Optimal control of semilinear elliptic equations with pointwise constraints on the gradient of the state

1993 ◽  
Vol 27 (1) ◽  
pp. 35-56 ◽  
Author(s):  
Eduardo Casas ◽  
Luis Alberto Fern�ndez
Keyword(s):  
Optimal Control ◽  
Elliptic Equations ◽  
The State ◽  
Semilinear Elliptic Equations ◽  
Semilinear Elliptic ◽  
Pointwise Constraints

scholarly journals Bi-objective optimal control of some PDEs: Nash equilibria and quasi-equilibria

2021 ◽  
Author(s):  
Enrique Enrique Fernandez Cara <cara@us. es> ◽  
Irene Marín-Gayte
Keyword(s):  
Optimal Control ◽  
Elliptic Equations ◽  
Nash Equilibria ◽  
Stokes Equations ◽  
Finite Element Approximation ◽  
Semilinear Elliptic Equations ◽  
Navier Stokes ◽  
Element Approximation ◽  
Semilinear Elliptic ◽  
Cost Functionals

This paper deals with the solution of some multi-objective optimal control problems for several PDEs: linear and semilinear elliptic equations and stationary Navier-Stokes systems.  Specifically, we look for Nash equilibria associated with standard cost functionals. For linear and semilinear elliptic equations, we prove the existence of equilibria and we deduce related optimality systems. For stationary Navier-Stokes equations, we prove the existence of Nash quasi-equilibria, i.e. solutions to the optimality system. In all cases, we present some iterative algorithms and, in some of them, we establish convergence results. For the existence and characterization of Nash quasi-equilibria in the Navier-Stokes case, we use the formalism of Dubovitskii and Milyutin. In this context, we also present a finite element approximation and we illustrate the techniques with numerical experiments.

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