Dynamic boundary control game with a star of vibrating strings

Sonja Steffensen, Martin Gugat

Consider a star-shaped network of vibrating strings.
Each string is governed by the wave equation.
At the central node, the states are coupled by algebraic node conditions
in such a way that the energy is conserved.
At each boundary node of the network there is
a player that performs Dirichlet boundary control action
and in this way influences the system state.
We consider the corresponding antagonistic game,
where each player minimizes her quadratic objective function
that is the sum of a control cost and
a tracking term for the final state. We show that under
suitable assumptions a unique Nash equilibrium exists
and give an explicit representation of the equilibrium strategies.