We get symbolic and numeric solutions developing a MAPLE® program which uses the initial velocity on the state variable of a wave equation as control function. Solution of this problem implies the minimization at the final time of the distance measured in a suitable norm between the solution of the problem and a given target. An iterative algorithm is constructed to compute the required optimal control as the limit of a suitable subsequence of controls. Results are tested with some numerical examples.

References

References
1.
Gugat
,
M.
,
2008
, “
Optimal Switching Boundary Control of a String to Rest in Finite Time
,”
Z. Angew. Math. Mech.
,
88
(
4
), pp.
283
305
.10.1002/zamm.200700154
2.
Hasanov
,
A.
,
2009
, “
Simultaneous Determination of the Source Terms in a Linear Hyperbolic Problem From the Final Overdetermination: Weak Solution Approach
,”
IMA J. Appl. Math.
,
74
, pp.
1
19
.10.1093/imamat/hxn042
3.
Mordukhovich
,
B. S.
, and
Raymond
,
J. P.
,
2004
, “
Dirichlet Boundary Control of Hyperbolic Equations in the Presence of State Constraints
,”
Appl. Math. Optim.
,
49
, pp.
145
157
.10.1007/s00245-003-0783-5
4.
Yamamoto
,
M.
,
1995
, “
Stability, Reconstruction Formula and Regularization for an Inverse Source Hyperbolic Problem by a Control Method
,”
Inverse Probl.
,
11
, pp.
481
496
.10.1088/0266-5611/11/2/013
5.
Zhang
,
X.
,
Zheng
,
C.
, and
Zuazua
,
E.
,
2009
, “
Time Discrete Wave Equations: Boundary Observability and Control
,”
Discrete Contin. Dyn. Syst.
,
23
(
1&2
), pp.
571
604
.10.3934/dcds.2009.23.571
6.
Lions
,
J. L.
,
1971
,
Optimal Control of Systems Governed by Partial Differential Equations
,
Springer-Verlag
,
New York
.
7.
Kowalewski
,
A.
,
2011
, “
Optimal Control via Initial State of an Infinite Order Time Delay Hyperbolic System
,”
Proceedings of the 18th International Conference on Process Control
,
14–17 June
,
Tatranska Lomnica, Slovakia
.
8.
Subaşı
,
M.
, and
Saraç
,
Y.
,
2012
, “
A Minimizer for Optimizing the Initial Velocity in a Wave Equation
,”
Optimization
,
61
(
3
), pp.
327
333
.10.1080/02331934.2010.511673
9.
Zuazua
,
E.
,
2004
, “
Optimal and Approximate Control of Finite-difference Approximation Schemes for the 1-D Wave Equation
,”
Rendiconti Mat., Ser. VIII, Tomo II
,
24
, pp.
201
237
.
10.
Yang
,
S. D.
,
2006
, “
Shooting Methods for Numerical Solutions of Exact Controllability Problems Constrained by Linear and Semilinear Wave Equations with Local Distributed Controls
,”
Appl. Math. Comput.
,
177
, pp.
128
148
.10.1016/j.amc.2005.10.042
11.
Lin
,
C. H.
,
Bruch
,
Jr.
,
J.
C.
,
Sloss
,
J. M.
,
Adali
,
S.
, and
Sadek
,
I. S.
,
2009
, “
Optimal Multi-interval Control of a Cantilever Beam by a Recursive Control Algorithm
,”
Opt. Control Appl. Methods
,
30
, pp.
399
414
.10.1002/oca.862
12.
Gerdts
,
M.
,
Greif
,
G.
, and
Pesch
,
H. J.
,
2008
, “
Numerical Optimal Control of the Wave Equation: Optimal Boundary Control of a String to Rest in Finite Time
,”
Math. Comput. Simul.
,
79
, pp.
1020
1032
.10.1016/j.matcom.2008.02.014
13.
Ladyzhenskaya
,
O. A.
,
1985
,
Boundary Value Problems in Mathematical Physics
,
Springer
,
New York
.
14.
Subaşı
,
M.
,
2002
, “
An Optimal Control Problem Governed by the Potential of a Linear Schrödinger Equation
,”
Appl. Math. Comput.
,
131–1
, pp.
95
106
.10.1016/S0096-3003(01)00161-8
15.
Subaşı
,
M.
,
2004
, “
A Variational Method of Optimal Control Problems for Nonlinear Schrödinger Equation
,”
Numer. Methods Partial Differ. Equ.
,
20
(
1
), pp.
82
89
.10.1002/num.10081
You do not currently have access to this content.