Welcome to the OPTPDE Problem Collection
Keywords: time optimal
Global classification: nonlinear
Functional: nonconvex nonlinear
Geometry: easy, fixed
Design: coupled via volume data
- linear parabolic operator of order 2.
- Defined on a 2-dim domain in 2-dim space
- Time dependent.
- box of order 0
- nonlinear convex, local of order 1
Submitted on 2018-04-20 by Lucas Bonifacius. Published on 2019-03-11
We have a simple example of a time-optimal control problem subject to the linear heat equation and pointwise bound constraints on the control. The goal is to steer the heat equation into an -ball centered at some desired state in the shortest time possible by an appropriate choice of the control. The time-optimal control problem can be transformed to a ﬁxed time interval and both versions are given below.
This particular problem utilizes a control function varying in time only. The exact solution is unknown, but numerical values are provided.
The control-action operator is deﬁned as
where and denote the characteristic functions on and .
The state equation is transformed to the reference time interval in order to deal with the variable time horizon; see [Bonifacius et al., 2018a, Section 3.1] for details. Thus, the transformed version of () reads
where the adjoint state is determined by
For the example, no analytical solution is known. However, numerical values from [Bonifacius et al., 2018a, Example 5.2] are provided. The state and adjoint state equations are discretized by means of the discontinuous Galerkin scheme in time (corresponding to a version of the implicit Euler method) and linear ﬁnite elements in space. This scheme is guaranteed to converge with a rate with denoting the temporal mesh size and the spatial mesh size; cf. [Bonifacius et al., 2018a, Corollary 4.16]. For further details on the implementation we refer to [Bonifacius et al., 2018a, Section 5].
The following table provides results for [Bonifacius et al., 2018a, Example 5.2] and they were provided by the authors for diﬀerent values of the control cost parameter , number of time steps and number of spatial nodes . The analysis for the case can be found in Bonifacius et al. [2018b].
L. Bonifacius, K. Pieper, and B. Vexler. A priori error estimates for space-time ﬁnite element discretization of parabolic time-optimal control problems. ArXiv e-prints, February 2018a. URL https://arxiv.org/abs/1802.00611.
L. Bonifacius, K. Pieper, and B. Vexler. Error estimates for space-time discretization of parabolic time-optimal control problems with bang-bang controls. ArXiv e-prints, September 2018b. URL https://arxiv.org/abs/1809.04886.