Contributions to the study of a class of optimal control problems on the orthogonal groups SO(3) and SO(4)
- Authors: Adams, Ross Montague
- Date: 2015
- Language: English
- Type: text , Thesis , Doctoral , PhD
- Identifier: http://hdl.handle.net/10962/64826 , vital:28608
- Description: In this thesis we investigate a class of invariant optimal control problems, and their associated quadratic Hamilton-Poisson systems, on the orthogonal groups SO(3) and SO(4). Specifically, we are concerned with the class of left-invariant control affine systems. We begin by classifying all cost-extended systems on SO(3) under cost equivalence. (Cost-extended systems are closely related to optimal control problems.) A classification of all quadratic Hamilton-Poisson systems on the (minus) Lie-Poisson space so(3)*, under affine equivalence, is also obtained. For the normal forms obtained in our classification (of Hamilton-Poisson systems) we investigate the (Lyapunov) stability nature of the equilibria using spectral and energy-Casimir methods. For a subclass of these systems, we obtain analytic expressions for the integral curves of the associated Hamiltonian vector fields in terms of (basic) Jacobi elliptic functions. The explicit relationship between the classification of cost-extended systems on SO(3) and the classification of quadratic Hamilton- Poisson systems on so(3)* is provided. On SO(4), a classification of all left-invariant control affine systems under L-equivalence is obtained. We then determine which of these representatives are controllable, thus obtaining a classification under detached feedback equivalence. We also obtain a partial classification of quadratic Hamilton-Poisson systems on the Lie-Poisson space so(4)*. An investigation of the stability nature of the equilibria for a subclass of these systems is also done. Several illustrative examples of optimal control problems on the orthogonal group SO(3) are provided. More specifically, we consider an optimal control problem corresponding to a representative of our classification (of cost-extended system) for each possible number of control inputs. For each of these problems, we obtain explicit expressions for the extremal trajectories on the homogeneous space S2 by projecting the extremal trajectories on the group SO(3). The examples provided show how our classifications of cost-extended systems and Hamilton-Poisson systems can be used to obtain the optimal controls and the extremal trajectories corresponding to a large class of optimal control problems on SO(3). An example of a four-input optimal control problem on SO(4) is also provided. This example is provided to show how the solutions of certain problems on SO(4) can be related to the solutions of certain optimal control problems on SO(3).
- Full Text:
- Date Issued: 2015
Invariant control systems and sub-Riemannian structures on lie groups: equivalence and isometries
- Authors: Biggs, Rory
- Date: 2015
- Language: English
- Type: text , Thesis , Doctoral , PhD
- Identifier: http://hdl.handle.net/10962/64815 , vital:28607
- Description: In this thesis we consider invariant optimal control problems and invariant sub-Riemannian structures on Lie groups. Primarily, we are concerned with the equivalence and classification of problems (resp. structures). In the first chapter, both the class of invariant optimal control problems and the class of invariant sub-Riemannian structures are organised as categories. The latter category is shown to be functorially equivalent to a subcategory of the former category. Via the Pontryagin Maximum Principle, we associate to each invariant optimal control problem (resp. invariant sub-Riemannian structure) a quadratic Hamilton-Poisson system on the associated Lie-Poisson space. These Hamiltonian systems are also organised as a category and hence the Pontryagin lift is realised as a contravariant functor. Basic properties of these categories are investigated. The rest of this thesis is concerned with the classification (and investigation) of certain subclasses of structures and systems. In the second chapter, the homogeneous positive semidefinite quadratic Hamilton-Poisson systems on three-dimensional Lie-Poisson spaces are classified up to compatibility with a linear isomorphism; a list of 23 normal forms is exhibited. In the third chapter, we classify the invariant sub-Riemannian structures in three dimensions and calculate the isometry group for each normal form. By comparing our results with known results, we show that most isometries (in three dimensions) are in fact the composition of a left translation and a Lie group isomorphism. In the fourth and last chapter of this thesis, we classify the sub-Riemannian and Riemannian structures on the (2n + 1)-dimensional Heisenberg groups. Furthermore, we find the isometry group and geodesics of each normal form.
- Full Text:
- Date Issued: 2015
The symmetry group of a model of hyperbolic plane geometry and some associated invariant optimal control problems
- Authors: Henninger, Helen Clare
- Date: 2012
- Subjects: Geometry , Symmetry groups , Symmetry (Mathematics)
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5432 , http://hdl.handle.net/10962/d1018232
- Description: In this thesis we study left-invariant control offine systems on the symmetry group of a. model of hyperbolic plane geometry, the matrix Lie group SO(1, 2)₀. We determine that there are 10 distinct classes of such control systems and for typical elements of two of these classes we provide solutions of the left-invariant optimal wntrol problem with quauratic costs. Under the identification of the Lie allgebra .so(l, 2) with Minkowski spacetime R¹̕'², we construct a controllabilility criterion for all left-invariant control affine systems on 50(1. 2)₀ which in the inhomogeneous case depends only on the presence or absence of an element in the image of the system's trace in R¹̕ ²which is identifiable using the inner product. For the solutions of both the optimal control problems, we provide explicit expressions in terms of Jacobi elliptic functions for the solutions of the reduced extremal equations and determine the nonlinear stability of the equilibrium points.
- Full Text:
- Date Issued: 2012
A study of a class of invariant optimal control problems on the Euclidean group SE(2)
- Authors: Adams, Ross Montague
- Date: 2011
- Subjects: Matrix groups Lie groups Extremal problems (Mathematics) Maximum principles (Mathematics) Hamilton-Jacobi equations Lyapunov stability
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5420 , http://hdl.handle.net/10962/d1006060
- Description: The aim of this thesis is to study a class of left-invariant optimal control problems on the matrix Lie group SE(2). We classify, under detached feedback equivalence, all controllable (left-invariant) control affine systems on SE(2). This result produces six types of control affine systems on SE(2). Hence, we study six associated left-invariant optimal control problems on SE(2). A left-invariant optimal control problem consists of minimizing a cost functional over the trajectory-control pairs of a left-invariant control system subject to appropriate boundary conditions. Each control problem is lifted from SE(2) to T*SE(2) ≅ SE(2) x se (2)*and then reduced to a problem on se (2)*. The maximum principle is used to obtain the optimal control and Hamiltonian corresponding to the normal extremals. Then we derive the (reduced) extremal equations on se (2)*. These equations are explicitly integrated by trigonometric and Jacobi elliptic functions. Finally, we fully classify, under Lyapunov stability, the equilibrium states of the normal extremal equations for each of the six types under consideration.
- Full Text:
- Date Issued: 2011