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Keywords: analytic solution
Global classification: convex
Functional: convex nonlinear
Geometry: easy, fixed
Design: coupled via boundary values 0th order
- linear hyperbolic operator of order 2.
- Defined on a 1-dim domain in 1-dim space
- Time dependent.
- box of order 0
Submitted on 2013-02-07 by Roland Herzog. Published on 2013-02-12
This is an exact controllability problem for the one-dimensional wave equation by Dirichlet boundary actuation. Such problems were thorougly analyzed in Gugat et al. , where the present problem appears as Example 1.
The given data is chosen in a way which admits an analytic solution.
This pair of optimal controls may not be unique, but it is the unique one with minimal norm. The corresponding displacement is piecewise linear, and the optimal velocity is piecewise constant on areas bounded by characteristic curves of the equation . Figure 0.1 displays the optimal controls. A plot of the optimal displacement is provided in [Gugat et al., 2005, Figure 2.1].
M. Gugat, G. Leugering, and G. Sklyar. -optimal boundary control for the wave equation. SIAM Journal on Control and Optimization, 44(1):49–74, 2005. ISSN 0363-0129. doi: 10.1137/S0363012903419212.