Minimum energy control
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In control theory the minimum energy control is the control u(t) 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 x(t0) = x0. One seeks an input u(t) so that the system will be in the state x1 at time t1, and for any other input 
, which also drives the system from x0 to x1 at time t1 the energy expenditure would be larger
To choose this input, first compute the controllability gramian
Assuming Wc is nonsingular (if and only if the system is controllable), the minimum energy control is then
![u(t) = -B^*e^{A^*(t_1-t)}W_c^{-1}(t_1)[e^{A(t_1-t_0)}x_0-x_1].](../../../math/d/5/2/d52583116faa1bdbb065f613ae1ed8fa.png)
Substitution into the solution
verifies the achievement of state x1 at t1.
