## wavebnd1 details:

Keywords: analytic solution

Global classification: convex

Functional: convex nonlinear

Geometry: easy, fixed

Design: coupled via boundary values 0th order

Differential operator:

• Wave:
• linear hyperbolic operator of order 2.
• Defined on a 1-dim domain in 1-dim space
• Time dependent.

Design constraints:

• none

State constraints:

• box of order 0

Mixed constraints:

• none

Submitted on 2013-02-07 by Roland Herzog. Published on 2013-02-12

## wavebnd1 description:

### Introduction

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. [2005], where the present problem appears as Example 1.

### Variables & Notation

#### Given Data

The given data is chosen in a way which admits an analytic solution.

### Optimality System

An optimality system is not provided in Gugat et al. [2005] but rather the exact solution is constructed analytically.

### Supplementary Material

An optimal control can be derived analytically from [Gugat et al.2005, Theorem 2.2]:

 ${f}_{1}\left(T-t\right)=\left\{\begin{array}{cc}\frac{{g}_{1}\left(t\right)+\stackrel{̂}{r}}{4},\phantom{\rule{1em}{0ex}}\hfill & t\in \left[0,0.25\right]\hfill \\ & \\ \frac{{g}_{1}\left(t\right)+\stackrel{̂}{r}}{3},\phantom{\rule{1em}{0ex}}\hfill & t\in \left[0.25,1\right]\hfill \\ & \\ \frac{-{g}_{2}\left(t-1\right)+\stackrel{̂}{r}}{4},\phantom{\rule{1em}{0ex}}\hfill & t\in \left[1,1.25\right]\hfill \\ & \\ \frac{-{g}_{2}\left(t-1\right)+\stackrel{̂}{r}}{3},\phantom{\rule{1em}{0ex}}\hfill & t\in \left[1.25,2\right]\hfill \\ & \\ \frac{{g}_{1}\left(t-2\right)+\stackrel{̂}{r}}{4},\phantom{\rule{1em}{0ex}}\hfill & t\in \left[2,2.25\right]\hfill \\ & \\ \frac{{g}_{1}\left(t-2\right)+\stackrel{̂}{r}}{3},\phantom{\rule{1em}{0ex}}\hfill & t\in \left[2.25,3\right]\hfill \\ & \\ \frac{-{g}_{2}\left(t-3\right)+\stackrel{̂}{r}}{4},\phantom{\rule{1em}{0ex}}\hfill & t\in \left[3,3.25\right]\hfill \end{array}\right\\phantom{\rule{2em}{0ex}}{f}_{2}\left(T-t\right)=\left\{\begin{array}{cc}\frac{{g}_{2}\left(t\right)-\stackrel{̂}{r}}{4},\phantom{\rule{1em}{0ex}}\hfill & t\in \left[0,0.25\right]\hfill \\ & \\ \frac{{g}_{2}\left(t\right)-\stackrel{̂}{r}}{3},\phantom{\rule{1em}{0ex}}\hfill & t\in \left[0.25,1\right]\hfill \\ & \\ \frac{-{g}_{1}\left(t-1\right)-\stackrel{̂}{r}}{4},\phantom{\rule{1em}{0ex}}\hfill & t\in \left[1,1.25\right]\hfill \\ & \\ \frac{-{g}_{1}\left(t-1\right)-\stackrel{̂}{r}}{3},\phantom{\rule{1em}{0ex}}\hfill & t\in \left[1.25,2\right]\hfill \\ & \\ \frac{{g}_{2}\left(t-2\right)-\stackrel{̂}{r}}{4},\phantom{\rule{1em}{0ex}}\hfill & t\in \left[2,2.25\right]\hfill \\ & \\ \frac{{g}_{2}\left(t-2\right)-\stackrel{̂}{r}}{3},\phantom{\rule{1em}{0ex}}\hfill & t\in \left[2.25,3\right]\hfill \\ & \\ \frac{-{g}_{1}\left(t-3\right)-\stackrel{̂}{r}}{4},\phantom{\rule{1em}{0ex}}\hfill & t\in \left[3,3.25\right].\hfill \end{array}\right\$

This pair of optimal controls may not be unique, but it is the unique one with minimal ${L}^{2}\left(0,T\right)$ norm. The corresponding displacement $y$ is piecewise linear, and the optimal velocity ${y}_{t}$ is piecewise constant on areas bounded by characteristic curves of the equation ${y}_{tt}={y}_{xx}$. Figure 0.1 displays the optimal controls. A plot of the optimal displacement is provided in [Gugat et al.2005, Figure 2.1].

### References

M. Gugat, G. Leugering, and G. Sklyar. ${L}^{p}$-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.