zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
Stochastic hybrid control. (English) Zbl 0967.93097
The authors consider a complicated version of controlled stochastic systems. The time t is measured continuously. The state of the system is represented by a continuous variable x and a discrete variable n. Also, the control has two parts, a continuous type control v that is a measurable stochastic process and a discrete-type (or impulse) control k that is a sequence of random variables. The key point is the set-interface D of which only the boundary is really used. Minimal and maximal set-interfaces are considered. When the state reaches the minimal set, a mandatory impulse (jump or switch) takes place, while if the state belongs to a maximal set, an optional impulse (jump or switch) may be applied, upon decision of the controller. Switching and jumps can be autonomous or totally controlled. A discounted marginal cost of the form f(x(t),n(t),v(t))exp(- 0 t c(x(s),n(s),v(s))ds) is introduced and a control problem consists in its minimization. The authors demonstrate that the dynamic programming approach leads to some involved quasi-variational inequality. If the system is non-degenerate then the classic treatment can be used for the solution of the control problem, otherwise, a way is to use the so-called viscosity solutions that are described in the last part of the paper.
93E20Optimal stochastic control (systems)
93B12Variable structure systems
49L25Viscosity solutions (infinite-dimensional problems)