Hessian Operators on Constraint Manifolds

SSRN Electronic Journal

Using the embedded gradient vector field method (see P. Birtea, D. Comȃnescu, Hessian operators on constraint manifolds, J Nonlinear Sci 25, 2015), we present a general formula for the Laplace-Beltrami operator defined on a constraint manifold, written in the ambient coordinates. Regarding the orthogonal group as a constraint submanifold of the Euclidean space of n × n matrices, we give an explicit formula for the Laplace-Beltrami operator on the orthogonal group using the ambient Euclidean coordinates. We apply this new formula for some relevant functions.

Advances in Mathematics, 2012

Siam Journal on Control and Optimization, 1984

In this paper we present a differential geometric approach to the Lagrange problem and the fixed time optimal control problem for nonlinear time-invariant control systems. We restrict attention to first order conditions for optimality and present a generalized Lagrange multiplier rule for restricted variational problems. Our treatment of the optimal control problem uses a recently proposed fibre bundle approach for the definition of nonlinear systems.

Bulletin des Sciences Mathématiques, 2020

We give an explicit construction of the Newton algorithm on orthogonal Stiefel manifolds. In order to do this we introduce a local frame appropriate for the computation of the Hessian matrix for a cost function defined on Stiefel manifolds. For a Brockett cost function defined on St 4 2 we give a classification of the critical points and we present numerical simulations of the Newton algorithm.

We study optimality properties of the smooth function tr ¡ £¡1Q£N ¡ 2M£¡1 ¢ , viewed as a function of £, with £ belonging to certain quadratic matrix Lie groups which are gen- eralizations of the orthogonal group. Some optimization matrix problems are formulated in terms of this function. Computational issues based on continuous algorithms are discussed.

IEEE Conference on Decision and Control and European Control Conference, 2011

The Lie group projection operator approach is an iterative scheme for solving continuous-time optimal control problems on Lie groups. This work details the approach for optimal control problems on T SO(3), the tangent bundle of the special orthogonal group SO(3). The dynamics of a rigid satellite is used as illustrative example. Numerical simulations are presented and discussed.

IEEE Transactions on Automatic Control, 2013

Many nonlinear systems of practical interest evolve on Lie groups or on manifolds acted upon by Lie groups. Examples range from aircraft and underwater vehicles to quantum mechanical systems. In this paper, we develop an algorithm for solving continuous time optimal control problems for systems evolving on (noncompact) Lie groups. This algorithm generalizes the projection operator approach for trajectory optimization originally developed for systems on vector spaces. Notions for generalizing system theoretic tools such as Riccati equations and linear and quadratic system approximations are developed. In this development, the covariant derivative of a map between two manifolds plays a key role in providing a chain rule for the required Lie group computations. An example optimal control problem on SO(3) is provided to highlight implementation details and to demonstrate the effectiveness of the method.

TURKISH JOURNAL OF MATHEMATICS, 2014

8th IFAC Symposium on Nonlinear Control Systems, 2010

In this paper, we investigate a generalization of the infinite time horizon linear quadratic regulator (LQR) for systems evolving on the special orthogonal group SO(3). Using Pontryagin's Maximum Principle, we derive the necessary conditions for optimality and the associated Hamiltonian equations. For a special class of weighting matrices, we show that the optimal feedback can be computed explicitly and we prove that the non differentiable value function is the viscosity solution of an appropriate Hamiltn-Jacobi-Bellman equation on SO(3). For arbitrary positive definite weighting matrices, numerical simulations allow us to explore the relationship between the optimal trajectories and weighting matrices, and in particular to highlight nontrivial non differentiability properties of the value function.

ESAIM: Control, Optimisation and Calculus of Variations, 2006

The motivation for this work is the real-time solution of a standard optimal control problem arising in robotics and aerospace applications. For example, the trajectory planning problem for air vehicles is naturally cast as an optimal control problem on the tangent bundle of the Lie Group SE(3), which is also a parallelizable Riemannian manifold. For an optimal control problem on the tangent bundle of such a manifold, we use frame co-ordinates and obtain first-order necessary conditions employing calculus of variations. The use of frame co-ordinates means that intrinsic quantities like the Levi-Civita connection and Riemannian curvature tensor appear in the equations for the co-states. The resulting equations are singularity-free and considerably simpler (from a numerical perspective) than those obtained using a local co-ordinates representation, and are thus better from a computational point of view. The first order necessary conditions result in a two point boundary value problem which we successfully solve by means of a Modified Simple Shooting Method.