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Sub-Riemannian sphere in Martinet flat case

A. Agrachev, B. Bonnard, M. Chyba, I. Kupka (2010)

ESAIM: Control, Optimisation and Calculus of Variations

This article deals with the local sub-Riemannian geometry on ℜ3, (D,g) where D is the distribution ker ω, ω being the Martinet one-form : dz - ½y2dxand g is a Riemannian metric on D. We prove that we can take g as a sum of squares adx2 + cd2. Then we analyze the flat case where a = c = 1. We parametrize the set of geodesics using elliptic integrals. This allows to compute the exponential mapping, the wave front, the conjugate and cut loci and the sub-Riemannian sphere. A direct consequence...

Symplectic Pontryagin approximations for optimal design

Jesper Carlsson, Mattias Sandberg, Anders Szepessy (2009)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

The powerful Hamilton-Jacobi theory is used for constructing regularizations and error estimates for optimal design problems. The constructed Pontryagin method is a simple and general method for optimal design and reconstruction: the first, analytical, step is to regularize the hamiltonian; next the solution to its stationary hamiltonian system, a nonlinear partial differential equation, is computed with the Newton method. The method is efficient for designs where the hamiltonian function can be...

Symplectic Pontryagin approximations for optimal design

Jesper Carlsson, Mattias Sandberg, Anders Szepessy (2008)

ESAIM: Mathematical Modelling and Numerical Analysis

The powerful Hamilton-Jacobi theory is used for constructing regularizations and error estimates for optimal design problems. The constructed Pontryagin method is a simple and general method for optimal design and reconstruction: the first, analytical, step is to regularize the Hamiltonian; next the solution to its stationary Hamiltonian system, a nonlinear partial differential equation, is computed with the Newton method. The method is efficient for designs where the Hamiltonian function...

Target achieving portfolio under model misspecification: quadratic optimization framework

Dariusz Zawisza (2012)

Applicationes Mathematicae

We incorporate model uncertainty into a quadratic portfolio optimization framework. We consider an incomplete continuous time market with a non-tradable stochastic factor. Two stochastic game problems are formulated and solved using Hamilton-Jacobi-Bellman-Isaacs equations. The proof of existence and uniqueness of a solution to the resulting semilinear PDE is also provided. The latter can be used to extend many portfolio optimization results.

The finite automata approaches in stringology

Jan Holub (2012)

Kybernetika

We present an overview of four approaches of the finite automata use in stringology: deterministic finite automaton, deterministic simulation of nondeterministic finite automaton, finite automaton as a model of computation, and compositions of finite automata solutions. We also show how the finite automata can process strings build over more complex alphabet than just single symbols (degenerate symbols, strings, variables).

The Hamilton-Cartan formalism in the calculus of variations

Hubert Goldschmidt, Shlomo Sternberg (1973)

Annales de l'institut Fourier

We give an exposition of the calculus of variations in several variables. The introduction of a linear differential form studied by Cartan makes possible an invariant treatment of the Hamiltonian formalism. Noether’s theorem, the Hamilton-Jacobi equation and the second variation are discussed and a Poisson bracket is defined.

The value function representing Hamilton–Jacobi equation with hamiltonian depending on value of solution

A. Misztela (2014)

ESAIM: Control, Optimisation and Calculus of Variations

In the paper we investigate the regularity of the value function representing Hamilton–Jacobi equation: − Ut + H(t, x, U, − Ux) = 0 with a final condition: U(T,x) = g(x). Hamilton–Jacobi equation, in which the Hamiltonian H depends on the value of solution U, is represented by the value function with more complicated structure than the value function in Bolza problem. This function is described with the use of some class of Mayer problems related to the optimal control theory and the calculus of...

The vanishing viscosity method in infinite dimensions

Piermarco Cannarsa, Giuseppe Da Prato (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

The vanishing viscosity method is adapted to the infinite dimensional case, by showing that the value function of a deterministic optimal control problem can be approximated by the solutions of suitable parabolic equations in Hilbert spaces.

Turnpike theorems by a value function approach

Alain Rapaport, Pierre Cartigny (2004)

ESAIM: Control, Optimisation and Calculus of Variations

Turnpike theorems deal with the optimality of trajectories reaching a singular solution, in calculus of variations or optimal control problems. For scalar calculus of variations problems in infinite horizon, linear with respect to the derivative, we use the theory of viscosity solutions of Hamilton-Jacobi equations to obtain a unique characterization of the value function. With this approach, we extend for the scalar case the classical result based on Green theorem, when there is uniqueness of the...

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