Before revision: For a linear time invariant (LTI) control system with initial state , one can pick 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 To choose this input, first compute the controllability gramian The minimum energy control is then If , these expressions simplify to and
After revision: For a linear time invariant (LTI) control system with initial state , one can pick 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 To choose this input, first compute the controllability gramian The minimum energy control is then Substitution into the solution verifies the achievement of state at .