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{{NumBlk|:|<math>\mathbf{x}_{i+1} = \mathbf{f}(\mathbf{x}_i,\mathbf{u}_i)</math>|{{EquationRef|1}}}}
describe the evolution of the state <math>\textstyle\mathbf{x}</math> given the control <math>\mathbf{u}</math> from time <math>i</math> to time <math>i+1</math>. The ''total cost'' <math>J_0</math> is the sum of running costs <math>\textstyle\ell</math> and final cost <math>\ell_f</math>, incurred when starting from state <math>\mathbf{x}</math> and applying the control sequence <math>\mathbf{U} \equiv \{\mathbf{u}_0,\mathbf{u}_1\dots,\mathbf{u}_{N-1}\}</math> until the horizon is reached:
:<math>J_0(\mathbf{x},\mathbf{U})=\sum_{i=0}^{N-1}\ell(\mathbf{x}_i,\mathbf{u}_i) + \ell_f(\mathbf{x}_N),</math>
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