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Nash equilibrium for a multiobjective control problem related to wastewater management

Néstor García-Chan, Rafael Muñoz-Sola, Miguel Ernesto Vázquez-Méndez (2009)

ESAIM: Control, Optimisation and Calculus of Variations

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).


Necessary conditions for weak lower semicontinuity on domains with infinite measure

Stefan Krömer (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We derive sharp necessary conditions for weak sequential lower semicontinuity of integral functionals on Sobolev spaces, with an integrand which only depends on the gradient of a scalar field over a domain in ℝ N . An emphasis is put on domains with infinite measure, and the integrand is allowed to assume the value + ∞ .

Necessary Optimality Conditions for a Lotka-Volterra Three Species System

N. C. Apreutesei (2010)

Mathematical Modelling of Natural Phenomena

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.

New a posteriori L ∞ ( L 2 ) and L 2 ( L 2 ) -error estimates of mixed finite element methods for general nonlinear parabolic optimal control problems

Zuliang Lu (2016)

Applications of Mathematics

We study new a posteriori error estimates of the mixed finite element methods for general optimal control problems governed by nonlinear parabolic equations. The state and the co-state are discretized by the high order Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. We derive a posteriori error estimates in L ∞ ( J ; L 2 ( Ω ) ) -norm and L 2 ( J ; L 2 ( Ω ) ) -norm for both the state, the co-state and the control approximation. Such estimates, which seem to be new, are an important...

New classes of analytic and Gevrey semigroups and applications

Angelo Favini, Roberto Triggiani (1993)

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

We consider the operator - A + i ⁢ B on a complex Hilbert space, where A is positive self-adjoint and B is self-adjoint, and where, moreover, « B is comparable to A α , α ≥ 1 », in a technical sense. Two applications are given.

New convexity conditions in the calculus of variations and compensated compactness theory

Krzysztof Chełmiński, Agnieszka Kałamajska (2006)

ESAIM: Control, Optimisation and Calculus of Variations

We consider the lower semicontinuous functional of the form I f ( u ) = ∫ Ω f ( u ) d x 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 to quasiconvex...

New convexity conditions in the calculus of variations and compensated compactness theory

Krzysztof Chełmiński, Agnieszka Kałamajska (2005)

ESAIM: Control, Optimisation and Calculus of Variations

We consider the lower semicontinuous functional of the form I f ( u ) = ∫ Ω f ( u ) d x 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...

New linking theorems

Martin Schechter (1998)

Rendiconti del Seminario Matematico della Università di Padova

New versions on Nikaidô's coincidence theorem

Liang-Ju Chu, Ching-Yan Lin (2002)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

In 1959, Nikaidô established a remarkable coincidence theorem in a compact Hausdorff topological space, to generalize and to give a unified treatment to the results of Gale regarding the existence of economic equilibrium and the theorems in game problems. The main purpose of the present paper is to deduce several generalized key results based on this very powerful result, together with some KKM property. Indeed, we shall simplify and reformulate a few coincidence theorems on acyclic multifunctions,...

Newton and conjugate gradient for harmonic maps from the disc into the sphere

Morgan Pierre (2004)

ESAIM: Control, Optimisation and Calculus of Variations

We compute numerically the minimizers of the Dirichlet energy E ( u ) = 1 2 ∫ B 2 | ∇ u | 2 d x among maps u : B 2 → S 2 from the unit disc into the unit sphere that satisfy a boundary condition and a degree condition. We use a Sobolev gradient algorithm for the minimization and we prove that its continuous version preserves the degree. For the discretization of the problem we use continuous P 1 finite elements. We propose an original mesh-refining strategy needed to preserve the degree with the discrete version of the algorithm (which is a preconditioned...

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