Introduces geometric control theory for students.
problem No specific problem addressed in the abstract.
method Expository presentation of geometric control theory.
result Suitable for advanced students with solid math background.
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48 results for “geometric control theory”
We use the methods of geometric control theory to study extremal trajectories of vertical rolling disk. We focus on the role of symmetries of the underlying geometric structures. We demonstrate the computations in the CAS Maple package DifferentialGeometry.
Study of 2D Lorentzian anti-de Sitter plane using geometric control theory.
problem Understanding extremal trajectories and reachable set on anti-de Sitter plane.
method Geometric control theory and differential geometry.
result Construction of optimal synthesis and description of Lorentzian distance.
Survey of recent developments in symmetric reductions and controls for Hamiltonian systems.
problem Understanding the internal relationships of geometric structures and controls in Hamiltonian systems with symmetry.
method Survey and introduction of recent developments in controlled Hamiltonian systems with symmetry.
result Reveals the relationships between geometric structures, nonholonomic constraints, dynamical vector fields, and controls.
Geometric framework for dynamic feedback linearization of control systems with symmetry.
problem Dynamic feedback linearization of control systems with symmetry.
method Geometric framework based on Lie symmetry, systematic procedure for all smooth, generic system trajectories.
result Sufficient condition for dynamic feedback linearizability obtained.
Just as an explicit parameterisation of system dynamics by state, i.e., a choice of coordinates, can impede the identification of general structure, so it is too with an explicit parameterisation of system dynamics by control. However, such explicit and fixed parameterisation by control is commonplace in control theory…
Some simple examples from quantum physics and control theory are used to illustrate the application of the theory of Lie systems. We will show, in particular, that for certain physical models both of the corresponding classical and quantum problems can be treated in a similar way, may be up to the replacement of the in…
Enhances quantum circuit synthesis using deep learning and geometric methods.
problem Optimizing quantum circuits for time efficiency.
method Combining deep learning with geometric control techniques.
result Improved time-optimal control in quantum circuit synthesis.
Some open problemsmath.OC
We discuss some challenging open problems in the geometric control theory and sub-Riemannian geometry.
We study local control of the mechanism with the growth vector (4,7). We study controllability and extremal trajectories on the nilpotent approximation as an example of the control theory on Lie group. We give solutions of the system an show examples of local extremal trajectories.
A close relationship between the classical Hamilton-Jacobi theory and the kinematic reduction of control systems by decoupling vector fields is shown in this paper. The geometric interpretation of this relationship relies on new mathematical techniques for mechanics defined on a skew-symmetric algebroid. This geometric…
We show an example providing a significance in geometric control theory of the existence of the dependence locus of a system of vector fields in particular, the generic appearance of non-trivial singular trajectories embedded in the dependence locus.
A general study of symmetries in optimal control theory is given, starting from the presymplectic description of this kind of system. Then, Noether's theorem, as well as the corresponding reduction procedure (based on the application of the Marsden-Weinstein theorem adapted to the presymplectic case) are stated both in…
Motion planning and control are key problems in a collection of robotic applications including the design of autonomous agile vehicles and of minimalist manipulators. These problems can be accurately formalized within the language of affine connections and of geometric control theory. In this paper we overview recent r…
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 …
The paper extends a geometric model using singular curves.
problem Understanding abnormal extremals in sub-Riemannian geometry.
method Analysis of singular curves and construction of a graded Lie algebra.
result A nilpotent graded Lie algebra is constructed isomorphic to .
Introduces optimization geometrodynamics for dynamic geometric optimization.
problem Gradient-based optimization methods struggle with changing geometric constraints.
method Optimization geometrodynamics separates invariant and improvable geometric mismatches.
result Dynamic geometric complexity measures the minimum geometric cost to reduce optimization difficulty.
Geometric problems are usually formulated by means of (exterior) differential systems. In this theory, one enriches the system by adding algebraic and differential constraints, and then looks for regular solutions. Here we adopt a dual approach, which consists to enrich a plane field, as this is often practised in cont…
The Geometric Alignment Tax: Tokenization vs. Continuous Geometry in Scientific Foundation Modelscs.LG
Foundation models fail to preserve continuous geometry, identified as the Geometric Alignment Tax.
problem Continuous geometry is lost in foundation models due to discrete categorical bottlenecks.
method Controlled ablations on synthetic systems and evaluation of 14 biological models using rate-distortion theory and MINE.
result Replacing cross-entropy with a continuous head reduces geometric distortion by up to 8.5x.
