wavebnd1 details:

Keywords: analytic solution

Global classification: convex

Functional: convex nonlinear

Geometry: easy, fixed

Design: coupled via boundary values 0th order

Differential operator:

Design constraints:

State constraints:

Mixed constraints:



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

Unknowns

f1,f2 L(0,T)control variables y state variable

Given Data

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

y0(x) = x 12target displacement y1(x) = 1 target velocity T = 3.25 final time Ω = (0,1) computational domain r̂ = 528 auxiliary  g1(t) = 14 auxiliary function g2(t) = 34 t auxiliary function.

Problem Description

Minimize maxf1L(0,T),f2L(0,T) s.t. ytt = yxx in Ω × (0,T) y(x,0) = 0 in Ω yt(x,0) = 0 in Ω y(x,T) = y0(x)in Ω yt(x,T) = y1(x)in Ω y(0,t) = f1(t)in (0,T) y(1,t) = f2(t)in (0,T).

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]:

f1(Tt) = g1(t)+r̂ 4 , t [0,0.25] g1(t)+r̂ 3 , t [0.25,1] g2(t1)+r̂ 4 ,t [1,1.25] g2(t1)+r̂ 3 ,t [1.25,2] g1(t2)+r̂ 4 , t [2,2.25] g1(t2)+r̂ 3 , t [2.25,3] g2(t3)+r̂ 4 ,t [3,3.25] f2(Tt) = g2(t)r̂ 4 , t [0,0.25] g2(t)r̂ 3 , t [0.25,1] g1(t1)r̂ 4 ,t [1,1.25] g1(t1)r̂ 3 ,t [1.25,2] g2(t2)r̂ 4 , t [2,2.25] g2(t2)r̂ 3 , t [2.25,3] g1(t3)r̂ 4 ,t [3,3.25].

This pair of optimal controls may not be unique, but it is the unique one with minimal L2(0,T) norm. The corresponding displacement y is piecewise linear, and the optimal velocity yt is piecewise constant on areas bounded by characteristic curves of the equation ytt = yxx. Figure 0.1 displays the optimal controls. A plot of the optimal displacement is provided in [Gugat et al.2005, Figure 2.1].


PIC

Figure 0.1: optimal controls

References

   M. Gugat, G. Leugering, and G. Sklyar. Lp-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.