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We devise multigrid preconditioners for linear-quadratic space-time distributed parabolic optimal control problems. While our method is rooted in earlier work on elliptic control, the temporal dimension presents new challenges in terms of algorithm design and quality. Our primary focus is on the cG(s)dG(r) discretizations which are based on functions that are continuous in space and discontinuous in time, but our technique is applicable to various other space-time finite element discretizations. We construct and analyse two kinds of multigrid preconditioners: the first is based on full coarsening in space and time, while the second is based on semi-coarsening in space only. Our analysis, in conjunction with numerical experiments, shows that both preconditioners are of optimal order with respect to the discretization in case of cG(1)dG(r) for r = 0, 1 and exhibit a suboptimal behaviour in time for Crank–Nicolson. We also show that, under certain conditions, the preconditioner using full space-time coarsening is more efficient than the one involving semi-coarsening in space, a phenomenon that has not been observed previously. Our numerical results confirm the theoretical findings.
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Analysis of Theoretical and Numerical Properties of Sequential Convex Programming for Continuous-Time Optimal Control
Sequential Convex Programming (SCP) has recently gained significant popularity as an effective method for solving optimal control problems and has been successfully applied in several different domains. However, the theoretical analysis of SCP has received comparatively limited attention, and it is often restricted to discrete-time formulations. In this paper, we present a unifying theoretical analysis of a fairly general class of SCP procedures for continuous-time optimal control problems. In addition to the derivation of convergence guarantees in a continuous-time setting, our analysis reveals two new numerical and practical insights. First, we show how one can more easily account for manifold-type constraints, which are a defining feature of optimal control of mechanical systems. Second, we show how our theoretical analysis can be leveraged to accelerate SCP-based optimal control methods by infusing techniques from indirect optimal control.
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- Award ID(s):
- 1931815
- PAR ID:
- 10377626
- Date Published:
- Journal Name:
- IEEE Transactions on Automatic Control
- ISSN:
- 0018-9286
- Page Range / eLocation ID:
- 1 to 16
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1109/TAC.2022.3207865
