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We propose a fourth-order extension of the geodesic problem arising in the continuity of the study of Riemannian cubics. We consider the variational problem in a Riemannian manifold and derive the Euler-Lagrange equation. For the special situation of Lie groups, we use Euler-Poincaré reduction to obtain the reduced equation on the Lie algebra.

We propose a fourth-order extension of the geodesic problem arising in the continuity of the study of Riemannian cubics. We consider the variational problem in a Riemannian manifold and derive the Euler-Lagrange equation. For the special situation of Lie groups, we use Euler-Poincaré reduction to obtain the reduced equation on the Lie algebra. [ABSTRACT FROM AUTHOR]

By constructing a series of perturbation functions through localization in the Fourier domain and using a symmetric form of the system, we show that the data-to-solution map for the Euler–Poincaré equations is nowhere uniformly continuous in B p , r s (R d) with s > max { 1 + d 2 , 3 2 } and (p , r) ∈ (1 , ∞) × [ 1 , ∞) . This improves our previous result (Li et al. in Nonlinear Anal RWA 63:103420, 2022) which shows the data-to-solution map for the Euler–Poincaré equations is non-uniformly continuous on a bounded subset of B p , r s (R d) near the origin. [ABSTRACT FROM AUTHOR]

We study the reduction by symmetry for optimality conditions in optimal control problems of left-invariant affine multi-agent control systems, with partial symmetry breaking cost functions for continuous-time and discrete-time systems. We recast the optimal control problem as a constrained variational problem with a partial symmetry breaking Lagrangian and obtain the reduced optimality conditions from a reduced variational principle via symmetry reduction techniques in both settings continuous-time, and discrete-time. We apply the results to a collision and obstacle avoidance problem for multiple vehicles evolving on $ SE(2) $ in the presence of a static obstacle.

We study the reduction by symmetry for optimality conditions in optimal control problems of left-invariant affine multi-agent control systems, with partial symmetry breaking cost functions for continuous-time and discrete-time systems. We recast the optimal control problem as a constrained variational problem with a partial symmetry breaking Lagrangian and obtain the reduced optimality conditions from a reduced variational principle via symmetry reduction techniques in both settings continuous-time, and discrete-time. We apply the results to a collision and obstacle avoidance problem for multiple vehicles evolving on S E (2) in the presence of a static obstacle. [ABSTRACT FROM AUTHOR]

The main result of this paper is a matched-pair decomposition of the space of symmetric contravariant tensors T Q . From this procedure two complementary Lie subalgebras of T Q under mutual interaction arise. Introducing a lift operator, the matched pair decomposition of the space of Hamiltonian vector fields is determined. According to this realization, the Euler–Poincaré flows on such spaces are decomposed into two subdynamics: one is the Euler–Poincaré formulation of isentropic fluid flows, and the other one corresponds with Euler–Poincaré equations on contravariant tensors of order n ⩾ 2 . [ABSTRACT FROM AUTHOR]

The main result of this paper is a matched-pair decomposition of the space of symmetric contravariant tensors TQ. From this procedure two complementary Lie subalgebras of TQ under mutual interaction arise. Introducing a lift operator, the matched pair decomposition of the space of Hamiltonian vector fields is determined. According to this realization, the Euler–Poincaré flows on such spaces are decomposed into two subdynamics: one is the Euler–Poincaré formulation of isentropic fluid flows, and the other one corresponds with Euler–Poincaré equations on contravariant tensors of order n⩾2.

We prove that the initial value problem for the Euler–Poincaré equations is not locally well-posed for initial data in the Besov space B p , ∞ σ whenever σ > 2 + max { 1 + d p , 3 2 }. By presenting a new construction of initial data u 0 , we prove the corresponding solution of Euler–Poincaré equations starting from u 0 is discontinuous at t = 0 in the norm of B p , ∞ σ , which implies the ill-posedness. Since this problem is locally well-posed in the Besov space B p , r σ for r < ∞ and the same σ , our result suggests that well-posedness does not hold at the endpoint r = ∞. [ABSTRACT FROM AUTHOR]

In recent works, beginning with [76] , several stochastic geophysical fluid dynamics (SGFD) models have been derived from variational principles. In this paper, we introduce a new framework for parametrization schemes (PS) in GFD. We derive a class of rough geophysical fluid dynamics (RGFD) models as critical points of rough action functionals using the theory of controlled rough paths. These RGFD models characterize Lagrangian trajectories in fluid dynamics as geometric rough paths (GRP) on the manifold of diffeomorphic maps. We formulate three constrained variational approaches for the derivation of these models. The first is the Clebsch formulation, in which the constraints are imposed as rough advection laws. The second is the Hamilton-Pontryagin formulation, in which the constraints are imposed as right-invariant rough vector fields. And the third is the Euler–Poincaré formulation, in which the variations are constrained. These constrained rough variational principles lead directly to the Lie–Poisson Hamiltonian formulation of fluid dynamics on GRP. The GRP framework preserves the geometric structure of fluid dynamics obtained by using Lie group reduction to pass from Lagrangian to Eulerian variational principles, yielding a rough formulation of the Kelvin circulation theorem. The rough formulation enhances its stochastic counterpart developed in [76] , and extended to semimartingales in [109]. For example, the rough-path variational approach includes non-Markovian perturbations of the Lagrangian fluid trajectories. In particular, memory effects can be introduced through a judicious choice of the rough path (e.g. a realization of a fractional Brownian motion). In the particular case when the rough path is a realization of a semimartingale, we recover the SGFD models in [76,109]. However, by eliminating the need for stochastic variational tools, we retain a pathwise interpretation of the Lagrangian trajectories. In contrast, the Lagrangian trajectories in the stochastic framework are described by stochastic integrals, which do not have a pathwise interpretation. Thus, the rough path formulation restores this property. [ABSTRACT FROM AUTHOR]