Optimal control of affine connection control systems from the point of view of Lie algebroids

The purpose of this paper is to use the framework of Lie algebroids to study optimal control problems (OCPs) for affine connection control systems (ACCSs) on Lie groups. In this context, the equations for critical trajectories of the problem are geometrically characterized as a Hamiltonian vector field.

Download Full-text

Optimal Control Problems for Affine Connection Control Systems: Characterization of Extremals

10.1063/1.2958162 ◽

2008 ◽

Cited By ~ 1

Author(s):

M. Barbero-Liñán ◽

M. C. Muñoz-Lecanda ◽

Rui Loja Fernandes ◽

Roger Picken

Keyword(s):

Optimal Control ◽

Control Systems ◽

Optimal Control Problems ◽

Affine Connection ◽

Control Problems

Download Full-text

Time-optimality by distance-optimality for parabolic control systems

ESAIM Mathematical Modelling and Numerical Analysis ◽

10.1051/m2an/2019046 ◽

2020 ◽

Vol 54 (1) ◽

pp. 79-103

Author(s):

Lucas Bonifacius ◽

Karl Kunisch

Keyword(s):

Optimal Control ◽

Control Systems ◽

Optimal Control Problems ◽

Time Optimal Control ◽

Control Problems ◽

Numerical Examples ◽

Algorithmic Solution ◽

Time Optimal ◽

Time Optimality

The equivalence of time-optimal and distance-optimal control problems is shown for a class of parabolic control systems. Based on this equivalence, an approach for the efficient algorithmic solution of time-optimal control problems is investigated. Numerical examples are provided to illustrate that the approach works well in practice.

Download Full-text

OPTIMAL CONTROL REALIZATIONS OF LAGRANGIAN SYSTEMS WITH SYMMETRY

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887811005853 ◽

2011 ◽

Vol 08 (07) ◽

pp. 1627-1651 ◽

Cited By ~ 1

Author(s):

M. DELGADO-TÉLLEZ ◽

A. IBORT ◽

T. RODRÍGUEZ DE LA PEÑA

Keyword(s):

Optimal Control ◽

Control Systems ◽

Optimal Control Problems ◽

Lagrangian System ◽

Control Problems ◽

Lagrangian Systems ◽

Explicit Realization ◽

Control Equation ◽

And Control

A new relation among a class of optimal control systems and Lagrangian systems with symmetry is discussed. It will be shown that a family of solutions of optimal control systems whose control equation are obtained by means of a group action are in correspondence with the solutions of a mechanical Lagrangian system with symmetry. This result also explains the equivalence of the class of Lagrangian systems with symmetry and optimal control problems discussed in [1, 2]. The explicit realization of this correspondence is obtained by a judicious use of Clebsch variables and Lin constraints, a technique originally developed to provide simple realizations of Lagrangian systems with symmetry. It is noteworthy to point out that this correspondence exchanges the role of state and control variables for control systems with the configuration and Clebsch variables for the corresponding Lagrangian system. These results are illustrated with various simple applications.

Download Full-text

Maxwell Points of Dynamical Control Systems Based on Vertical Rolling Disc—Numerical Solutions

Robotics ◽

10.3390/robotics10030088 ◽

2021 ◽

Vol 10 (3) ◽

pp. 88

Author(s):

Marek Stodola ◽

Matej Rajchl ◽

Martin Brablc ◽

Stanislav Frolík ◽

Václav Křivánek

Keyword(s):

Optimal Control ◽

Control Systems ◽

Optimal Control Problems ◽

Numerical Solutions ◽

Control Problems ◽

Affine Control Systems ◽

Dynamical Control ◽

And Control ◽

Analytical Approaches ◽

Maxwell Points

We study two nilpotent affine control systems derived from the dynamic and control of a vertical rolling disc that is a simplification of a differential drive wheeled mobile robot. For both systems, their controllable Lie algebras are calculated and optimal control problems are formulated, and their Hamiltonian systems of ODEs are derived using the Pontryagin maximum principle. These optimal control problems completely determine the energetically optimal trajectories between two states. Then, a novel numerical algorithm based on optimisation for finding the Maxwell points is presented and tested on these control systems. The results show that the use of such numerical methods can be beneficial in cases where common analytical approaches fail or are impractical.

Download Full-text

On distributed parameter control systems in the abnormal case and in the case of nonoperator equality constraints

Journal of Applied Mathematics and Stochastic Analysis ◽

10.1155/s1048953393000139 ◽

1993 ◽

Vol 6 (2) ◽

pp. 137-151

Author(s):

Urszula Ledzewicz

Keyword(s):

Optimal Control ◽

Control Systems ◽

Optimal Control Problems ◽

Equality Constraints ◽

Extremum Principle ◽

Distributed Parameter ◽

Control Problems ◽

Parameter Control ◽

Distributed Parameter Control ◽

Lusternik Theorem

In this paper, a general distributed parameter control problem in Banach spaces with integral cost functional and with given initial and terminal data is considered. An extension of the Dubovitskii-Milyutin method to the case of nonregular operator equality constraints, based on Avakov's generalization of the Lusternik theorem, is presented. This result is applied to obtain an extension of the Extremum Principle for the case of abnormal optimal control problems. Then a version of this problem with nonoperator equality constraints is discussed and the Extremum Principle for this problem is presented.

Download Full-text

SOME APPLICATIONS OF QUASI-VELOCITIES IN OPTIMAL CONTROL

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887811005427 ◽

2011 ◽

Vol 08 (04) ◽

pp. 835-851 ◽

Cited By ~ 5

Author(s):

LÍGIA ABRUNHEIRO ◽

MARGARIDA CAMARINHA ◽

JOSÉ F. CARIÑENA ◽

JESÚS CLEMENTE-GALLARDO ◽

EDUARDO MARTÍNEZ ◽

...

Keyword(s):

Optimal Control ◽

Optimal Control Problems ◽

Nonholonomic Systems ◽

Lie Algebroids ◽

Control Problems ◽

Cost Functional

In this paper we study optimal control problems for nonholonomic systems defined on Lie algebroids by using quasi-velocities. We consider both kinematic, i.e. systems whose cost functional depends only on position and velocities, and dynamic optimal control problems, i.e. systems whose cost functional depends also on accelerations. The formulation of the problem directly at the level of Lie algebroids turns out to be the correct framework to explain in detail similar results appeared recently [20]. We also provide several examples to illustrate our construction.

Download Full-text