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Traffic flow is modeled by a conservation law describing the density of cars. It is assumed that each driver chooses his own departure time in order to minimize the sum of a departure and an arrival cost. There are N groups of drivers, The i-th group consists of κi drivers, sharing the same departure and arrival costs ϕi(t),ψi(t). For any given population sizes κ1,...,κn, we prove the existence of a Nash equilibrium solution, where no driver can lower his own total cost by choosing a different departure...
This paper is concerned with mathematical modelling in the management of a wastewater treatment system. The problem is formulated as looking for a Nash equilibrium of a multiobjective pointwise control problem of a
parabolic equation. Existence of solution is proved and a first order optimality system is obtained. Moreover, a numerical method to solve this system is detailed and numerical results are shown in a realistic situation posed in the estuary of Vigo (Spain).
The goal of this paper is to prove the first and second order optimality conditions for some control problems governed by semilinear elliptic equations with pointwise control constraints and finitely many equality and inequality pointwise state constraints. To carry out the analysis we formulate a regularity assumption which is equivalent to the first order optimality conditions. Though the presence of pointwise state constraints leads to a discontinuous adjoint state, we prove that the optimal...
The goal of this paper is to prove the first and second order
optimality conditions for some control problems governed by
semilinear elliptic equations with pointwise control constraints
and finitely many equality and inequality pointwise state
constraints. To carry out the analysis we formulate a regularity
assumption which is equivalent to the first order optimality
conditions. Though the presence of pointwise state constraints
leads to a discontinuous adjoint state, we prove that the optimal
control...
We present necessary conditions for linear noncooperative N-player delta dynamic games on an arbitrary time scale. Necessary conditions for an open-loop Nash-equilibrium and for a memoryless perfect state Nash-equilibrium are proved.
In the paper we present second-order necessary conditions for constrained vector optimization problems in infinite-dimensional spaces. In this way we generalize some corresponding results obtained earlier.
An optimal control problem is studied
for a Lotka-Volterra system of three differential equations. It
models an ecosystem of three species which coexist. The species
are supposed to be separated from each others. Mathematically,
this is modeled with the aid of two control variables. Some
necessary conditions of optimality are found in order to maximize
the total number of individuals at the end of a given time
interval.
We study a two-dimensional model for micromagnetics, which consists in an energy functional over -valued vector fields. Bounded-energy configurations tend to be planar, except in small regions which can be described as vortices (Bloch lines in physics). As the characteristic “exchange-length” tends to 0, they converge to planar divergence-free unit norm vector fields which jump along line singularities. We derive lower bounds for the energy, which are explicit functions of the jumps of the limit....
We study a two-dimensional model for micromagnetics, which consists in an energy functional over S2-valued vector fields. Bounded-energy configurations tend to be planar, except in small regions which can be described as vortices (Bloch lines in physics). As the characteristic “exchange-length” tends to 0, they converge to planar divergence-free unit norm vector fields which jump along line singularities. We derive lower bounds for the energy, which are explicit functions of the jumps of the limit....
We consider the lower semicontinuous functional of the form where satisfies a given conservation law defined by differential operator of degree one with constant coefficients. We show that under certain constraints the well known Murat and Tartar’s -convexity condition for the integrand extends to the new geometric conditions satisfied on four dimensional symplexes. Similar conditions on three dimensional symplexes were recently obtained by the second author. New conditions apply to quasiconvex...
We consider the lower semicontinuous functional of the form
where u satisfies a given
conservation law defined by differential operator of degree one
with constant coefficients. We show that under certain constraints
the well known Murat and Tartar's Λ-convexity condition
for the integrand f extends to the new geometric conditions
satisfied on four dimensional symplexes. Similar conditions on
three dimensional symplexes were recently obtained by the second
author. New conditions apply...
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