Abstract
The purpose of our paper is to study a class of left-invariant, drift-free optimal control problem on the Lie group ISO(3,1). The left-invariant, drift-free optimal control problems involves finding a trajectory-control pair on ISO(3,1), which minimize a cost function and satisfies the given dynamical constrains and boundary conditions in a fixed time. The problem is lifted to the cotangent bundle T*G using the optimal Hamiltonian on G*, where the maximum principle yields the optimal control. We use energy methods (Arnold’s method, in this case) to give sufficient conditions fornonlinear stability of the equilibrium states. Around this equilibrium states we might be able to find the periodical orbits using Moser's theorem, as future work. For the some unstable equilibrium states, a quadratic control is considered in order to stabilize the dynamics.
Highlights
The Poincaré group ISO(3,1) was first defined by Minkowski (1908) as the group of space-time isometries
In the second paragraph we define an optimal control problem on iso(3,1) and we find the controls that minimize a quadratic cost function
The last paragraphs are dedicated to the stability; first, we analyse the spectral and nonlinear stability of some equilibrium states using energy-Casimir methods; at the last, we use a quadratic cost function to stabilize a family of equilibrium states for which the energy-Casimir methods were inconclusive
Summary
The Poincaré group ISO(3,1) was first defined by Minkowski (1908) as the group of space-time isometries. Due to its big importance in quantum theory of fields [1,2,3,4,5], we are interested to study an optimal control problem on this Lie group. In the second paragraph we define an optimal control problem on iso(3,1) and we find the controls that minimize a quadratic cost function. Let us consider , , , , the usual generators of spatial rotations, boosts, space translations, and time translation respectively, of the Poincaré inhomogeneous Lie algebra iso(3,1) (see [9]); the nonzero brackets are given by. The system is obtained by applying Krishnaprasad's theorem (see [6,7, 8]) to the optimal Hamiltonian given by: Proposition 2 The dynamics (3) has the following Poisson realization: 3,1 , Π , , where:. Remark Following [9], the Lie-Poisson structure Π admits two linear independent Casimir operators:
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