Related papers: Geometric numerical methods for Lie systems and th…
In this article, a unified approach to obtain symplectic integrators on T*G from Lie group integrators on a Lie group G is presented. The approach is worked out in detail for symplectic integrators based on Runge--Kutta--Munthe-Kaas methods…
A new format for commutator-free Lie group methods is proposed based on explicit classical Runge-Kutta schemes. In this format exponentials are reused at every stage and the storage is required only for two quantities: the right hand side…
The integration of gyroscope measurements is an essential task for most navigation systems. Modern vehicles typically use strapdown systems, such that gyro integration requires coning compensation to account for the sensor's rotation during…
This work represents a PhD thesis concerning three main topics. The first one deals with the study and applications of Lie systems with compatible geometric structures, e.g. symplectic, Poisson, Dirac, Jacobi, among others. Many new Lie…
This work presents a newly renovated approach to the analysis of second-order Riccati equations from the point of view of the theory of Lie systems. We show that these equations can be mapped into Lie systems through certain Legendre…
Geometric integration of non-autonomous classical engineering problems, such as rotor dynamics, is investigated. It is shown, both numerically and by backward error analysis, that geometric (structure preserving) integration algorithms are…
In this paper we study, from a variational and geometrical point of view, second-order variational problems on Lie groupoids and the construction of variational integrators for optimal control problems. First, we develop variational…
For a right-invariant system on a compact Lie group G, I present two methods to design a control to drive the state from the identity to any element of the group. The first method, under appropriate assumptions, achieves exact control to…
In this second part about dynamics of atomic system we revisit the logic application of $SU(2)$ dynamics. We reiterate that solution of quantum dynamics systems can be represented geometrically. Such geometric representations of solutions…
Classical mathematical techniques such as discrete integration, gradient descent optimization, and state estimation (exemplified by the Runge-Kutta method, Gauss-Newton minimization, and extended Kalman filter or EKF, respectively), rely on…
This paper concerns the numerical procedure for solving hybrid optimal control problems with sliding modes. The proposed procedure has several features which distinguishes it from the other procedures for the problem. First of all a sliding…
Gamma distributed delay differential equations (DDEs) arise naturally in many modelling applications. However, appropriate numerical methods for generic Gamma distributed DDEs are not currently available. Accordingly, modellers often resort…
The integration of the equations of motion in gravitational dynamical systems -- either in our Solar System or for extra-solar planetary system -- being non integrable in the global case, is usually performed by means of numerical…
In Part I of this series we presented the general ideas of applying group-algebraic methods for describing quantum systems. The treatment was there very "ascetic" in that only the structure of a locally compact topological group was used.…
We review some recent results of the theory of Lie systems in order to apply such results to study Ermakov systems. The fundamental properties of Ermakov systems, i.e. their superposition rules, the Lewis-Ermakov invariants, etc., are found…
In this paper we construct higher-order variational integrators for a class of degenerate systems described by Lagrangians that are linear in velocities. We analyze the geometry underlying such systems and develop the appropriate theory for…
Lie-integration is one of the most efficient algorithms for numerical integration of ordinary differential equations if high precision is needed for longer terms. The method is based on the computation of the Taylor-coefficients of the…
The Lie-point symmetry method is used to find some closed-form solutions for a constitutive equation modeling stress in elastic materials. The partial differential equation (PDE), which involves a power law with arbitrary exponent n, was…
This paper formulates optimal control problems for rigid bodies in a geometric manner and it presents computational procedures based on this geometric formulation for numerically solving these optimal control problems. The dynamics of each…
The applications of geometric control theory methods on Lie groups and homogeneous spaces to the theory of quantum computations are investigated. These methods are shown to be very useful for the problem of constructing an universal set of…