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In [[control theory]], a '''controlled invariant subspace''' of the [[state space representation]] of some system is a [[Euclidean subspace|subspace]] such that, if the state of the system is initially in the subspace, it is possible to control the system so that the state is in the subspace at all times. This concept was introduced by Giuseppe Basile and Giovanni Marro {{harv|Basile|Marro|1969}}.

== Definition ==
Consider a linear system described by the differential equation
:<math> \dot{\mathbf{x}}(t) = A\mathbf{x}(t) + B\mathbf{u}(t). </math>
Here, '''x'''(''t'') ∈ '''R'''''n'' denotes the state of the system and '''u'''(''t'') ∈ '''R'''''p'' is the input. The matrices ''A'' and ''B'' have size ''n'' × ''n'' and ''n'' × ''p'' respectively.

A subspace ''V'' ⊂ '''R'''''n'' is a ''controlled invariant subspace'' if for any '''x'''(0) ∈ ''V'', there is an input '''u'''(''t'') such that '''x'''(''t'') ∈ ''V'' for all nonnegative ''t''.

== Properties ==
A subspace ''V'' ⊂ '''R'''''n'' is a controlled invariant subspace if and only if ''AV'' ⊂ ''V'' + Im ''B''. If ''V'' is a controlled invariant subspace, then there exists a matrix ''K'' such that the input '''u'''(''t'') = ''K'''''x'''(''t'') keeps the state within ''V''; this is a simple feedback control {{harv|Ghosh|1985|loc=Thm 1.1}}.

==References==
* {{Citation | last1=Basile | first1=Giuseppe | last2=Marro | first2=Giovanni | title=Controlled and conditioned invariant subspaces in linear system theory | doi=10.1007/BF00931370 | year=1969 | journal=Journal of Optimization Theory and Applications | volume=3 | issue=5 | pages=306–315}}.
* {{Citation | last1=Ghosh | first1=Bijoy K. | title=Proceedings of the 24th IEEE Conference on Decision and Control | publisher=IEEE | doi=10.1109/CDC.1985.268620 | year=1985 | chapter=Controlled invariant and feedback controlled invariant subspaces in the design of a generalized dynamical system | pages=872–873}}.
* {{Citation | last1=Basile | first1=Giuseppe | last2=Marro | first2=Giovanni | title=Controlled and Conditioned Invariants in Linear System Theory | publisher=Englewood Cliffs : Prentice-Hall | year=1992}}.

[[Category:Invariant subspaces]]
[[Category:Operator theory]]
[[Category:Linear algebra]]
[[Category:Control theory]]

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