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Optimal control problems on parallelizable Riemannian manifolds: theory and applications

Ram V. Iyer, Raymond Holsapple, David Doman (2005)

ESAIM: Control, Optimisation and Calculus of Variations

The motivation for this work is the real-time solution of a standard optimal control problem arising in robotics and aerospace applications. For example, the trajectory planning problem for air vehicles is naturally cast as an optimal control problem on the tangent bundle of the Lie Group SE(3), which is also a parallelizable Riemannian manifold. For an optimal control problem on the tangent bundle of such a manifold, we use frame co-ordinates and obtain first-order necessary conditions...

Optimal design of turbines with an attached mass

Boris P. Belinskiy, C. Maeve McCarthy, Terry J. Walters (2003)

ESAIM: Control, Optimisation and Calculus of Variations

We minimize, with respect to shape, the moment of inertia of a turbine having the given lowest eigenfrequency of the torsional oscillations. The necessary conditions of optimality in conjunction with certain physical parameters admit a unique optimal design.

Optimal design of turbines with an attached mass

Boris P. Belinskiy, C. Maeve McCarthy, Terry J. Walters (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We minimize, with respect to shape, the moment of inertia of a turbine having the given lowest eigenfrequency of the torsional oscillations. The necessary conditions of optimality in conjunction with certain physical parameters admit a unique optimal design.

Optimality Conditions for D.C. Vector Optimization Problems under D.C. Constraints

Gadhi, N., Metrane, A. (2004)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: Primary 90C29; Secondary 49K30.In this paper, we establish necessary optimality conditions and sufficient optimality conditions for D.C. vector optimization problems under D.C. constraints. Under additional conditions, some results of [9] and [15] are also recovered.

Regularity of displacement solutions in Hencky plasticity. II: The main result

Jarosław L. Bojarski (2011)

Applicationes Mathematicae

The aim of this paper is to study the problem of regularity of displacement solutions in Hencky plasticity. Here, a non-homogeneous material is considered, where the elastic-plastic properties change discontinuously. In the first part, we have found the extremal relation between the displacement formulation defined on the space of bounded deformation and the stress formulation of the variational problem in Hencky plasticity. In the second part, we prove that the displacement...

Regularity of displacement solutions in Hencky plasticity. I: The extremal relation

Jarosław L. Bojarski (2011)

Applicationes Mathematicae

The aim of this paper is to study the problem of regularity of displacement solutions in Hencky plasticity. A non-homogeneous material whose elastic-plastic properties change discontinuously is considered. We find (in an explicit form) the extremal relation between the displacement formulation (defined on the space of bounded deformation) and the stress formulation of the variational problem in Hencky plasticity. This extremal relation is used in the proof of the regularity of displacements. ...

Regularity of solutions in plasticity. I: Continuum

Jarosław L. Bojarski (2003)

Applicationes Mathematicae

The aim of this paper is to study the problem of regularity of solutions in Hencky plasticity. We consider a non-homogeneous material whose elastic-plastic properties change discontinuously. We prove that the displacement solutions belong to the space L D ( Ω ) ≡ u ∈ L ¹ ( Ω , ℝ ⁿ ) | ∇ u + ( ∇ u ) T ∈ L ¹ ( Ω , ℝ n × n ) if the stress solution is continuous and belongs to the interior of the set of admissible stresses, at each point. The part of the functional which describes the work of boundary forces is relaxed.

Regularity of solutions in plasticity. II: Plates

Jarosław L. Bojarski (2004)

Applicationes Mathematicae

The aim of this paper is to study the problem of regularity of displacement solutions in Hencky plasticity. We consider a plate made of a non-homogeneous material whose elastic-plastic properties change discontinuously. We prove that the displacement solutions belong to the space W 2 , 1 ( Ω ) if the stress solution is continuous and belongs to the interior of the set of admissible stresses, at each point. The part of the functional which describes the work of boundary forces is relaxed.

Structure of stable solutions of a one-dimensional variational problem

Nung Kwan Yip (2006)

ESAIM: Control, Optimisation and Calculus of Variations

We prove the periodicity of all H2-local minimizers with low energy for a one-dimensional higher order variational problem. The results extend and complement an earlier work of Stefan Müller which concerns the structure of global minimizer. The energy functional studied in this work is motivated by the investigation of coherent solid phase transformations and the competition between the effects from regularization and formation of small scale structures. With a special choice of a bilinear double...

Subriemannian geodesics of Carnot groups of step 3

Kanghai Tan, Xiaoping Yang (2013)

ESAIM: Control, Optimisation and Calculus of Variations

In Carnot groups of step  ≤ 3, all subriemannian geodesics are proved to be normal. The proof is based on a reduction argument and the Goh condition for minimality of singular curves. The Goh condition is deduced from a reformulation and a calculus of the end-point mapping which boils down to the graded structures of Carnot groups.

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