# Minimum energy control

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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.

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

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

Substitution into the solution

verifies the achievement of state at .