Before revision: 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 Substitution into the solution verifies the achievement of state at .
After revision: 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 Substitution into the solution verifies the achievement of state at .