Time Delay in Optimal Control Loops for Wave Equations

Gugat, Martin; Leugering, Günter

ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 1, pp. 13-37.

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 $\delta >0$ 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 $y_x(t,1)=u(t-\delta )$ for $t>\delta$, 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 $\delta$ 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 $\delta$, such that at the time $T+\delta$, 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.

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