In control theory, the minimum energy control is the control
that will bring a linear time invariant system to a desired state with a minimum expenditure of energy.
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Let the linear time invariant (LTI) system be


with initial state
. One seeks an input
so that the system will be in the state
at time
, and for any other input
, which also drives the system from
to
at time
, the energy expenditure would be larger, i.e.,

To choose this input, first compute the controllability Gramian

Assuming
is nonsingular (if and only if the system is controllable), the minimum energy control is then
![{\displaystyle u(t)=-B^{*}e^{A^{*}(t_{1}-t)}W_{c}^{-1}(t_{1})[e^{A(t_{1}-t_{0})}x_{0}-x_{1}].}](//wikimedia.org/api/rest_v1/media/math/render/svg/98bdfa82bd82ab506cfdfd6bf4923c268efa80ab)
Substitution into the solution

verifies the achievement of state
at
.