zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Fractional Hamilton-Jacobi equation for the optimal control of nonrandom fractional dynamics with fractional cost function. (English) Zbl 1111.49014
Summary: By using the variational calculus of fractional order, one derives a Hamilton-Jacobi equation and a Lagrangian variational approach to the optimal control of one-dimensional fractional dynamics with fractional cost function. It is shown that these two methods are equivalent, as a result of the Lagrange’s characteristics method (a new approach) for solving nonlinear fractional partial differential equations. The key of this results is the fractional Taylor’s series f(x+h)=E α (h α D α )f(x) where E α (·) is the Mittag-Leffler function.
MSC:
49K15Optimal control problems with ODE (optimality conditions)
90C39Dynamic programming
60G18Self-similar processes
49L20Dynamic programming method (infinite-dimensional problems)
49K20Optimal control problems with PDE (optimality conditions)
References:
[1]L. Decreusefond and A.S. Ustunel,Stochastic analysis of the fractional Brownian motion, Potential Anal., 1999(10), 177–214
[2]T.E. Duncan, Y. Hu and B. Pasik-Duncan,Stochastic calculus for fractional Brownian motion, I. Theory, SIAM J. Control Optim. 38 (2000) 582–612 · Zbl 0947.60061 · doi:10.1137/S036301299834171X
[3]Y. Hu and B. Øksendal,Fractional white noise calculus and applications to finance, Infinite Dim. Anal. Quantum Probab. Related Topics 6 (2003), 1–32 · Zbl 1045.60072 · doi:10.1142/S0219025703001110
[4]G. Jumarie,Stochastic differential equations with fractional Brownian motion input, Int. J. Syst. Sc., 1993(6) 1113–1132
[5]G. Jumarie,Fractional Brownian motions via random walk in the complex plane and via fractional derivative. Comparison and further results on their Fokker-Planck equations, Chaos, Solitons and Fractals, 2004 (4), 907–925
[6]G. Jumarie,On the representation of fractional Brownian motion as an integral with respect to (dt)α, Applied Mathematics Letters, 2005 (18), 739–648
[7]G. Jumarie,On the solution of the stochastic differential equation of exponential growth driven by fractional Brownian motion, Applied Mathematics Letters, 2005 (18), 817–826
[8]G. Jumarie,A non-random variational approach to stochastic linear quadratic Gaussian optimization involving fractional noises (FLQG), Journal of Applied Mathematics and Computing, 2005 (1–2), 19–32
[9]G. Jumarie,Merton’s model of optimal portfolio in a Black-Scholes market driven by a fractional Brownian motion with short-range dependence, Insurance : Mathematics and Economics, to appear
[10]H. Kober,On fractional integrals and derivatives, Quart. J. Math. Oxford, 1940(11), 193–215
[11]A.V. Letnivov,Theory of differentiation of fractional order, Math. Sb., 1868(3), 1–7
[12]J. Liouville,Sur le calcul des differentielles indices quelconques (in french) J. Ecole Polytechnique, 1832(13), 71
[13]B.B. Mandelbrot and J.W. van Ness,Fractional Brownian motions, fractional noises and applications, SIAM Rev., 1968(10), 442–437
[14]B.B. Mandelbrot and R. Cioczek-Georges,A class of micropuls es and antipersistent fractional Brownian motions, Stochastic Processes and their Applications, 1995(60), 1–18
[15]B.B. Mandelbrot and R. Cioczek-Georges,Alternative micropulses and fractional Brownian motion, Stochastic Processes and their Applications, 1996(64), 143–152
[16]T. J. Osler,Taylor’s series generalized for fractional derivatives and applications, SIAM. J. Mathematical Analysis, 1971(2), No 1, 37–47