimeGreen}\bm{\mu}},{\color{DarkGreen}\bm{u}},t) = \bm{f}({\color{blue}\bm{x}},{\color{DarkGreen}\bm{u}},{\color{DeepPink}\bm{p}},t) \\[1em] \bm{z}^\prime &= \dfrac{\partial H}{\partial{\color{LimeGreen}\bm{\mu}}}({\color{blue}\bm{x}},{\color{OrangeRed}\bm{\lambda}},{\color{LimeGreen}\bm{\mu}},{\color{DarkGreen}\bm{u}},t) = \bm{g}({\color{blue}\bm{x}},{\color{DarkGreen}\bm{u}},{\color{DeepPink}\bm{p}},t) \end{aligned}

Variation of \delta{\color{SaddleBrown}\bm{\eta}} e {\color{DodgerBlue}\bm{\omega}}

\begin{aligned} \bm{\Psi}\big({\color{blue}\bm{x}}(a),{\color{blue}\bm{x}}(b),{\color{DeepPink}\bm{p}}\big)+\bm{z}(b) &= \bm{0}\\[1em] \bm{b}\big({\color{blue}\bm{x}}(a),{\color{blue}\bm{x}}(b),{\color{DeepPink}\bm{p}}\big) &= \bm{0} \end{aligned}

eliminate z(t)

\bm{z}(b)=\underbrace{\bm{z}(a)}_{=\bm{0}}+\int_a^b \bm{g}({\color{blue}\bm{x}},{\color{DarkGreen}\bm{u}},{\color{DeepPink}\bm{p}},t)\, \mathrm{d}t

thus, BC become \bm{\Psi}\big({\color{blue}\bm{x}}(a),{\color{blue}\bm{x}}(b),{\color{DeepPink}\bm{p}}\big)+ \int_a^b \bm{g}({\color{blue}\bm{x}},{\color{DarkGreen}\bm{u}},{\color{DeepPink}\bm{p}},t)\,\mathrm{d}t = \bm{0}

The BVP

BVP after variation

The original ODE

\large {\color{blue}\bm{x}}^\prime = \bm{f}({\color{blue}\bm{x}},{\color{DarkGreen}\bm{u}},{\color{DeepPink}\bm{p}},t)

Adjoint equations

\large {\color{OrangeRed}\bm{\lambda}}^\prime = -\dfrac{\partial H}{\partial{\color{blue}\bm{x}}}({\color{blue}\bm{x}},{\color{OrangeRed}\bm{\lambda}},{\color{LimeGreen}\bm{\mu}},{\color{DarkGreen}\bm{u}},t)

Original Boundary Conditions

\large \bm{b}\big({\color{blue}\bm{x}}(a),{\color{blue}\bm{x}}(b),{\color{DeepPink}\bm{p}}\big) = \bm{0}

Adjoint Boundary Conditions

\large \begin{aligned} \dfrac{\partial B}{\partial{\color{blue}\bm{x}}_a}\big({\color{blue}\bm{x}}(a),{\color{blue}\bm{x}}(b),{\color{DeepPink}\bm{p}},{\color{DodgerBlue}\bm{\omega}},{\color{SaddleBrown}\bm{\eta}}\big) +{\color{OrangeRed}\bm{\lambda}}(a) & =\bm{0} \\[1em] \dfrac{\partial B}{\partial{\color{blue}\bm{x}}_b}\big({\color{blue}\bm{x}}(a),{\color{blue}\bm{x}}(b),{\color{DeepPink}\bm{p}},{\color{DodgerBlue}\bm{\omega}},{\color{SaddleBrown}\bm{\eta}}\big) -{\color{OrangeRed}\bm{\lambda}}(b) & =\bm{0} \end{aligned}

Integral Constraints

\large \bm{\Psi}\big({\color{blue}\bm{x}}(a),{\color{blue}\bm{x}}(b),{\color{DeepPink}\bm{p}}\big)+ \int_a^b \bm{g}({\color{blue}\bm{x}},\bm{u},{\color{DeepPink}\bm{p}},t)\,\mathrm{d}t = \bm{0}

Optimization Parameters

\large \dfrac{\partial B}{\partial{\color{DeepPink}\bm{p}}}\big({\color{blue}\bm{x}}(a),{\color{blue}\bm{x}}(b),{\color{DeepPink}\bm{p}},{\color{DodgerBlue}\bm{\omega}},{\color{SaddleBrown}\bm{\eta}}\big) + \int_a^b \dfrac{\partial H}{\partial{\color{DeepPink}\bm{p}}}\big({\color{blue}\bm{x}},{\color{OrangeRed}\bm{\lambda}},{\color{SaddleBrown}\bm{\eta}},{\color{DarkGreen}\bm{u}},t\big) \,\mathrm{d}t = \bm{0}

Pontryagin maximum principle (minimum principle)

\large {\color{DarkGreen}\bm{u}}(t) = \mathop{\textrm{arg min}}\limits_{\bm{v}\in\mathcal{U}} H\big({\color{blue}\bm{x}}(t), {\color{OrangeRed}\bm{\lambda}}(t), {\color{SaddleBrown}\bm{\eta}},\bm{v},t\big)

References

Betts, John T. 2010. Practical Methods for Optimal Control Using Nonlinear Programming. 3rd ed. Society for Industrial; Applied Mathematics.

Bryson, Arthur E., and Yu-Chi Ho. 1975. Applied Optimal Control: Optimization, Estimation, and Control. Wiley.

Kirk, Donald E. 2004. Optimal Control Theory: An Introduction. Dover Publications.

Liberzon, Daniel. 2012. Calculus of Variations and Optimal Control Theory: A Concise Introduction. Princeton University Press.