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Keywords: flow control, analytic solution
Global classification: nonlinear-quadratic
Functional: convex quadratic
Geometry: easy, fixed
Design: coupled via initial data
- nonlinear hyperbolic operator of order 1.
- Defined on a 1-dim domain in 1-dim space
- Time dependent.
Submitted on 2013-02-20 by Stefan Ulbrich. Published on 2013-02-24
This is an optimal control problem for entropy solutions of the inviscid one-dimensional Burgers equation considered in Castro et al. . The control acts as initial data and the objective function is a tracking type functional at end time with discontinuous desired state.
This problem appears as [Castro et al., 2008, Section 7, Experiment 1]. Suitable numerical schemes for the Burgers equation are given in [Castro et al., 2008, Section 3]. Note that the state is expected to have shock discontinuities.
Instead of formulating the Burgers equation on , the equation can be restricted to , and then and can be described as constant boundary data at . This leads to a very similar problem with the same optimal solution.
A globally optimal state and control are known analytically:
Note that the value of the objective is zero for this solution. The optimal state contains a shock which moves through the domain but does not reach the boundary of within the time interval .
The performance of a gradient descent method and an alternating descent method in combination with various discretizations of the state equation is described in [Castro et al., 2008, Section 7]. The authors use
as their Experiment 2.
C. Castro, F. Palacios, and E. Zuazua. An alternating descent method for the optimal control of the inviscid Burgers equation in the presence of shocks. Mathematical Models & Methods in Applied Sciences, 18(3):369–416, 2008. doi: 10.1142/S0218202508002723.