Page 191 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 191
RATOR SEMIGROUPS AND EVOLUTION EQUATIONS (MS-29)
Observability for Non-Autonomous Systems
Fabian Gabel, fabian.gabel@tuhh.de
Hamburg University of Technology, Germany
Coauthors: Clemens Bombach, Christian Seifert, Martin Tautenhahn
We study non-autonomous abstract Cauchy problems
x˙ (t) = A(t)x(t) , y(t) = C(t)x(t) , t > 0 , x(0) = x0 ∈ X ,
where A(t) : D(A) → X is a strongly measurable family of operators on a Banach space X
and C(t) ∈ L(X, Y ) is a family of bounded observation operators from X to a Banach space
Y.
For measurable subsets E ⊆ (0, T ), T > 0, we provide sufficient conditions such that the
Cauchy problem satisfies a final state observability estimate
1/r
x(T ) X y(t) r dt , r ∈ [1, ∞) ,
Y
E
where an analogous estimate holds for the case r = ∞.
An application of the above result to families of strongly elliptic differential operators A(t)
and observation operators
C(t)u := 1Ω(t)u , Ω(t) ⊆ Rd , u ∈ Lp(Rd) ,
is presented. In this setting, we give sufficient and necessary geometric conditions on the family
of sets (Ω(t)) such that the corresponding Cauchy problem satisfies a final state observability
estimate.
Chernoff approximation of operator semigroups and applications
Yana Kinderknecht (Butko), yanabutko@yandex.ru
Saarland University, Germany
We present a method to approximate operator semigroups with the help of the Chernoff the-
orem. We discuss different approaches to construct Chernoff approximations for semigroups,
generated by Markov processes, and for Schrödinger groups. This method provides simultane-
ousely some numerical schemes for PDEs and pseudo-differential equations (in particular, the
operator splitting method), Euler–Maruyama schemes for the corresponding SDEs and other
Markov chain approximations to the corresponding Markov processes, can be understood as a
numerical path integration method. In some cases, Chernoff approximations have the form of
limits of n iterated integrals of elementary functions as n → ∞ (in this case, they are called
Feynman formulae) and can be used for direct computations and simulations of stochastic pro-
cesses. The limits in Feynman formulae sometimes coincide with (or give rise to) path integrals
with respect to probability measures (such path integrals are usually called Feynman-Kac for-
mulae) or with respect to Feynman type pseudomeasures. Therefore, Feynman formulae can be
used to approximate the corresponding path integrals and to establish relations between differ-
ent path integrals.
In this talk, we discuss Chernoff approximations for semigroups generated by Feller pro-
cesses in Rd. We are also interested in constructing Chernoff approximations for semigroups,
189
Observability for Non-Autonomous Systems
Fabian Gabel, fabian.gabel@tuhh.de
Hamburg University of Technology, Germany
Coauthors: Clemens Bombach, Christian Seifert, Martin Tautenhahn
We study non-autonomous abstract Cauchy problems
x˙ (t) = A(t)x(t) , y(t) = C(t)x(t) , t > 0 , x(0) = x0 ∈ X ,
where A(t) : D(A) → X is a strongly measurable family of operators on a Banach space X
and C(t) ∈ L(X, Y ) is a family of bounded observation operators from X to a Banach space
Y.
For measurable subsets E ⊆ (0, T ), T > 0, we provide sufficient conditions such that the
Cauchy problem satisfies a final state observability estimate
1/r
x(T ) X y(t) r dt , r ∈ [1, ∞) ,
Y
E
where an analogous estimate holds for the case r = ∞.
An application of the above result to families of strongly elliptic differential operators A(t)
and observation operators
C(t)u := 1Ω(t)u , Ω(t) ⊆ Rd , u ∈ Lp(Rd) ,
is presented. In this setting, we give sufficient and necessary geometric conditions on the family
of sets (Ω(t)) such that the corresponding Cauchy problem satisfies a final state observability
estimate.
Chernoff approximation of operator semigroups and applications
Yana Kinderknecht (Butko), yanabutko@yandex.ru
Saarland University, Germany
We present a method to approximate operator semigroups with the help of the Chernoff the-
orem. We discuss different approaches to construct Chernoff approximations for semigroups,
generated by Markov processes, and for Schrödinger groups. This method provides simultane-
ousely some numerical schemes for PDEs and pseudo-differential equations (in particular, the
operator splitting method), Euler–Maruyama schemes for the corresponding SDEs and other
Markov chain approximations to the corresponding Markov processes, can be understood as a
numerical path integration method. In some cases, Chernoff approximations have the form of
limits of n iterated integrals of elementary functions as n → ∞ (in this case, they are called
Feynman formulae) and can be used for direct computations and simulations of stochastic pro-
cesses. The limits in Feynman formulae sometimes coincide with (or give rise to) path integrals
with respect to probability measures (such path integrals are usually called Feynman-Kac for-
mulae) or with respect to Feynman type pseudomeasures. Therefore, Feynman formulae can be
used to approximate the corresponding path integrals and to establish relations between differ-
ent path integrals.
In this talk, we discuss Chernoff approximations for semigroups generated by Feller pro-
cesses in Rd. We are also interested in constructing Chernoff approximations for semigroups,
189
