## Time delay in optimal control loops for wave equations

Department of Mathematics, Friedrich-Alexander Universität Erlangen-Nürnberg,
Cauerstr. 11, 91058 Erlangen, Germany.

martin.gugat@fau.de; guenter.leugering@fau.de

Received:
7
November
2014

Accepted:
7
July
2015

In optimal control loops delays can occur, for example through transmission *via
*digital communication channels. Such delays influence the state that is generated
by the implemented control. We study the effect of a delay in the implementation of
*L*^{2}-norm minimal Neumann boundary controls
for the wave equation. The optimal controls are computed as solutions of problems of exact
optimal control, that is if they are implemented without delay, they steer the system to a
position of rest in a given finite time *T*. We show that arbitrarily small delays
*δ**>*
0 can have a destabilizing effect in the sense that we can find initial
states such that if the optimal control *u* is implemented in the form *y*_{x}(*t,*1) =
*u*(*t* − *δ*) for
*t**>**δ*, the
energy of the system state at the terminal time *T* is almost twice as big as the initial energy. We
also show that for more regular initial states, the effect of a delay in the
implementation of the optimal control is bounded above in the sense that for initial
positions with derivatives of *BV*-regularity and initial velocities with
*BV*-regularity, the terminal energy is bounded above by
the delay *δ*
multiplied with a factor that depends on the BV-norm of the initial data. We show that for
more general hyperbolic optimal exact control problems the situation is similar. For
systems that have arbitrarily large eigenvalues, we can find terminal times
*T* and
arbitrarily small time delays *δ*, such that at the time *T* + *δ*, in
the optimal control loop with delay the norm of the state is twice as large as the
corresponding norm for the initial state. Moreover, if the initial state satisfies an
additional regularity condition, there is an upper bound for the effect of time delay of
the order of the delay with a constant that depends on the initial state only.

Mathematics Subject Classification: 49J20 / 35L53 / 35L05

Key words: PDE constrained optimization / optimal control / delay / wave equation / boundary control / energy / BV-regularity / hyperbolic system / exact controllability with BV-regularity

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