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2023 | OriginalPaper | Chapter

3. Dynamic Programming

Authors : Gjerrit Meinsma, Arjan van der Schaft

Published in: A Course on Optimal Control

Publisher: Springer Nature Switzerland

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Abstract

The minimum principle was developed in the Soviet Union in the late fifties of the previous century.

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previous chapter Minimum Principle

next chapter Linear Quadratic Control

C. Carathéodory. Variationsrechnung und partielle Differentialgleichungen erster Ordnung. B.G. Teubner, Leipzig, 1935.

“Königsweg der Variationsrechnung” in H. Boerner, Caratheodorys Eingang zur variationsrechnung, Jahresbericht der Deutschen Mathematiker Vereinigung, 56 (1953), 31—58; see H.J. Pesch, Caratheodory’s royal road of the Calculus of Variations: Missed exits to the Maximum Principle of Optimal Control Theory, AIMS.

Controlling a system with an input \({{} \texttt {u}}\) (optimal or not) that depends on \({{} \texttt {x}}\) is known as closed-loop control, and the resulting system is known as the closed-loop system. Controlling the system with a given time function \({{} \texttt {u}}(t)\) is called open-loop control.

Outside the scope of this book, but still: let [x] denote the dimension of a quantity x. For example, \([t]=\text {time}\). From \(\dot{{{} \texttt {x}}}={{} \texttt {u}}\), it follows that \([u]=[x][t]^{-1}\). Also, the expression \(x^2+\rho ^2u^2\) implies that \(\rho ^2u^2\) has the same dimension as \(x^2\). Hence, \([\rho ]=[t]\), and then \([V]=[J]=[x]^2[t]\). This suggests that \(V(x,t)=x^2P(t)\). In fact, application of the Buckingham \(\pi \)-theorem (not part of this course) shows that V(x, t) must have the form \(V(x,t)=x^2 \rho G((t-T)/\rho )\) for some dimensionless function \(G:\mathbb {R}\rightarrow \mathbb {R}\).

A stabilizing input is an input that steers the state to a given equilibrium. Better would have been to call it “asymptotically stabilizing” input or “attracting” input, but “stabilizing” is the standard in the literature.

Title: Dynamic Programming
Authors: Gjerrit Meinsma
Arjan van der Schaft
Publisher: Springer Nature Switzerland
Book: A Course on Optimal Control
Print ISBN: 978-3-031-36654-3

Electronic ISBN: 978-3-031-36655-0

Copyright Year: 2023
DOI: https://doi.org/10.1007/978-3-031-36655-0_3