76 Energy Optimization in Process Systems ¨ ¨ resulting from the fundamental N other's theorem (N other, 1918, 1971; Caviglia and Morro, 1987; Caviglia, 1988) are most important results which have con- tributed significantly to our deeper understanding of the meaning of known physical laws. Comparing the continuous and discrete results of dynamic optimization, one may formally regard the continuous algorithm of Maximum Principle, Equa- tions (2.105)­(2.108), as a limit of the discrete result, Equations (2.32)­(2.35), when the number of stages tends to infinity. Yet it should be realized that the discrete Maximum Principle requires stronger assumptions than the continu- ous one. Briefly, the discrete case assumptions must include convexity of rate functions and constraining sets. On the contrary, the sketch of the derivation of the Maximum Principle from the HJB equation (2.82) has been obtained here under severe differentiability assumptions for the HJB solution R(x, t). In fact, in many examples R(x, t) is not differentiable and the HJB equation does not admit classical (smooth) solutions at all. These cases are outlined in the following section. 2.8. VISCOSITY SOLUTIONS AND NON-SMOOTH ANALYSES Hamilton­Jacobi equations are non-linear equations that arise in many fields