Abstract
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.
Highlights
We consider bi-objective optimal control problems for various PDEs and systems
We deal with the stationary Navier-Stokes system, that is, the equations satisfied by the time-independent velocity field and pressure of an incompressible viscous fluid flow
We prove the existence of Nash equilibria, we provide an optimality system and we present some iterative algorithms
Summary
We consider bi-objective optimal control problems for various PDEs and systems. Our aims are to prove existence, characterize efficiently the equilibria and, compute numerical solutions to these multi-objective control problems. They are very important from the mathematical viewpoint and appear frequently in the applications; for some previous works on the subject, see for instance [2]. We will apply the so called formalism of Dubovitskii and Milyutin This approach was introduced in the context of mathematical programming and has been succesfully applied to the solution of many optimal control problems for ODEs since the 70’s. We prove the existence of Nash equilibria, we provide an optimality system and we present some iterative algorithms.
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More From: ESAIM: Control, Optimisation and Calculus of Variations
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