Welcome to the OPTPDE Problem Collection
todist1 details:
Keywords: time optimal, analytic solution
Global classification: nonlinear
Functional: nonconvex nonlinear
Geometry: easy, fixed
Design: coupled via volume data
Differential operator:
- Heat:
- linear parabolic operator of order 2.
- Defined on a 2-dim domain in 2-dim space
- Time dependent.
Design constraints:
- box of order 0
State constraints:
- nonlinear convex, local of order 1
Mixed constraints:
- none
Submitted on 2018-04-20 by Lucas Bonifacius. Published on 2019-03-11
todist1 description:
Introduction
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 distributed in space and time. It features an analytically given optimal solution.
The problem has been used as a numerical test in [Bonifacius et al., 2018, Example 5.1].
Variables & Notation
Unknowns
Given Data
Problem Description
() |
The state equation can be transformed to the reference time interval in order to deal with the variable time horizon; see [Bonifacius et al., 2018, Section 3.1] for details. Thus, the transformed version of () reads
() |
Note that the problems () and () are equivalent. The unknowns for the transformed problem () are and .
Optimality System
The first-order necessary optimality conditions for () are formally given as follows: for given local minimizers , , there exists Lagrange multipliers and such that
and the adjoint state is determined by
It can be shown that the above optimality conditions are satisfied in the given example, see, [Bonifacius et al., 2018, Theorem 3.10].
Supplementary Material
For this example, a local optimal solution (with verified second-order sufficient optimality conditions) is known analytically and given in [Bonifacius et al., 2018, Example 5.1]. The optimal state, adjoint state, and control for the transformed problem () are
where the parameters and , the optimal time and multiplier are given by
References
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 2018. URL https://arxiv.org/abs/1802.00611.