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This section records for later reference some basic facts about linear systems of ordinary differential equations.
DEFINITION. Let X(.) : R → Mn×n be the unique solution of the matrix ODE
We call X(.) a fundamental solution, and sometimes write
the last formula being the definition of the exponential etM. Observe that
X−1 (t) = X(−t).
THEOREM 1.1 (SOLVING LINEAR SYSTEMS OF ODE).
(i) The unique solution of the homogeneous system of ODE
(ii) The unique solution of the nonhomogeneous system
This expression is the variation of parameters formula.
References
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[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.
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