Harbir Antil - Research

Optimal Control for Free Boundary Problems

For a description of PDE constrained optimization problems we refer to: link

A prototype free boundary problem can be found at link

A free boundary problem with curvature flow i.e., a Laplace equation inside the bulk and a Young-Laplace on the free boundary. Further on the free boundary there is an intimate relation between the curvature and surface tension. link (courtesy: Patrick Sodré)

Goal

We consider a PDE-constrained optimization problem governed by a free boundary problem. The state system is based on coupling the Laplace equation in the bulk with a Young-Laplace equation on the free boundary to account for surface tension, as proposed by P. Saavedra and L. R. Scott (see below). This amounts to solving a second order system both in the bulk and on the interface. Our analysis hinges on a convex constraint on the control such that the state constraints are always satisfied. Using only first order regularity we show that the control to state operator is twice Fr\'echet differentiable. We improve slightly the regularity of the state variables and exploit this to show existence of a control together with second order sufficient optimality conditions.

Finally we prove the a-priori optimal order of convergence for the control O(h).

Collaborators

Ricardo Nochetto (University of Maryland)
Patrick Sodré (University of Maryland)

Achievements

We have derived the second order sufficient optimality conditions.
We have proved the optimal order of convergence for the control.
We have derived new regularity results for Stokes problem.
We have provided control of prescribed mean curvature.

Related Publications

H. Antil, and S. W. Walker.
Optimal Control of a Degenerate PDE for Surface Shape.
DOI: 10.1007/s00245-017-9407-3. Applied Mathematics & Optimization, 2017 link

H. Antil, R. H. Nochetto, and P. Sodré.
The Stokes problem with Navier slip boundary condition: Minimal fractional Sobolev regularity of the domain.
Submitted to SIAM Journal on Mathematical Analysis (2015).

H. Antil, R. H. Nochetto, and P. Sodré.
Optimal Control of a Free Boundary Problem: Analysis with Second Order Sufficient Conditions.
SIAM Journal of Control and Optimization (2014), 52(5):2771--2799. arxiv

H. Antil, R. H. Nochetto, and P. Sodré.
Optimal Control of a Free Boundary Problem with Surface Tension Effects: A Priori Error Analysis.
SIAM Journal of Numerical Analysis (2015), 53(5):2279–2306. arxiv

Literature

P. Saavedra, L. R. Scott.
Variational Formulation of a Model Free-Boundary Problem . Mathematics of Computation (1991), 57(196):451--475. link