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Regularity along optimal trajectories of the value function of a Mayer problem

Carlo Sinestrari (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We consider an optimal control problem of Mayer type and prove that, under suitable conditions on the system, the value function is differentiable along optimal trajectories, except possibly at the endpoints. We provide counterexamples to show that this property may fail to hold if some of our conditions are violated. We then apply our regularity result to derive optimality conditions for the trajectories of the system.

Regularity and optimal control of quasicoupled and coupled heating processes

Jiří Jarušek (1996)

Applications of Mathematics

Sufficient conditions for the stresses in the threedimensional linearized coupled thermoelastic system including viscoelasticity to be continuous and bounded are derived and optimization of heating processes described by quasicoupled or partially linearized coupled thermoelastic systems with constraints on stresses is treated. Due to the consideration of heating regimes being “as nonregular as possible” and because of the well-known lack of results concerning the classical regularity of solutions...

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.

Regularization of linear least squares problems by total bounded variation

G. Chavent, K. Kunisch (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We consider the problem : (P) Minimize λ 2 over u ∈ K ∩ X, where α≥ 0, β > 0, K is a closed convex subset of L2(Ω), and the last additive term denotes the BV-seminorm of u, T is a linear operator from L2 ∩ BV into the observation space Y. We formulate necessary optimality conditions for (P). Then we show that (P) admits, for given regularization parameters α and β, solutions which depend in a stable manner on the data z. Finally we study the asymptotic behavior when α = β → 0. The regularized...

Relaxation of optimal control problems in Lp-SPACES

Nadir Arada (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We consider control problems governed by semilinear parabolic equations with pointwise state constraints and controls in an Lp-space (p < ∞). We construct a correct relaxed problem, prove some relaxation results, and derive necessary optimality conditions.

Relaxation of optimal control problems in 𝖫 𝗉 -spaces

Nadir Arada (2001)

ESAIM: Control, Optimisation and Calculus of Variations

We consider control problems governed by semilinear parabolic equations with pointwise state constraints and controls in an L p -space ( p &lt; ∞ ). We construct a correct relaxed problem, prove some relaxation results, and derive necessary optimality conditions.

Relaxation of vectorial variational problems

Tomáš Roubíček (1995)

Mathematica Bohemica

Multidimensional vectorial non-quasiconvex variational problems are relaxed by means of a generalized-Young-functional technique. Selective first-order optimality conditions, having the form of an Euler-Weiestrass condition involving minors, are formulated in a special, rather a model case when the potential has a polyconvex quasiconvexification.

Riemannian metrics on 2D-manifolds related to the Euler−Poinsot rigid body motion

Bernard Bonnard, Olivier Cots, Jean-Baptiste Pomet, Nataliya Shcherbakova (2014)

ESAIM: Control, Optimisation and Calculus of Variations

The Euler−Poinsot rigid body motion is a standard mechanical system and it is a model for left-invariant Riemannian metrics on SO(3). In this article using the Serret−Andoyer variables we parameterize the solutions and compute the Jacobi fields in relation with the conjugate locus evaluation. Moreover, the metric can be restricted to a 2D-surface, and the conjugate points of this metric are evaluated using recent works on surfaces of revolution. Another related 2D-metric on S2 associated to the...

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