Editing Controllability (section)

== Continuous linear systems ==

Consider the [[continuous time|continuous]] [[linear]] system <ref group="note">A [[linear time-invariant system]] behaves the same but with the coefficients being constant in time.</ref>

: <math>\dot{\mathbf{x}}(t) = A(t) \mathbf{x}(t) + B(t) \mathbf{u}(t)</math>
: <math>\mathbf{y}(t) = C(t) \mathbf{x}(t) + D(t) \mathbf{u}(t).</math>

There exists a control <math>u</math> from state <math>x_0</math> at time <math>t_0</math> to state <math>x_1</math> at time <math>t_1 > t_0</math> if and only if <math>x_1 - \phi(t_0,t_1)x_0</math> is in the [[column space]] of
: <math>W(t_0,t_1) = \int_{t_0}^{t_1} \phi(t_0,t)B(t)B(t)^{T}\phi(t_0,t)^{T} dt</math>
where <math>\phi</math> is the [[state-transition matrix]], and <math>W(t_0,t_1)</math> is the [[Controllability Gramian]].

In fact, if <math>\eta_0</math> is a solution to <math>W(t_0,t_1)\eta = x_1 - \phi(t_0,t_1)x_0</math> then a control given by <math>u(t) = -B(t)^{T}\phi(t_0,t)^{T}\eta_0</math> would make the desired transfer.

Note that the matrix <math>W</math> defined as above has the following properties:
* <math>W(t_0,t_1)</math> is [[symmetric matrix|symmetric]]
* <math>W(t_0,t_1)</math> is [[positive semidefinite matrix|positive semidefinite]] for <math>t_1 \geq t_0</math>
* <math>W(t_0,t_1)</math> satisfies the linear [[matrix differential equation]]
:: <math>\frac{d}{dt}W(t,t_1) = A(t)W(t,t_1)+W(t,t_1)A(t)^{T}-B(t)B(t)^{T}, \; W(t_1,t_1) = 0</math>
* <math>W(t_0,t_1)</math> satisfies the equation
:: <math>W(t_0,t_1) = W(t_0,t) + \phi(t_0,t)W(t,t_1)\phi(t_0,t)^{T}</math><ref>{{cite book|first=Roger W.|last=Brockett|title=Finite Dimensional Linear Systems|publisher=John Wiley & Sons|year=1970|isbn=978-0-471-10585-5}}</ref>