Averaged control |
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| Affiliation: | 1. Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153, Japan;2. Aix Marseille Univ, Université de Toulon, CNRS, CPT, Marseille, France;3. Peoples'' Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation;1. Center for Applied Mathematics, Tianjin University, Tianjin 300072, China;2. School of Mathematics and Physics, China University of Geosciences, Wuhan 430074, China;3. School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China |
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| Abstract: | 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. |
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| Keywords: | Parametrized ODE and PDE Averaged controllability Averaged observability Averaged optimal control |
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