arXiv subjects

Anthony Bloch

Publications and source records attributed to Anthony Bloch.

60 records · Page 4Linked to original sources

arXiv2017

Optimal Control Problems with Symmetry Breaking Cost Functions

We investigate symmetry reduction of optimal control problems for left-invariant control systems on Lie groups, with partial symmetry breaking cost functions. Our approach emphasizes the role of variational principles and considers a discrete-time setting as well as the standard continuous-time formulation. Specifically, we recast the optimal control problem as a constrained variational problem with a partial symmetry breaking Lagrangian and obtain the Euler--Poincaré equations from a variational principle. By applying a Legendre transformation to it, we recover the Lie-Poisson equations obtained by A. D. Borum [Master's Thesis, University of Illinois at Urbana-Champaign, 2015] in the same context. We also discretize the variational principle in time and obtain the discrete-time Lie-Poisson equations. We illustrate the theory with some practical examples including a motion planning problem in the presence of an obstacle.

math.OC↗

arXiv2016

MPC on manifolds with an application to the control of spacecraft attitude on SO(3)

We develop a model predictive control (MPC) design for systems with discrete-time dynamics evolving on smooth manifolds. We show that the properties of conventional MPC for dynamics evolving on $\mathbb R^n$ are preserved and we develop a design procedure for achieving similar properties. We also demonstrate that for discrete-time dynamics on manifolds with Euler characteristic not equal to 1, there do not exist globally stabilizing, continuous control laws. The MPC law is able to achieve global asymptotic stability on these manifolds, because the MPC law may be discontinuous. We apply the method to spacecraft attitude control, where the spacecraft attitude evolves on the Lie group SO(3) and for which a continuous globally stabilizing control law does not exist. In this case, the MPC law is discontinuous and achieves global stability.

math.OC↗

arXiv2014

A geometric approach to the optimal control of nonholnomic mechanical systems

In this paper, we describe a constrained Lagrangian and Hamiltonian formalism for the optimal control of nonholonomic mechanical systems. In particular, we aim to minimize a cost functional, given initial and final conditions where the controlled dynamics is given by nonholonomic mechanical system. In our paper, the controlled equations are derived using a basis of vector fields adapted to the nonholonomic distribution and the Riemannian metric determined by the kinetic energy. Given a cost function, the optimal control problem is understood as a constrained problem or equivalently, under some mild regularity conditions, as a Hamiltonian problem on the cotangent bundle of the nonholonomic distribution. A suitable Lagrangian submanifold is also shown to lead to the correct dynamics. We demonstrate our techniques in several examples including a continuously variable transmission problem and motion planning for obstacle avoidance problems.

math.OC↗

arXiv2012

Decoherence Control and Purification of Two-dimensional Quantum Density Matrices under Lindblad Dissipation

Control of quantum dissipative systems can be challenging because control variables are typically part of the system Hamiltonian, which can only generate motion along unitary orbits of the system. To transit between orbits, one must harness the dissipation super-operator. To separate the inter-orbit dynamics from the Hamiltonian dynamics for a generic two-dimensional system, we project the Lindblad master equation onto the set of spectra of the density matrix, and we interpret the location along the orbit to be a new control variable. The resulting differential equation allows us to analyze the controllability of a general two-dimensional Lindblad system, particularly systems where the dissipative term has an anti-symmetric part. We extend this to categorize the possible purifiable systems in two dimensions.

quant-ph↗

arXiv2007

Hill's Equation with Random Forcing Terms

Motivated by a class of orbit problems in astrophysics, this paper considers solutions to Hill's equation with forcing strength parameters that vary from cycle to cycle. The results are generalized to include period variations from cycle to cycle. The development of the solutions to the differential equation is governed by a discrete map. For the general case of Hill's equation in the unstable limit, we consider separately the case of purely positive matrix elements and those with mixed signs; we then find exact expressions, bounds, and working estimates for the growth rates. We also find exact expressions, estimates, and bounds for the infinite products of several $2 \times 2$ matrices with random variables in the matrix elements. In the limit of sharply spiked forcing terms (the delta function limit), we find analytic solutions for each cycle and for the discrete map that matches solutions from cycle to cycle; for this case we find the growth rates and the condition for instability in the limit of large forcing strength, as well as the widths of the stable/unstable zones.

math-ph↗

arXiv1993

Dissipation Induced Instabilities

The main goal of this paper is to prove that if the energy-momentum (or energy-Casimir) method predicts formal instability of a relative equilibrium in a Hamiltonian system with symmetry, then with the addition of dissipation, the relative equilibrium becomes spectrally and hence linearly and nonlinearly unstable. The energy-momentum method assumes that one is in the context of a mechanical system with a given symmetry group. Our result assumes that the dissipation chosen does not destroy the conservation law associated with the given symmetry group---thus, we consider internal dissipation. This also includes the special case of systems with no symmetry and ordinary equilibria. The theorem is proved by combining the techniques of Chetaev, who proved instability theorems using a special Chetaev-Lyapunov function, with those of Hahn, which enable one to strengthen the Chetaev results from Lyapunov instability to spectral instability. The main achievement is to strengthen Chetaev's methods to the context of the block diagonalization version of the energy momentum method given by Lewis, Marsden, Posbergh, and Simo. However, we also give the eigenvalue movement formulae of Krein, MacKay and others both in general and adapted to the context of the normal form of the linearized equations given by the block diagonal form, as provided by the energy-momentum method. A number of specific examples, such as the rigid body with internal rotors, are provided to illustrate the results.

chao-dyn↗