Evolution Equations In Hilbert Spaces Research Articles

On duality between estimation and control for linear stochastic functional evolution equations in Hilbert spaces

This paper treats a linear least-squares estimation problem in which both the state and observation processes are governed by infinite-dimensional linear stochastic functional evolution equations. The main result obtained in this paper is expressed in terms of a duality principle between the estimation and a linear-quadratic control problem. As a by-product, the representation of the solution process of the equation is also obtained.

An approximation theorem of wong-zakai type for nonlinear stochastic partial differential equations

We present an extension of the Wong-Zakai approximation theorem for nonlinear 984 given by the Wiener process and a martingale. By approximating these disturbances we obtain in the limit equation the Ito correction term for the infinite dimensional case. Such form of the correction term connected with the Wiener process was proved in the author's papers [21] and [22], where the approximation theorem for semilinear stochastic evolution equations in Hilbert spaces was studied. Our model here is similar as the one considered by Pardoux [17]

Semilinear degenerate parabolic systems and distributed capacitance models

A two-scale microstructure model of current flow in a medium with continuously distributed capacitance is extended to include nonlinearities in the conductance across the interface between the local capacitors and the global conducting medium. The resulting degenerate system of partial differential equations is shown to be in the form of a semilinear parabolic evolution equation in Hilbert space. It is shown directly that such an equation is equivalent to a subgradient flow and, hence, displays the appropriate parabolic regularizing effects. Various limiting cases are identified and the corresponding convergence results obtained by letting selected parameters tend to infinity.

On weak solutions of stochastic equations in Hilbert spaces

We consider nonlinear stochastic evolution equation in Hilbert space. A simple new proof of existence of weak solution is given provided C0-semigroup governing the linear part of this equation is compact. Cylindrical noise is allowed. This generalizes previous results of Viot and Metivier.

Existence of periodic solutions for nonlinear evolution equations in Hilbert spaces

In this paper, we consider the existence and multiplicity of periodic solutions of the problem u ′ + A u ∋ g ( t , u ) u’ + Au \ni g(t,u) where A A is a subdifferential of a convex function defined in a Hilbert space H H and g : R × H → H g:R \times H \to H is a Carathéodory function periodic with respect to the first variable.

On uniqueness in law of solutions to stochastic evolution equations in hilbert spaces

Uniqueness in law is proved for a stochastic semilinear evolution equation with additive noise and continuous nonlinearity. We apply here an infinite dimensional version of a method due to Stroock and Varadhan. * This work was done during this author's stay at UNSW supported by

A note on nonlinear stochastic equations in Hilbert spaces

We consider nonlinear stochastic evolution equation in Hilbert space. A simple proof of existence of strong and weak solution is given provided semigroup governing the linear part of this equation is analytic and compact. Cylindrical noise is allowed.

Fubini theorem for anticipating stochastic integrals in Hilbert space

Let (X,l,μ) be a measure space, letW be a cylindrical Hilbert-Wiener process, and letϕ be an anticipating integrable process-valued function onX. We prove, under natural assumptions onϕ, that there exists a measurable version Yx,x eX, of the anticipating integral ofϕ(x) such that the integral ∫x Yxμ(dx) is a version of the anticipating integral of ∫X ϕ(x)μ(dx). We apply this anticipating Fubini theorem to study solutions of a class of stochastic evolution equations in Hilbert space.

An extension of the Wong-Zakai theorem for stochastic evolution equations in Hilbert spaces

In this paper we examine an approximation theorem of the Wong–Zakai type for stochastic evolution equations in a Hilbert space with the noise being the generalized derivative of the Wiener process with values in another Hilbert space. As a consequence of the approximation of the Wiener process we get in the limit equation the Ito correction term for the infinite dimensional case. The obtained result includes the case of stochastic delay equations. The uniqueness and existence of solutions are guaranteed by known theorems for the mild solutions

On the semigroup approach to stochastic evolution equations

Bilinear stochastic evolution equations in Hilbert spaces with unbounded operators are studied by means of the semigroup approach. [BGeneral abstract results of regularity and well posedness are proved and applied to several more concrete problems, like classical variational equations, parabolic equations in non-variational form, non-homogeneous boundary value problems and feedback boundary control problems, and finally some equations which are not of parabolic type

The Dynamic Programming Equation for the Time-Optimal Control Problem in Infinite Dimensions

The existence and uniqueness of a viscosity solution for the Bellman equation associated with the time-optimal control problem for a semilinear evolution equation in Hilbert space is provided. Applications to time-optimal control problems governed by parabolic equations are given.

