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On the equivalence of variational problems. II

Jan Chrastina (1993)

Archivum Mathematicum

Elements of general theory of infinitely prolonged underdetermined systems of ordinary differential equations are outlined and applied to the equivalence of one-dimensional constrained variational integrals. The relevant infinite-dimensional variant of Cartan’s moving frame method expressed in quite elementary terms proves to be surprisingly efficient in solution of particular equivalence problems, however, most of the principal questions of the general theory remains unanswered. New concepts of...

On the existence of a weak solution of the boundary value problem for the equilibrium of a shallow shell reinforced with stiffening ribs

Igor Bock, Ján Lovíšek (1978)

Aplikace matematiky

The existence and the unicity of a weak solution of the boundary value problem for a shallow shell reinforced with stiffening ribs is proved by the direct variational method. The boundary value problem is solved in the space W ( Ω ) ⊂ H 0 1 ( Ω ) × H 0 1 ( Ω ) × H 0 2 ( Ω ) , on which the corresponding bilinear form is coercive. A finite element method for numerical solution is introduced. The approximate solutions converge to a weak solution in the space Q ( Ω ) .

On variational approach to the Hamilton-Jacobi PDE

Jan H. Chabrowski, Ke Wei Zhang (1993)

Commentationes Mathematicae Universitatis Carolinae

In this paper we construct a minimizing sequence for the problem (1). In particular, we show that for any subsolution of the Hamilton-Jacobi equation ( * ) there exists a minimizing sequence weakly convergent to this subsolution. The variational problem (1) arises from the theory of computer vision equations.

Robust portfolio selection under exponential preferences

Dariusz Zawisza (2010)

Applicationes Mathematicae

We consider an incomplete market with an untradable stochastic factor and a robust investment problem based on the CARA utility. We formulate it as a stochastic differential game problem, and use Hamilton-Jacobi-Bellman-Isaacs equations to derive an explicit representation of the robust optimal portfolio; the HJBI equation is transformed using a substitution of the Cole-Hopf type. Not only the pure investment problem, but also a problem of robust hedging is taken into account: an agent tries to...

Some results for an optimal control problem with a semilinear state equation

Fausto Gozzi (1988)

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

We consider a quadratic control problem with a semilinear state equation depending on a small parameter ϵ . We show that the optimal control is a regular function of such parameter.

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...

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