Related papers: Geometry of optimal control problems and Hamiltoni…
In this survey, we review the classical Hamilton Jacobi theory from a geometric point of view in different geometric backgrounds. We propose a Hamilton Jacobi equation for different geometric structures attending to one particular…
These notes reproduce the content of a short, 50-minutes, survey talk given at the Nice University in September, 2004. We added a few topics that have not been touched on in the lecture by lack of time.
These are the notes for my lectures at the Trento summer school held September 1997. The aim of the lectures is to provide an introduction to real algebraic surfaces using the minimal model program. This leads to a fairly complete…
A quantum theory in a finite-dimensional Hilbert space can be geometrically formulated as a proper Hamiltonian theory as explained in [2, 3, 7, 8]. From this point of view a quantum system can be described in a classical-like framework…
These notes represent a much expanded and updated version of the \textquotedblleft mini course\textquotedblright that the author gave at the ETH (Z\"{u}rich) and the University of Z\"{u}rich in February of 1995. The purpose of these notes…
Markov Chain Monte Carlo methods have revolutionised mathematical computation and enabled statistical inference within many previously intractable models. In this context, Hamiltonian dynamics have been proposed as an efficient way of…
Contraction properties of the Riccati operator are studied within the context of non-stationary linear-quadratic optimal control. A lifting approach is used to obtain a bound on the rate of strict contraction, with respect to the Riemannian…
In this paper, we apply the geometric Hamilton--Jacobi theory to obtain solutions of Hamiltonian systems in Classical Mechanics, that are either compatible with a cosymplectic or a contact structure. As it is well known, the first structure…
Recently, there has been an increasing interest in modelling and computation of physical systems with neural networks. Hamiltonian systems are an elegant and compact formalism in classical mechanics, where the dynamics is fully determined…
We consider continuous-state and continuous-time control problems where the admissible trajectories of the system are constrained to remain on a union of half-planes which share a common straight line. This set will be named a junction. We…
We address the problem of optimal path planning for a simple nonholonomic vehicle in the presence of obstacles. Most current approaches are either split hierarchically into global path planning and local collision avoidance, or neglect some…
These are expanded notes from a four lecture mini-course given by the author at the Spring School on Non-archimedean geometry and Eigenvarieties, held at the University of Heidelberg in March 2023. The course discusses coherent sheaves,…
In these notes of lectures at the 2004 Summer School of Mathematical Physics in Ravello, Italy, the author develops an approach to calculus in which more efficient choices of limits are taken at key points of the development. For example,…
In physics, two systems that radically differ at short scales can exhibit strikingly similar macroscopic behaviour: they are part of the same long-distance universality class. Here we apply this viewpoint to geometry and initiate a program…
We extend the results of Riemannian geometry over finite groups and provide a full classification of all linear connections for the minimal noncommutative differential calculus over a finite cyclic group. We solve the torsion-free and…
This paper tackles Hamiltonian chaos by means of elementary tools of Riemannian geometry. More precisely, a Hamiltonian flow is identified with a geodesic flow on configuration space-time endowed with a suitable metric due to Eisenhart.…
These notes study the dynamics of iterated holomorphic mappings from a Riemann surface to itself, concentrating on the classical case of rational maps of the Riemann sphere. They are based on introductory lectures given at Stony Brook…
In dissipative ordinary differential equation systems different time scales cause anisotropic phase volume contraction along solution trajectories. Model reduction methods exploit this for simplifying chemical kinetics via a time scale…
We consider the motion planning of an object in a Riemannian manifold where the object is steered from an initial point to a final point utilizing optimal control. Considering Pontryagin Minimization Principle we compute the Optimal…
In this second paper of the series we specify general theory developed in the first paper. Here we study the structure of Jacobi fields in the case of an analytic system and piece-wise analytic control. Moreover, we consider only…