Invariant optimal control on the three-dimensional semi-Euclidean group: control affine and quadratic Hamilton-Poisson systems
- Authors: Barrett, Dennis Ian
- Date: 2014
- Subjects: Automorphisms , Symmetry (Mathematics) , Lyapunov stability , Geometry, Riemannian , Geometry, Affine , Elliptic functions
- Language: English
- Type: text , Thesis , Masters , MSc
- Identifier: http://hdl.handle.net/10962/64805 , vital:28605
- Description: In this thesis we consider invariant control systems and Hamilton-Poisson systems on the three dimensional semi-Euclidean group SE(1,1). We first classify the left-invariant control affine systems (under detached feedback equivalence). We provide a complete list of normal forms, as well as classifying conditions. As a corollary to this classification, we derive controllability criteria for control affine systems on SE(1,1). Secondly, we consider quadratic Hamilton-Poisson systems on the (minus) Lie-Poisson space se(1,1)*. These systems are classified up to an affine isomorphism. Six normal forms are identified for the homogeneous case, whereas sixteen representatives (including several infinite families) are obtained for the inhomogeneous systems. Thereafter we consider the stability and integration of the normal forms obtained. For all homogeneous systems, and a subclass of inhomogeneous systems, we perform a complete stability analysis and derive explicit expressions for all integral curves. (The extremal controls of a large class of optimal control problems on SE(1,1) are linearly related to these integral curves.) Lastly, we discuss the Riemannian and sub-Riemannian problems. The (left-invariant) Riemannian and sub-Riemannian structures on SE(1,1) are classified, up to isometric group automorphisms and scaling. Explicit expressions for the geodesics of the normalised structures are found.
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- Authors: Barrett, Dennis Ian
- Date: 2014
- Subjects: Automorphisms , Symmetry (Mathematics) , Lyapunov stability , Geometry, Riemannian , Geometry, Affine , Elliptic functions
- Language: English
- Type: text , Thesis , Masters , MSc
- Identifier: http://hdl.handle.net/10962/64805 , vital:28605
- Description: In this thesis we consider invariant control systems and Hamilton-Poisson systems on the three dimensional semi-Euclidean group SE(1,1). We first classify the left-invariant control affine systems (under detached feedback equivalence). We provide a complete list of normal forms, as well as classifying conditions. As a corollary to this classification, we derive controllability criteria for control affine systems on SE(1,1). Secondly, we consider quadratic Hamilton-Poisson systems on the (minus) Lie-Poisson space se(1,1)*. These systems are classified up to an affine isomorphism. Six normal forms are identified for the homogeneous case, whereas sixteen representatives (including several infinite families) are obtained for the inhomogeneous systems. Thereafter we consider the stability and integration of the normal forms obtained. For all homogeneous systems, and a subclass of inhomogeneous systems, we perform a complete stability analysis and derive explicit expressions for all integral curves. (The extremal controls of a large class of optimal control problems on SE(1,1) are linearly related to these integral curves.) Lastly, we discuss the Riemannian and sub-Riemannian problems. The (left-invariant) Riemannian and sub-Riemannian structures on SE(1,1) are classified, up to isometric group automorphisms and scaling. Explicit expressions for the geodesics of the normalised structures are found.
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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:
- 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.
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