Large time control and turnpike properties for wave equations

Zuazua E. Large time control and turnpike properties for wave equations
Annual Reviews in Control DOI: 10.1016/j.arcontrol.2017.04.002

Abstract: In the last decades mathematical control theory has been extensively developed to handle various models, including Ordinary and Partial Differential Equations (ODE and PDE), both of deterministic and stochastic nature, discrete and hybrid systems.
However, little attention has been paid to the length of the time horizon of control, which is necessarily long in many applications, and to how it affects the nature of controls and controlled trajectories. The turnpike property refers precisely to those aspects and stresses the fact that, often, optimal controls and trajectories, in long time intervals, undergo some relevant asymptotic simplification property ensuring that, during most of the time-horizon of control, optimal pairs remain close to the steady-state optimal one.
Due to the intrinsic finite velocity of propagation and the oscillatory nature of solutions of the free wave equation, optimal controls for waves are typically of oscillatory nature. But, despite this, as we shall see, under suitable coercivity conditions on the cost functional to be minimised and when controllability holds, the turnpike property is also fulfilled for the wave equation.
When this occurs, the approximation of the time-depending control problem by the steady-state one is justified, a fact that is often employed in applications to reduce the computational cost.
We present some recent results of this nature for the wave equation and other closely related conservative systems, and discuss some other related issues and a number of relevant open problems that arise in this field.

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