If μ= 0, we can divide by μ and convert to the formulation (1.2). And if ∇g(x∗) = 0, we can take λ = 1, μ = 0, making assertion (1.3) correct (if not particularly useful).
References
[B-CD] M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkhauser, 1997.
[B-J] N. Barron and R. Jensen, The Pontryagin maximum principle from dynamic programming and viscosity solutions to first-order partial differential equations, Transactions AMS 298 (1986), 635–641.
[C1] F. Clarke, Optimization and Nonsmooth Analysis, Wiley-Interscience, 1983.
[C2] F. Clarke, Methods of Dynamic and Nonsmooth Optimization, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, 1989.
[Cr] B. D. Craven, Control and Optimization, Chapman & Hall, 1995.
[E] L. C. Evans, An Introduction to Stochastic Differential Equations, lecture notes avail-able at http://math.berkeley.edu/˜ evans/SDE.course.pdf.
[F-R] W. Fleming and R. Rishel, Deterministic and Stochastic Optimal Control, Springer, 1975.
[F-S] W. Fleming and M. Soner, Controlled Markov Processes and Viscosity Solutions, Springer, 1993.
[H] L. Hocking, Optimal Control: An Introduction to the Theory with Applications, OxfordUniversity Press, 1991.
[I] R. Isaacs, Differential Games: A mathematical theory with applications to warfare and pursuit, control and optimization, Wiley, 1965 (reprinted by Dover in 1999).
[K] G. Knowles, An Introduction to Applied Optimal Control, Academic Press, 1981.
[Kr] N. V. Krylov, Controlled Diffusion Processes, Springer, 1980.
[L-M] E. B. Lee and L. Markus, Foundations of Optimal Control Theory, Wiley, 1967.
[L] J. Lewin, Differential Games: Theory and methods for solving game problems with singular surfaces, Springer, 1994.
[M-S] J. Macki and A. Strauss, Introduction to Optimal Control Theory, Springer, 1982.
[O] B. K. Oksendal, Stochastic Differential Equations: An Introduction with Applications, 4th ed., Springer, 1995.
[O-W] G. Oster and E. O. Wilson, Caste and Ecology in Social Insects, Princeton UniversityPress.
[P-B-G-M] L. S. Pontryagin, V. G. Boltyanski, R. S. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Interscience, 1962.
[T] William J. Terrell, Some fundamental control theory I: Controllability, observability, and duality, American Math Monthly 106 (1999), 705–719.
