DeepAI AI Chat
Log In Sign Up

Optimal Control of the Kirchhoff Equation

by   Masoumeh Hashemi, et al.

We consider an optimal control problem for the steady-state Kirchhoff equation, a prototype for nonlocal partial differential equations, different from fractional powers of closed operators. Existence and uniqueness of solutions of the state equation, existence of global optimal solutions, differentiability of the control-to-state map and first-order necessary optimality conditions are established. The aforementioned results require the controls to be functions in H^1 and subject to pointwise upper and lower bounds. In order to obtain the Newton differentiability of the optimality conditions, we employ a Moreau-Yosida-type penalty approach to treat the control constraints and study its convergence. The first-order optimality conditions of the regularized problems are shown to be Newton diffentiable, and a generalized Newton method is detailed. A discretization of the optimal control problem by piecewise linear finite elements is proposed and numerical results are presented.


Error estimates for a pointwise tracking optimal control problem of a semilinear elliptic equation

We consider a pointwise tracking optimal control problem for a semilinea...

Deep learning as optimal control problems: models and numerical methods

We consider recent work of Haber and Ruthotto 2017 and Chang et al. 2018...

Optimal control of a quasilinear parabolic equation and its time discretization

In this paper we discuss the optimal control of a quasilinear parabolic ...

Image sequence interpolation using optimal control

The problem of the generation of an intermediate image between two given...

Computing Optimal Control of Cascading Failure in DC Networks

We consider discrete-time dynamics, for cascading failure in DC networks...

Optimal Control of Hughes' Model for Pedestrian Flow via Local Attraction

We discuss the control of a human crowd whose dynamics is governed by a ...

Optimal Control of Sliding Droplets using the Contact Angle Distribution

Controlling the shape and position of moving and pinned droplets on a so...