A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
The purpose of this paper is to use the framework of Lie algebroids to study optimal control problems for affine connection control systems on Lie groups. In this context, the equations for critical trajectories of the problem are geometrically characterized as a Hamiltonian vector field.
Unified geometric formulation of Maxwell-Vlasov system using presymplectic and symmetry reduction.
The paper presents the geometry of Lie algebroids and its applications to optimal control. The first part deals with the theory of Lie algebroids, connections on Lie algebroids and dynamical systems defined on Lie algebroids (mainly Lagrangian and Hamiltonian systems). In the second part we use the framework of Lie alg…
Affine hamiltonians are defined in the paper and their study is based especially on the fact that in the hyperregular case they are dual objects of lagrangians defined on affine bundles, by mean of natural Legendre maps. The variational problems for affine hamiltonians and lagrangians of order are studied, re…
This paper develops a Hamiltonian reduction method for field theories over affine principal bundles.
The paper studies time-optimal problems on specific Lie groups, describing orbits and integrals.
An affine Cartan calculus is developed. The concepts of special affine bundles and special affine duality are introduced. The canonical isomorphisms, fundamental for Lagrangian and Hamiltonian formulations of the dynamics in the affine setting are proved.
We describe the structure of the Lie groups endowed with a left-invariant symplectic form, called symplectic Lie groups, in terms of semi-direct products of Lie groups, symplectic reduction and principal bundles with affine fiber. This description is particularly nice if the group is Hamiltonian, that is, if the left c…
We introduce a notion of a weak Poisson structure on a manifold modeled on a locally convex space. This is done by specifying a Poisson bracket on a subalgebra $\cA \subeq C^\infty(M)$ which has to satisfy a non-degeneracy condition (the differentials of elements of $\cA$ separate tangent vectors) and we postulate …
Survey of recent developments in symmetric reductions and controls for Hamiltonian systems.
We find necessary and sufficient conditions for a local geodesic flow of an affine connection on a surface to admit a linear first integral. The conditions are expressed in terms of two scalar invariants of differential orders 3 and 4 in the connection. We use this result to find explicit obstructions to the existence …
A general, consistent and complete framework for geometrical formulation of mechanical systems is proposed, based on certain structures on affine bundles (affgebroids) that generalize Lie algebras and Lie algebroids. This scheme covers and unifies various geometrical approaches to mechanics in the Lagrangian and Hamilt…
Differentiable simulations control molecular Hamiltonians for desired outcomes.
New samplers improve MCMC efficiency in high dimensions.
Random features enhance control of complex systems.
Develops control and observer methods for complex systems.
A frame independent formulation of analytical mechanics in the Newtonian space-time is presented. The differential geometry of affine values i.e., the differential geometry in which affine bundles replace vector bundles and sections of one dimensional affine bundles replace functions on manifolds, is used. Lagrangian a…
Motivated by the ubiquity of control-affine systems in optimal control theory, we investigate the geometry of point-affine control systems with metric structures in dimensions two and three. We compute local isometric invariants for point-affine distributions of constant type with metric structures for systems with 2 s…
Unified theory of -expectations derived from chaotic dynamics.
In this paper, from the viewpoint of completeness of Marsden-Weinstein reduction, we illustrate how to give the definitions of a controlled Hamiltonian (CH) system and a reducible controlled Hamiltonian system with symmetry; and how to describe the dynamics of a CH system and the controlled Hamiltonian equivalence; as …
We show that discrete lattices are bi-Hamiltonian, using geometric realizations of discretizations of the Adler-Gel'fand-Dikii flows as local evolutions of arc length-parametrized polygons in centro-affine space. We prove the compatibility of two known Hamiltonian structure defined on the space of geometric invar…
Polynomial Duistermaat-Heckman measure on symplectic groupoid quotients.
