Second class constraints and the consistency of optimal control theory in phase space

Mauricio Contreras G., Juan Pablo Peña and Rodrigo Aros

Physica A: Statistical Mechanics and its Applications, 2021, vol. 562, issue C

Abstract: As has been shown in the literature (Rothe and Rothe, 2010; Rothe and Scholtz, 2003), the description of a mechanical system in terms of canonical transformations together with the Hamilton–Jacobi equation for the S function is ill-defined when the system has second class constraints. In this case, Carathéodory’s integrability conditions are violated and either the corresponding Hamilton–Jacobi equation cannot be solved or their solutions do not describe the system at all. This can be remedied by enlarging the phase space so that the constraints become first class in the extended space. Another way to approach this problem, is to apply the Rothe–Scholtz method discussed in Rothe and Rothe (2010) and Rothe and Scholtz (2003), so that the constraints themselves become variables of a new canonical transformation. This method works when the elements of the Dirac matrix are constant. On the other hand, it has been shown that optimal control theory can be written in phase space as a mechanical system with second class restrictions (Itami, 2001; Hojman, 0000; Contreras et al., 2017; Contreras and Peña 2018). This implies that the description of control theory can become inconsistent in terms of the Hamilton–Jacobi equation. In this article we will use the description of Rothe–Scholtz to analyse a subclass of LQ linear-quadratic problems whose Dirac matrix is constant and to check if the integrability conditions can be fulfilled so as to not get inconsistencies.

Keywords: Optimal control theory; Hamilton–Jacobi–Bellman equation; Dirac’s method; Second class constrained systems; Canonical transformations; Carathéodory integration conditions (search for similar items in EconPapers)
Date: 2021
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/S0378437120307032
Full text for ScienceDirect subscribers only. Journal offers the option of making the article available online on Science direct for a fee of $3,000

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eee:phsmap:v:562:y:2021:i:c:s0378437120307032

DOI: 10.1016/j.physa.2020.125335

Access Statistics for this article

Physica A: Statistical Mechanics and its Applications is currently edited by K. A. Dawson, J. O. Indekeu, H.E. Stanley and C. Tsallis

More articles in Physica A: Statistical Mechanics and its Applications from Elsevier
Bibliographic data for series maintained by Catherine Liu ().