On some regularity properties of solutions to stochastic evolution equations in Hilbert spaces

On some regularity properties of solutions to stochastic evolution equations in Hilbert spaces

Stability and stabilizability of stochastic evolution equations on Hilbert spaces

The stochastic evolution equations in Hilbert space with state- and control-dependent noise are dealt with. For linear stochastic evolution equations with state-dependent noise, the necessary and sufficient conditions for asymptotic mean-square stability are discussed. For linear and Lure-type non-linear stochastic evolution equations with state- and control-dependent noise, the conditions are derived for which there exists a feedback control law such that the closed-loop system is asymptotically stable in a mean-square sense. Attention is also paid to the case in which there is only state-dependent noise or only control-dependent noise and to the case in which the noise intensity is arbitrarily large.

Multivalued perturbations of subdifferential type evolution equations in Hilbert spaces

In this paper we study the multivalued evolution equation − x ̇ (t) ϵ∂ϑ(x(t)) + F(t, x(t)), x(0) = x 0 , where ϑ: X → R ̄ is a proper, convex, lower semicontinuous (l.s.c.) function, F(·, ·) is a multivalued perturbation, and X is an infinite dimensional, separable Hilbert space. We have an existence result for F(·, ·) being nonconvex valued, and another for F(·, ·) being convex valued but not closed valued. When ϑ = δ K = indicator function of a compact, convex set K, we obtain some extensions of earlier results by Moreau and Henry. Then using the Kuratowski-Mosco convergence of sets and the τ-convergence of functions, we prove a well posedness result for the evolution inclusion we are studying. Also we consider a random version of it and prove the existence of a random solution. Finally we present applications to problems in partial differential equations.

A Free Boundary Value Problem Modeling Thermal Oxidation of Silicon

Using results on evolution equations in Hilbert spaces a problem describing the diffusion of an oxidant through an oxide layer and the growth of this layer caused by the oxidation of silicon is solved. Moreover, estimates for the growth of the thickness of the oxide layer being of practical interest are given.

∗-Solutions of evolution equations in Hilbert space

Equation (1) is linear when g is independent of U. We refer to Pazy [9] and Tanabe [lo] for linear evolution equations with the operators A(t) assumed to be infinitesimal generators of C,-semigroups. A strong solution u of (1) requires u(t) E g(A(t)) for each t. It is possible that there are no strong solutions, but solutions of weaker types exist which do not need the abovementioned requirement (see [9, lo] in the

Minimax regulators for evolution equations in Hilbert spaces, under indeterminacy conditions

Minimax regulators for evolution equations in Hilbert spaces, under indeterminacy conditions

Characteristics functionals of randomly excited physical systems

This paper deals with charateristics functionals of the solutions of a wide class of stochastic differential equations describing random physical processes. First, a construction of the characteristics functional is shown in a general case of the stochastic evolution equation in Hilbert space. Then making use of the construction presented, the analysis of three random physical processes is performed: the random harmonic oscillator, random wave process and random heat conduction. Special attention in these applications is focused on the cases where an excitation process has the form of a sequence of randomly arriving impulses.

Identifiability of Linear Systems in Hilbert Spaces

The identifiability problem is discussed for linear dynamical systems described by first and second order evolution equations in Hilbert spaces. The unknowns are the initial values and the operators appearing in the system equations and a number of identifiability conditions are established within the framework of linear operator theory. These are applied to various classes of partial differential equations on bounded and unbounded spatial domains to obtain the constant and spatially varying parameter identifiability conditions for such systems.

Some results on linear stochastic evolution equations in hilbert spaces by the semi–groups method

(1983). Some results on linear stochastic evolution equations in hilbert spaces by the semi–groups method. Stochastic Analysis and Applications: Vol. 1, No. 1, pp. 57-88.