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### A class of self-concordant functions on Riemannian manifolds.

Balkan Journal of Geometry and its Applications (BJGA)

### A simple proof of the characterization of functions of low Aviles Giga energy on a ball via regularity

ESAIM: Control, Optimisation and Calculus of Variations

The Aviles Giga functional is a well known second order functional that forms a model for blistering and in a certain regime liquid crystals, a related functional models thin magnetized films. Given Lipschitz domain Ω ⊂ ℝ2the functional is ${I}_{}\left(u\right)=\frac{1}{2}{\int }_{\Omega }^{-1}{|1-|Du|}^{2}{|}^{2}+{|{D}^{2}u|}^{2}\mathrm{d}z$ I ϵ ( u ) = 1 2 ∫ Ω ϵ -1 1 − Du 2 2 + ϵ D 2 u 2 d z whereubelongs to the subset of functions in ${W}_{0}^{2,2}\left(\Omega \right)$W02,2(Ω) whose gradient (in the sense of trace) satisfiesDu(x)·ηx = 1 where ηx is the inward pointing unit normal to ∂Ω at x. In [Ann. Sc. Norm. Super. Pisa Cl....

### A simple proof of the characterization of functions of low Aviles Giga energy on a ball via regularity

ESAIM: Control, Optimisation and Calculus of Variations

The Aviles Giga functional is a well known second order functional that forms a model for blistering and in a certain regime liquid crystals, a related functional models thin magnetized films. Given Lipschitz domain Ω ⊂ ℝ2 the functional is ${\mathit{I}}_{\mathit{ϵ}}\mathrm{\left(}\mathit{u}\mathrm{\right)}\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}}{{}^{\mathrm{\int }}}_{\mathit{\Omega }}{\mathit{ϵ}}^{-1}{\left|\mathrm{1}\mathrm{-}{\left|\mathit{Du}\right|}^{\mathrm{2}}\right|}^{\mathrm{2}}\mathrm{+}\mathit{ϵ}{\left|{\mathit{D}}^{\mathrm{2}}\mathit{u}\right|}^{\mathrm{2}}\mathrm{d}\mathit{z}$ where u belongs to the subset of functions in ${\mathit{W}}_{\mathrm{0}}^{\mathrm{2}\mathit{,}\mathrm{2}}\mathrm{\left(}\mathit{\Omega }\mathrm{\right)}$ whose gradient (in the sense of trace) satisfies Du(x)·ηx = 1 where ηx is the inward pointing unit normal ...

### Chattering variational limits of control systems.

Forum mathematicum

### Discrete mechanics and optimal control: An analysis

ESAIM: Control, Optimisation and Calculus of Variations

The optimal control of a mechanical system is of crucial importance in many application areas. Typical examples are the determination of a time-minimal path in vehicle dynamics, a minimal energy trajectory in space mission design, or optimal motion sequences in robotics and biomechanics. In most cases, some sort of discretization of the original, infinite-dimensional optimization problem has to be performed in order to make the problem amenable to computations. The approach proposed in this paper...

### Discrete mechanics and optimal control: An analysis*

ESAIM: Control, Optimisation and Calculus of Variations

The optimal control of a mechanical system is of crucial importance in many application areas. Typical examples are the determination of a time-minimal path in vehicle dynamics, a minimal energy trajectory in space mission design, or optimal motion sequences in robotics and biomechanics. In most cases, some sort of discretization of the original, infinite-dimensional optimization problem has to be performed in order to make the problem amenable to computations. The approach proposed in this paper...

### Limit theorems for a variational problem arising in computer vision

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

### Lower semicontinuity of variational integrals on elliptic complexes

Studia Mathematica

We prove a lower semicontinuity result for variational integrals associated with a given first order elliptic complex, extending, in this general setting, a well known result in the case ${}^{\text{'}}\left(ℝⁿ,ℝ\right){\to }^{\nabla }\text{'}\left(ℝⁿ,ℝⁿ\right){\to }^{curl}\text{'}\left(ℝⁿ,{ℝ}^{n×n}\right)$.

### On accessibility of bilinear systems

Czechoslovak Mathematical Journal

### On fuzzy input data and the worst scenario method

Applications of Mathematics

In practice, input data entering a state problem are almost always uncertain to some extent. Thus it is natural to consider a set ${𝒰}_{\mathrm{a}d}$ of admissible input data instead of a fixed and unique input. The worst scenario method takes into account all states generated by ${𝒰}_{\mathrm{a}d}$ and maximizes a functional criterion reflecting a particular feature of the state solution, as local stress, displacement, or temperature, for instance. An increase in the criterion value indicates a deterioration in the featured quantity....

### On the optimal control of implicit systems

ESAIM: Control, Optimisation and Calculus of Variations

### On the optimal control of implicit systems

ESAIM: Control, Optimisation and Calculus of Variations

In this paper we consider the well-known implicit Lagrange problem: find a trajectory solution of an underdetermined implicit differential equation, satisfying some boundary conditions and which is a minimum of the integral of a Lagrangian. In the tangent bundle of the surrounding manifold X, we define the geometric framework of q-pi- submanifold. This is an extension of the geometric framework of pi- submanifold, defined by Rabier and Rheinboldt for determined implicit differential equations,...

