601 research outputs found
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
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
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
The fundamental gap of a kind of two dimensional sub-elliptic operator
This paper is concerned at the minimization fundamental gap problem for a
class of two-dimensional degenerate sub-elliptic operators. We establish
existence results for weak solutions, Sobolev embedding theorem and spectral
theory of sub-elliptic operators. We provide the existence and characterization
theorems for extremizing potentials when is subject to
norm constraint
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
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