# Optimal Control of a Degenerate PDE for Surface Shape

## Abstract

Controlling the shapes of surfaces provides a novel way to direct self-assembly of colloidal particles on those surfaces and may be useful for material design. This motivates the investigation of an optimal control problem for surface shape in this paper. Specifically, we consider an objective (tracking) functional for surface shape with the prescribed mean curvature equation in graph form as a state constraint. The control variable is the prescribed curvature. We prove existence of an optimal control, and using improved regularity estimates, we show sufficient differentiability to make sense of the first order optimality conditions. This allows us to rigorously compute the gradient of the objective functional for both the continuous and discrete (finite element) formulations of the problem. Numerical results are shown to illustrate the minimizers and optimal controls on different domains.

- Authors:

- George Mason University, Department of Mathematical Sciences (United States)
- Louisiana State University, Department of Mathematics and Center for Computation and Technology (CCT) (United States)

- Publication Date:

- OSTI Identifier:
- 22749762

- Resource Type:
- Journal Article

- Journal Name:
- Applied Mathematics and Optimization

- Additional Journal Information:
- Journal Volume: 78; Journal Issue: 2; Other Information: Copyright (c) 2018 Springer Science+Business Media, LLC, part of Springer Nature; Article Copyright (c) 2017 Springer Science+Business Media New York; http://www.springer-ny.com; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0095-4616

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; FINITE ELEMENT METHOD; OPTIMAL CONTROL; PARTIAL DIFFERENTIAL EQUATIONS

### Citation Formats

```
Antil, Harbir, and Walker, Shawn W., E-mail: walker@math.lsu.edu.
```*Optimal Control of a Degenerate PDE for Surface Shape*. United States: N. p., 2018.
Web. doi:10.1007/S00245-017-9407-3.

```
Antil, Harbir, & Walker, Shawn W., E-mail: walker@math.lsu.edu.
```*Optimal Control of a Degenerate PDE for Surface Shape*. United States. doi:10.1007/S00245-017-9407-3.

```
Antil, Harbir, and Walker, Shawn W., E-mail: walker@math.lsu.edu. Mon .
"Optimal Control of a Degenerate PDE for Surface Shape". United States. doi:10.1007/S00245-017-9407-3.
```

```
@article{osti_22749762,
```

title = {Optimal Control of a Degenerate PDE for Surface Shape},

author = {Antil, Harbir and Walker, Shawn W., E-mail: walker@math.lsu.edu},

abstractNote = {Controlling the shapes of surfaces provides a novel way to direct self-assembly of colloidal particles on those surfaces and may be useful for material design. This motivates the investigation of an optimal control problem for surface shape in this paper. Specifically, we consider an objective (tracking) functional for surface shape with the prescribed mean curvature equation in graph form as a state constraint. The control variable is the prescribed curvature. We prove existence of an optimal control, and using improved regularity estimates, we show sufficient differentiability to make sense of the first order optimality conditions. This allows us to rigorously compute the gradient of the objective functional for both the continuous and discrete (finite element) formulations of the problem. Numerical results are shown to illustrate the minimizers and optimal controls on different domains.},

doi = {10.1007/S00245-017-9407-3},

journal = {Applied Mathematics and Optimization},

issn = {0095-4616},

number = 2,

volume = 78,

place = {United States},

year = {2018},

month = {10}

}