TY - JOUR

T1 - Second class constraints and the consistency of optimal control theory in phase space

AU - Contreras G., Mauricio

AU - Peña, Juan Pablo

AU - Aros, Rodrigo

PY - 2021/1/15

Y1 - 2021/1/15

N2 - 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.

AB - 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.

KW - Canonical transformations

KW - Carathéodory integration conditions

KW - Dirac's method

KW - Hamilton–Jacobi–Bellman equation

KW - Optimal control theory

KW - Second class constrained systems

UR - http://www.scopus.com/inward/record.url?scp=85092096924&partnerID=8YFLogxK

U2 - 10.1016/j.physa.2020.125335

DO - 10.1016/j.physa.2020.125335

M3 - Article

AN - SCOPUS:85092096924

VL - 562

JO - Physica A: Statistical Mechanics and its Applications

JF - Physica A: Statistical Mechanics and its Applications

SN - 0378-4371

M1 - 125335

ER -