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In control theory, the minimum energy control is the control u {\displaystyle u} that will bring a linear time invariant system to a desired state with a minimum expenditure of energy.
Let the linear time invariant system be
with initial state x = x 0 {\displaystyle x=x_{0}}. One seeks an input u {\displaystyle u} so that the system will be in the state x 1 {\displaystyle x_{1}} at time t 1 {\displaystyle t_{1}} , and for any other input u ¯ {\displaystyle {\bar {u}}} , which also drives the system from x 0 {\displaystyle x_{0}} to x 1 {\displaystyle x_{1}} at time t 1 {\displaystyle t_{1}} , the energy expenditure would be larger, i.e.,
To choose this input, first compute the controllability Gramian