Space-time Galerkin POD with application in optimal control of semilinear partial differential equations. (English) Zbl 1392.35323
Summary: In the context of Galerkin discretizations of a partial differential equation (PDE), the modes of the classical method of proper orthogonal decomposition (POD) can be interpreted as the ansatz and trial functions of a low-dimensional Galerkin scheme. If one also considers a Galerkin method for the time integration, one can similarly define a POD reduction of the temporal component. This has been described earlier but not expanded upon – probably because the reduced time discretization globalizes time, which is computationally inefficient. However, in finite-time optimal control systems, time is a global variable and there is no disadvantage from using a POD reduced Galerkin scheme in time. In this paper, we provide a newly developed generalized theory for space-time Galerkin POD, prove its optimality in the relevant function spaces, show its application for the optimal control of nonlinear PDEs, and, by means of a numerical example with Burgers’ equation, discuss the competitiveness by comparing to standard approaches.
MSC:
| 35Q93 | PDEs in connection with control and optimization |
| 65L60 | Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations |
| 49M25 | Discrete approximations in optimal control |
| 65M22 | Numerical solution of discretized equations for initial value and initial-boundary value problems involving PDEs |
| 93C20 | Control/observation systems governed by partial differential equations |
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