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Averaged control. (English) Zbl 1309.93029

Summary: We analyze the problem of controlling parameter-dependent systems. We introduce the notion of averaged control according to which the quantity of interest is the average of the states with respect to the parameter.
First, we consider the problem of controllability for linear finite-dimensional systems and show that a necessary and sufficient condition for averaged controllability is an averaged rank condition, in the spirit of the classical rank condition for linear control systems, but involving averaged momenta of any order of the matrices generating the dynamics and representing the control action.
We also describe some open problems and directions of possible research, in particular on the average controllability of evolution partial differential equations. In this context we analyze also the averaged version of a classical optimal control problem for a parameter dependent elliptic equation and derive the corresponding optimality system.

MSC:

93B05 Controllability
93B07 Observability
93C15 Control/observation systems governed by ordinary differential equations
93C20 Control/observation systems governed by partial differential equations
93C05 Linear systems in control theory
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