Algorithm reduces control regret for unknown systems.
problem Minimizing control regret for unknown linear systems.
method Novel geometric exploration strategy and polynomial-time algorithms.
result First polynomial-time algorithms with optimal regret bounds.
We treat the vakonomic dynamics with general constraints within a new geometric framework which will be appropriate to study optimal control problems. We compare our formulation with Vershik-Gershkovich one in the case of linear constraints. We show how nonholonomic mechanics also admits a new geometrical description w…
Funções elíticasmath.DG
Karcher reimagined elliptic functions using geometry.
problem Understanding and controlling elliptic functions.
method Geometrical approach to rewrite elliptic function theory.
result Optimal control over elliptic function behavior and image values.
Ergodic solenoidal homologymath.DG
We define generalized currents associated with immersions of abstract solenoids with a transversal measure. We realize geometrically the full real homology of a compact manifold with these generalized currents, and more precisely with immersions of minimal uniquely ergodic solenoids. This makes precise and geometric De…
Sub-Finsler structures from the time-optimal control viewpoint for some nilpotent distributionsmath.DG
In this paper we study the sub-Finsler geometry as a time-optimal control problem. In particular, we consider non-smooth and non-strictly convex sub-Finsler structures associated with the Heisenberg, Grushin, and Martinet distributions. Motivated by problems in geometric group theory, we characterize extremal curves, d…
We determine the algebraic structure underlying the geometric complex associated to a link in Bar-Natan's geometric formalism of Khovanov's link homology theory (n=2). We find an isomorphism of complexes which reduces the complex to one in a simpler category. This reduction enables us to specify exactly the amount of i…
Study of control problems on Carnot groups with SO(3) symmetry using geometric algebra.
problem Control problems on Carnot groups with SO(3) symmetry.
method Geometric algebra approach to understand geodesics and develop a control algorithm.
result New algorithm for local control developed.
The paper proves index theorems for graph-based optimal control problems.
problem Optimal control problems on graphs with constraints.
method Proves Morse index theorems for a broad class of variational problems on graphs.
result Formulas compute the difference of Hessians related to different graphs or boundary conditions.
Understanding and inverse design of implicit bias in stochastic learning: a geometric perspectivecs.LG
Geometric framework explains and controls implicit bias in machine learning.
problem Understanding and controlling the selection of solutions in overparameterized models.
method Developed a theoretical and constructive framework based on geometric corrections induced by gradient noise and continuous symmetries of the loss.
result Computed the induced bias across various architectures and enabled inverse design to shape the bias.
Wave maps from circle to Riemannian manifold: global controllability is equivalent to homotopymath.AP
Wave maps from circle to manifold controllable if homotopy classes match.
problem Global controllability of wave maps from circle to Riemannian manifolds.
method Characterization of controllability via homotopy classes, uniform-time global controllability between steady states, quantitative exponential stability.
result Global controllability is equivalent to homotopy class of data.
Geometric finiteness theory for essential surfaces in knot exteriors with geometric bounds.
problem Understanding the topology of essential surfaces in knot exteriors with geometric constraints.
method Developed a relative geometric finiteness theory using bounded geometry and thickness conditions.
result Every bounded-geometry slice contains only finitely many pair-isotopy classes, and topology is recoverable from finite geometric data.
Deep residual networks can approximate any continuous function using control theory.
problem Universal approximation capabilities of deep residual neural networks.
method Relating residual networks to control systems and using Lie algebraic techniques.
result Deep residual networks with adequately deep layers can approximate any continuous function on a compact set.
Geometric theory explains substitutability in market outcomes based on production constraints.
problem Understanding substitutability in markets with structured feasible products.
method Modeling the set of feasible products as a compact Riemannian manifold to study intrinsic geometry and its effects on substitutability.
result Intrinsic geometry of the feasible set governs substitutability and market outcomes, with curvature controlling technological substitution elasticity.
Study homotopy groups in GIT quotients using transversality methods.
problem Homotopy groups of stable loci in affine GIT.
method Infinite-dimensional transversality framework extended to GIT setting.
result Generic homotopies avoid unstable and strictly semistable strata under certain conditions.
