523 research outputs found
Observability Inequality of Backward Stochastic Heat Equations for Measurable Sets and Its Applications
This paper aims to provide directly the observability inequality of backward
stochastic heat equations for measurable sets. As an immediate application, the
null controllability of the forward heat equations is obtained. Moreover, an
interesting relaxed optimal actuator location problem is formulated, and the
existence of its solution is proved. Finally, the solution is characterized by
a Nash equilibrium of the associated game problem
Optimal Actuator Location of the Minimum Norm Controls for Stochastic Heat Equations
In this paper, we study the approximate controllability for the stochastic
heat equation over measurable sets, and the optimal actuator location of the
minimum norm controls. We formulate a relaxed optimization problem for both
actuator location and its corresponding minimum norm control into a two-person
zero sum game problem and develop a sufficient and necessary condition for the
optimal solution via Nash equilibrium. At last, we prove that the relaxed
optimal solution is an optimal actuator location for the classical problem
Unique Continuation for Stochastic Heat Equations
We establish a unique continuation property for stochastic heat equations
evolving in a bounded domain . Our result shows that the value of the
solution can be determined uniquely by means of its value on an arbitrary open
subdomain of at any given positive time constant. Further, when is
convex and bounded, we also give a quantitative version of the unique
continuation property. As applications, we get an observability estimate for
stochastic heat equations, an approximate result and a null controllability
result for a backward stochastic heat equation
Observability inequalities for the backward stochastic evolution equations and their applications
The present article delves into the investigation of observability
inequalities pertaining to backward stochastic evolution equations. We employ a
combination of spectral inequalities, interpolation inequalities, and the
telegraph series method as our primary tools to directly establish
observability inequalities. Furthermore, we explore three specific equations as
application examples: a stochastic degenerate equation, a stochastic fourth
order parabolic equation and a stochastic heat equation. It is noteworthy that
these equations can be rendered null controllability with only one control in
the drift term to each system
Quantitative uniqueness estimates for stochastic parabolic equations on the whole Euclidean space
In this paper, a quantitative estimate of unique continuation for the
stochastic heat equation with bounded potentials on the whole Euclidean space
is established. This paper generalizes the earlier results in [29] and [17]
from a bounded domain to an unbounded one. The proof is based on the locally
parabolic-type frequency function method. An observability estimate from
measurable sets in time for the same equation is also derived.Comment: 26 page
Norm and time optimal control problems of stochastic heat equations
This paper investigates the norm and time optimal control problems for
stochastic heat equations. We begin by presenting a characterization of the
norm optimal control, followed by a discussion of its properties. We then
explore the equivalence between the norm optimal control and time optimal
control, and subsequently establish the bang-bang property of the time optimal
control. These problems, to the best of our knowledge, are among the first to
discuss in the stochastic case
Exact Controllability of Linear Stochastic Differential Equations and Related Problems
A notion of -exact controllability is introduced for linear controlled
(forward) stochastic differential equations, for which several sufficient
conditions are established. Further, it is proved that the -exact
controllability, the validity of an observability inequality for the adjoint
equation, the solvability of an optimization problem, and the solvability of an
-type norm optimal control problem are all equivalent
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