Error Estimates for the Optimal Control of a Parabolic Fractional PDE
In this work, we consider the integral definition of the fractional Laplacian and analyze a linear-quadratic optimal control problem for the so-called fractional heat equation; control constraints are also considered. We derive existence and uniqueness results, first order optimality conditions, and regularity estimates for the optimal variables. To discretize the state equation we propose a fully discrete scheme that relies on an implicit finite difference discretization in time combined with a piecewise linear finite element discretization in space. We derive stability results and a novel $$L^2(0,T;L^2(\Omega))$$ a priori error estimate. On the basis of the aforementioned solution technique, we propose a fully discrete scheme for our optimal control problem that discretizes the control variable with piecewise constant functions, and we derive a priori error estimates for it. We illustrate the theory with one- and two-dimensional numerical experiments.
- Research Organization:
- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
- Sponsoring Organization:
- USDOE National Nuclear Security Administration (NNSA); USDOE Laboratory Directed Research and Development (LDRD) Program; Spanish National Fund for Scientific and Technological Development (FONDECYT)
- Grant/Contract Number:
- NA0003525; 11180193
- OSTI ID:
- 1828020
- Report Number(s):
- SAND-2020-14257J; 697280
- Journal Information:
- SIAM Journal on Numerical Analysis, Vol. 59, Issue 2; ISSN 0036-1429
- Publisher:
- Society for Industrial and Applied Mathematics (SIAM)Copyright Statement
- Country of Publication:
- United States
- Language:
- English
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