Study Riemannian metrics on lens spaces, find cut loci and diameters.
problem Understanding Riemannian metrics on lens spaces and their geometric properties.
method Geometric control theory methods applied to axisymmetric metrics.
result Cut loci and cut times converge to sub-Riemannian structure's values.
Study Riemannian metrics on lens spaces, find cut loci and diameters.
problem Analyzing Riemannian metrics on lens spaces.
method Geometric control theory methods.
result Cut loci and cut times converge to sub-Riemannian structure's cut locus and time.
We solve the Plateau problem for marginally outer trapped surfaces in general Cauchy data sets. We employ the Perron method and tools from geometric measure theory to force and control a blow-up of Jang's equation. Substantial new geometric insights regarding the lower order properties of marginally outer trapped surfa…
Develops a new exponential map for time-varying vector fields.
problem Lack of global flows for general time-varying vector fields.
method Categorical development of spaces of vector fields and flows, allowing for systematic localisation.
result Derives the homeomorphism of the exponential map for vector fields with measurable time-dependence.
Unified geometric formulation of Maxwell-Vlasov system using presymplectic and symmetry reduction.
problem Unified geometric formulation of Maxwell-Vlasov system.
method Skinner-Rusk formalism, presymplectic geometry, reduction by diffeomorphism group, affine Hamiltonian controls.
result Unified geometric structure unifying Lagrangian, Hamiltonian, gauge, reduction, and control-theoretic aspects.
Bistable structures associated with non-linear deformation behavior, exemplified by the Venus flytrap and slap bracelet, can switch between different functional shapes upon actuation. Despite numerous efforts in modeling such large deformation behavior of shells, the roles of mechanical and nonlinear geometric effects …
An optimal control problem associated with the dynamics of the orientation of a bipolar molecule in the plane can be understood by means of tools in differential geometry. For first time in the literature -symplectic formalism is used to provide the optimal control problems associated to some families of partial dif…
A geometric approach to differential game theory is illustrated. The parallel pursuit is considered as a two-player zero-sum differential game. The optimal strategies of each player is designed based on Riemann-Finsler geometry. Our approach incorporates a closed loop optimal control and the presentation is familiar wi…
Study of Gödel Universe as Lie group with specific metric.
problem Characterize geodesics in the Gödel Universe.
method Geometric theory of optimal control applied to Lie groups with left-invariant Lorentz metrics.
result No closed timelike or isotropic geodesics in the Gödel Universe.
Study explores optimal portfolio control in financial markets with transaction costs.
problem Optimal portfolio control in financial markets with proportional transaction costs.
method Geometric approach to financial markets, set-valued techniques, stochastic Mayer control problem.
result Continuity of the optimal value and control under price approximations in a multi-asset framework.
A geometric theory explains loss functions for robust representation learning.
problem Treats robustness, domain adaptation, and sensor drift as separate literatures.
method Estimates covariance Sigma_task and uses it to pin Jacobian penalties.
result Proves optimality and necessity of range coverage for penalty matrices.
The left-invariant sub-Riemannian problem on the Engel group is considered. The problem gives the nilpotent approximation to generic nonholonomic systems in four-dimensional space with two-dimensional control, for instance to a system which describes motion of mobile robot with a trailer. The global optimality of extre…
Optimal control problems on Riemannian manifolds are solved by penalizing constraint violations.
problem Optimal control problems with velocity constraints on Riemannian manifolds.
method Penalizing constraint violations and showing convergence to hard-constrained solutions.
result Solutions to soft-constrained problems converge to solutions of hard-constrained problems as penalty parameter increases.
The fundamental theorem of the theory of optimal control, the Pontryagin maximum principle (PMP), is extended to the setting of almost Lie (AL) algebroids, geometrical objects generalizing Lie algebroids. This formulation of the PMP yields, in particular, a scheme comprising reductions of optimal control problems simil…
The Rusk-Skinner formalism was developed in order to give a geometrical unified formalism for describing mechanical systems. It incorporates all the characteristics of Lagrangian and Hamiltonian descriptions of these systems (including dynamical equations and solutions, constraints, Legendre map, evolution operators, e…