### Partial regularity for anisotropic functionals of higher order

ESAIM: Control, Optimisation and Calculus of Variations

We prove a ${C}^{k,\alpha }$ partial regularity result for local minimizers of variational integrals of the type $I\left(u\right)={\int }_{\Omega }f\left({D}^{k}u\left(x\right)\right)\mathrm{d}x$, assuming that the integrand f satisfies (p,q) growth conditions.

### Partial regularity of minimizers of higher order integrals with (p, q)-growth

ESAIM: Control, Optimisation and Calculus of Variations

We consider higher order functionals of the form$F\left[u\right]=\underset{\Omega }{\int }f\left({D}^{m}u\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}x\phantom{\rule{2.0em}{0ex}}\text{for}\phantom{\rule{4.0pt}{0ex}}u:{ℝ}^{n}\supset \Omega \to {ℝ}^{N},$ where the integrand $f:{⨀}^{m}\left({ℝ}^{n},{ℝ}^{N}\right)\to ℝ$, m≥ 1 is strictly quasiconvex and satisfies a non-standard growth condition. More precisely we assume that f fulfills the (p, q)-growth condition ${\gamma |A|}^{p}\le f\left(A\right)\le {L\left(1+|A|}^{q}\right)\phantom{\rule{2.0em}{0ex}}\text{for}\phantom{\rule{4.0pt}{0ex}}\text{all}\phantom{\rule{4.0pt}{0ex}}A\in {⨀}^{m}\left({ℝ}^{n},{ℝ}^{N}\right),$with γ, L > 0 and $1. We study minimizers of the functional $F\left[·\right]$ and prove a partial ${C}_{\mathrm{loc}}^{m,\alpha }$-regularity result.

### Partial regularity of minimizers of higher order integrals with (p, q)-growth

ESAIM: Control, Optimisation and Calculus of Variations

We consider higher order functionals of the form$F\left[u\right]=\underset{\Omega }{\int }f\left({D}^{m}u\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}x\phantom{\rule{2.0em}{0ex}}\text{for}\phantom{\rule{4.0pt}{0ex}}u:{ℝ}^{n}\supset \Omega \to {ℝ}^{N},$ where the integrand $f:{⨀}^{m}\left({ℝ}^{n},{ℝ}^{N}\right)\to ℝ$, m≥ 1 is strictly quasiconvex and satisfies a non-standard growth condition. More precisely we assume that f fulfills the (p, q)-growth condition ${\gamma |A|}^{p}\le f\left(A\right)\le {L\left(1+|A|}^{q}\right)\phantom{\rule{2.0em}{0ex}}\text{for}\phantom{\rule{4.0pt}{0ex}}\text{all}\phantom{\rule{4.0pt}{0ex}}A\in {⨀}^{m}\left({ℝ}^{n},{ℝ}^{N}\right),$with γ, L > 0 and $1. We study minimizers of the functional $F\left[·\right]$ and prove a partial ${C}_{\mathrm{loc}}^{m,\alpha }$-regularity result.

### Regularity results for an optimal design problem with a volume constraint

ESAIM: Control, Optimisation and Calculus of Variations

Regularity results for minimal configurations of variational problems involving both bulk and surface energies and subject to a volume constraint are established. The bulk energies are convex functions with p-power growth, but are otherwise not subjected to any further structure conditions. For a minimal configuration (u,E), Hölder continuity of the function u is proved as well as partial regularity of the boundary of the minimal set E. Moreover, full regularity of the boundary of the minimal set...

### Sensitivity examination of the simulation result of discrete event dynamic systems with perturbation analysis.

Qüestiió

Simulation completed with perturbation analysis provides a new approach for the optimal control of queuing network type systems. The objective of this paper is to calculate the sensitivity range of finite zero-order perturbation, that is, to determine the maximum and minimum size of perturbation within which zero-order propagation rules can be applied. By the introduction of the concept of virtual queue and first and second level no-input and full-output matrices, an algorithm is provided which...

Metrika

### The symmetry reduction of variational integrals

Mathematica Bohemica

The Routh reduction of cyclic variables in the Lagrange function and the Jacobi-Maupertuis principle of constant energy systems are generalized. The article deals with one-dimensional variational integral subject to differential constraints, the Lagrange variational problem, that admits the Lie group of symmetries. Reduction to the orbit space is investigated in the absolute sense relieved of all accidental structures. In particular, the widest possible coordinate-free approach to the underdetermined...

### The symmetry reduction of variational integrals, complement

Mathematica Bohemica

Some open problems appearing in the primary article on the symmetry reduction are solved. A new and quite simple coordinate-free definition of Poincaré-Cartan forms and the substance of divergence symmetries (quasisymmetries) are clarified. The unbeliavable uniqueness and therefore the global existence of Poincaré-Cartan forms without any uncertain multipliers for the Lagrange variational problems are worth extra mentioning.

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