Constantin Udriste

Controllability of a Nonholonomic Macroeconomic System

This paper studies optimal control problems and sub-Riemannian geometry on a nonholonomic macroeconomic system. The main results show that a nonholonomic macroeconomic system is controllable either by trajectories of a single-time driftless control system (single-time bang–bang controls), or by nonholonomic geodesics or by sheets of a two-time driftless control system (two-time bang–bang controls). They are strongly connected to the possibility of describing a nonholonomic macroeconomic system via a Gibbs–Pfaff equation or by four associated vector fields, based on a contact structure of the state space and our isomorphism between thermodynamics and macroeconomics that praises three laws of a nonholonomic macroeconomic system.

Geometric modeling in probability and statistics

Part I: The Geometry of Statistical Models.- Statistical Models.- Explicit Examples.- Entropy on Statistical Models.- Kullback-Leibler Relative Entropy.- Informational Energy.- Maximum Entropy Distributions.- Part II: Statistical Manifolds.- An Introduction to Manifolds.- Dualistic Structure.- Dual Volume Elements.- Dual Laplacians.- Contrast Functions Geometry.- Contrast Functions on Statistical Models.- Statistical Submanifolds.- Appendix A: Information Geometry Calculator.

Maxwell Geometric Dynamics

This paper describes the least squares approximations of the solutions of Maxwell PDEs via their Euler-Lagrange prolongations. Section 5.1 recalls the variational problem in electrodynamics. Section 69.2 describes the Ibragimov-Maxwell Lagrangian. Section 69.3 gives the Ibragimov-Udriste-Maxwell Lagrangian and finds its Euler-Lagrange PDEs system. Section 5.4 finds the Euler-Lagrange PDEs system associated to Udriste-Maxwell Lagrangian, and shows that the waves are particular solutions. Section 69.5 studies the discrete Maxwell geometric dynamics. Section 69.6 performs Von Neumann analysis of the associated difference scheme. Section 69.7 addresses an open problem regarding our theory in the context of differential forms on a Riemannian manifold. Section 69.8 underlines the importance of the least squares Lagrangian.

From integral manifolds and metrics to potential maps

Our paper contains two main results: (1) the integral manifolds of a distribution together with two Riemann metrics produce potential maps which are in fact least squares approximations of the starting integral manifolds; (2) the least squares energy admits extremals satisfying periodic boundary conditions. Section 1 contains historical and bibliographical notes. Section 2 analyses some elements of the geometry produced on the jet bundle of order one by a semi-Riemann Sasaki-like metric. Section 3 describes the maximal integral manifolds of a distribution as solutions of a PDEs system of order one. Section 4 studies Poisson-like second-order prolongations of first order PDE systems and formulates the Lorentz-Udri

Finsler-Lagrange-Hamilton Structures Associated to Control Systems

Section 1 analyses the relations between the second fundamental form of an indicatrix and the Finsler metric produced by the fundamental function, and visualizes some indicatrices using MAPLE 6 codes. Section 2 shows how a control system induces a Finsler structure defined on a subset of the tangent bundle, such that the cost value of a curve solution of the control system is just the length of the curve. Therefore, Finsler geodesics are solutions of the optimal control problem as curves making minimal the cost. Section 3 describes the geometric dynamics produced by a control system and a metric. Section 4 introduces the idea of Finsler gradient flow in optimization problems. The Finsler, Lagrange or Hamilton structures associated to optimal control systems constitute the common denominator of Sections 2–4, the problem of usefulness of Finsler geometry for optimal control systems is still open.

Semivectorial Bilevel Optimization on Riemannian Manifolds

In this paper, we deal with the semivectorial bilevel problem in the Riemannian setting. The upper level is a scalar optimization problem to be solved by the leader, and the lower level is a multiobjective optimization problem to be solved by several followers acting in a cooperative way inside the greatest coalition and choosing among Pareto solutions with respect to a given ordering cone. For the so-called optimistic problem, when the followers choice among their best responses is the most favorable for the leader, we give optimality conditions. Also for the so-called pessimistic problem, when there is no cooperation between the leader and the followers, and the followers choice may be the worst for the leader, we present an existence result.

HAMILTONIAN APPROACHES OF FIELD THEORY

Received 18 February 2004 We extend some results and concepts of single-time covariant Hamiltonian field theory to the new context of multitime covariant Hamiltonian theory. In this sense, we point out the role of the polysymplectic structure δ⊗J, we prove that the dual action is indefinite, we find the eigenvalues and the eigenfunctions of the operator (δ⊗J)(∂/∂t) with periodic boundary conditions, and we obtain interesting inequalities relating functionals created by the new context. As an important example for physics and differential geometry, we study the multitime Yang-Mills-Witten Hamiltonian, extending the Legendre transformation in a suitable way. Our original results are accompanied by well-known relations between Lagrangian and Hamiltonian, and by geometrical explanations regarding the Yang-Mills-Witten Lagrangian.

Extrema constrained by a family of curves and local extrema

This paper considers the connections between the local extrema of a function f:D→R and the local extrema of the restrictions of f to specific subsets of D. In particular, such subsets may be parametrized curves, integral manifolds of a Pfaff system, Pfaff inequations. The paper shows the existence of C1 or C2-curves containing a given sequence of points. Such curves are then exploited to establish the connections between the local extrema of f and the local extrema of f constrained by the family of C1 or C2-curves. Surprisingly, what is true for C1-curves fails to be true in part for C2-curves. Sufficient conditions are given for a point to be a global minimum point of a convex function with respect to a family of curves.

Convex Functions on Riemannian Manifolds

In this chapter we present, in a systematic manner, the basic concepts and theorems regarding the Riemannian convexity of real functions.

Convex Programs on Finsler Manifolds

Generally a metric structure (Euclidean, Riemannian, Finslerian, Lagrangian, Hamiltonian and their generalizations) suitable selected on a given manifold produces convexity of sets and of real functions via geodesies, but till now were studied intesively only the cases of the Euclidean and Riemannian metrics ([6], [11] and the references). Though basic ideas in the Finslerian convexity were presented by the author in [11], [15], this type of convexity has still a lot of open problems. That’s way the present paper refers again to the convex functions on a Finsler manifold (§2), adding theory of the convex programs (§3), the theory of dual programs (§4), and the Kuhn-Tucker theorem (§5).