In this paper, we investigate the existence of a subclass of quotients of affine connection control systems, which preserve the mechanical structures. Both local and global sufficient and necessary conditions are given for the geodesically accessible affine connection control systems such that they can admit this subcl…
Motivated by control-affine systems in optimal control theory, we introduce the notion of a point-affine distribution on a manifold X - i.e., an affine distribution F together with a distinguished vector field contained in F. We compute local invariants for point-affine distributions of constant type when dim(X)=n, ran…
This work is devoted to a systematic study of symplectic convexity for integrable Hamiltonian systems with elliptic and focus-focus singularities. A distinctive feature of these systems is that their base spaces are still smooth manifolds (with boundary and corners), similarly to the toric case, but their associated in…
The geometrical structure known as the Tulczyjew triple has proved to be very useful in describing mechanical systems, even those with singular Lagrangians or subject to constraints. Starting from basic concepts of variational calculus, we construct the Tulczyjew triple for first-order Field Theory. The important featu…
Bi-Hamiltonian structures are of great importance in the theory of integrable Hamiltonian systems. The notion of compatibility of symplectic structures is a key aspect of bi-Hamiltonian systems. Because of this, a few different notions of compatibility have been introduced. In this paper we show that, under some additi…
In this paper, we consider nonlinear PDEs in a port-Hamiltonian setting based on an underlying jet-bundle structure. We restrict ourselves to systems with 1-dimensional spatial domain and 2nd-order Hamiltonian including certain dissipation models that can be incorporated in the port- Hamiltonian framework by means of a…
We give the following results for Pinkall's central affine curve flow on the plane: (i) a systematic and simple way to construct the known higher commuting curve flows, conservation laws, and a bi-Hamiltonian structure, (ii) Baecklund transformations and a permutability formula, (iii) infinitely many families of explic…
Dissipative SymODEN learns dynamics with dissipation and control from data.
Clarifies relation for solving control-affine Schrödinger bridge problems.
This paper classifies Hamiltonian actions by symplectic groupoids using Delzant subspaces.
The geometrical structure known as Tulczyjew triple has been used with success in analytical mechanics and first order field theory to describe a wide range of physical systems including Lagrangian/Hamiltonian systems with constraints and/or sources, or with singular Lagrangian. Starting from the first principles of th…
We geometrically describe optimal control problems in terms of Morse families in the Hamiltonian framework. These geometric structures allow us to recover the classical first order necessary conditions for optimality and the starting point to run an integrability algorithm. Moreover the integrability algorithm is adapt…
Study magnetic Hamiltonian systems with constraints, deriving Hamilton-Jacobi equations.
We investigate necessary and sufficient conditions under which a general nonlinear affine control system with outputs can be written as a gradient control system corresponding to some pseudo-Riemannian metric defined on the state space. The results rely on a suitable notion of compatibility of the system with respect t…
Physics-informed learning framework for pH systems and EB-PBC control.
This paper tackles data-efficient nonlinear control in Hamiltonian systems using symplectic geometry.
We construct a sequence of commuting central affine curve flows on invariant under the action of and prove the following results: (a) The central affine curvatures of a solution of the j-th central affine curve flow is a solution of the j-th flow of Gelfand-Dickey (GD) hierarchy on the s…
The paper is devoted to the local classification of generic control-affine systems on an n-dimensional manifold with scalar input for any n>3 or with two inputs for n=4 and n=5, up to state-feedback transformations, preserving the affine structure. First using the Poincare series of moduli numbers we introduce the intr…
This text presents some basic notions in symplectic geometry, Poisson geometry, Hamiltonian systems, Lie algebras and Lie groups actions on symplectic or Poisson manifolds, momentum maps and their use for the reduction of Hamiltonian systems. It should be accessible to readers with a general knowledge of basic notions …
The paper designs neural networks with assurance for controlling nonlinear systems.
In this paper, we put the issue of dynamic equivalence of control systems in the context of pullbacks of coframings on infinite jet bundles over the state manifolds. While much attention has been given to differentially flat systems, i.e. systems dynamically equivalent to linear control systems, the advantage of this a…
New algorithm solves Schrödinger bridge problem with mismatched channels.
Recurrent neural networks (RNNs) have gained a great deal of attention in solving sequential learning problems. The learning of long-term dependencies, however, remains challenging due to the problem of a vanishing or exploding hidden states gradient. By exploring further the recently established connections between RN…
The aim of this paper is to prove that a control affine system on a manifold is equivalent by diffeomorphism to a linear system on a Lie group or a homogeneous space if and only the vector fields of the system are complete and generate a finite dimensional Lie algebra. A vector field on a connected Lie group is linear …
In this paper, we give precisely the geometric constraint conditions of canonical symplectic form and regular reduced symplectic forms for the dynamical vector fields of a regular controlled Hamiltonian (RCH) system and its regular reduced systems, which are called the Type I and Type II of Hamilton-Jacobi equations. A…
The paper extends a theorem about momentum maps to singular symplectic spaces.