Bozenna Pasik-Duncan

FRACTIONAL BROWNIAN MOTION AND STOCHASTIC EQUATIONS IN HILBERT SPACES

In this paper, stochastic differential equations in a Hilbert space with a standard, cylindrical fractional Brownian motion with the Hurst parameter in the interval (1/2,1) are investigated. Existence and uniqueness of mild solutions, continuity of the sample paths and state space regularity of the solutions, and the existence of limiting measures are verified. The equivalence of the probability laws for the solution evaluated at different times and different initial conditions and the convergence of these probability laws to the limiting probability are verified. These results are applied to specific stochastic parabolic and hyperbolic differential equations. The solution of a specific parabolic equation with the fractional Brownian motion only in the boundary condition is shown to have many results that are analogues of the results for a fractional Brownian motion in the domain.

A Kiefer-Wolfowitz algorithm with randomized differences

A Kiefer-Wolfowitz or simultaneous perturbation algorithm that uses either one-sided or two-sided randomized differences and truncations at randomly varying bounds is given in this paper. At each iteration of the algorithm only two observations are required in contrast to 2l observations, where l is the dimension, in the classical algorithm. The algorithm given is shown to be convergent under only some mild conditions. The rate of convergence and asymptotic normality of the algorithm are also established.

Adaptive control of continuous-time linear stochastic systems

An adaptive control problem for linear, continuous-time stochastic systems is described and solved in this paper. A solution of the adaptive control problem means that the family of maximum likelihood estimators is shown to be strongly consistent and the average costs are shown to converge to the optimal average costs. The unknown parameters in the model appear affinely in the drift term of the stochastic differential equation. The assumptions that are made for the solution are natural and verifiable. A recursive equation is given for the maximum likelihood estimates.

Adaptive continuous-time linear quadratic Gaussian control

The adaptive linear quadratic Gaussian control problem, where the linear transformation of the state A and the linear transformation of the control B are unknown, is solved assuming only that (A, B) is controllable and (A, Q/sub 1//sup 1/2/) is observable, where Q/sub 1/ determines the quadratic form for the state in the integrand of the cost functional. A weighted least squares algorithm is modified by using a random regularization to ensure that the family of estimated models is uniformly controllable and observable. A diminishing excitation is used with the adaptive control to ensure that the family of estimates is strongly consistent. A lagged certainty equivalence control using this family of estimates is shown to be self-optimizing for an ergodic, quadratic cost functional.

Adaptive Boundary and Point Control of Linear Stochastic Distributed Parameter Systems

An adaptive control problem for the boundary or the point control of a linear stochastic distributed parameter system is formulated and solved in this paper. The distributed parameter system is modeled by an evolution equation with an infinitesimal generator for an analytic semigroup. Since there is boundary or point control, the linear transformation for the control in the state equation is also an unbounded operator. The unknown parameters in the model appear affinely in both the infinitesimal generator of the semigroup and the linear transformation of the control. Strong consistency is verified for a family of least squares estimates of the unknown parameters. An Ito formula is established for smooth functions of the solution of this linear stochastic distributed parameter system with boundary or point control. The certainty equivalence adaptive control is shown to be self-tuning by using the continuity of the solution of a stationary Riccati equation as a function of parameters in a uniform operator topology. For a quadratic cost functional of the state and the control, the certainty equivalence control is shown to be self-optimizing; that is, the family of average costs converges to the optimal ergodic cost. Some examples of stochastic parabolic problems with boundary control and a structurally damped plate with random loading and point control are described that satisfy the assumptions for the adaptive control problem solved in this paper.

STOCHASTIC INTEGRATION FOR FRACTIONAL BROWNIAN MOTION IN A HILBERT SPACE

A Hilbert space-valued stochastic integration is defined for an integrator that is a cylindrical fractional Brownian motion in a Hilbert space and an operator-valued integrand. Since the integrator is not a semimartingale for the fractional Brownian motions that are considered, a different definition of integration is required. Both deterministic and stochastic operator-valued integrands are used. The approach uses some ideas from Malliavin calculus. In addition to the definition of stochastic integration, an Ito formula is given for smooth functions of some processes that are obtained by the stochastic integration.

Semilinear Stochastic Equations in a Hilbert Space with a Fractional Brownian Motion

The solutions of a family of semilinear stochastic equations in a Hilbert space with a fractional Brownian motion are investigated. The nonlinear term in these equations has primarily only a growth condition assumption. An arbitrary member of the family of fractional Brownian motions can be used in these equations. Existence and uniqueness for both weak and mild solutions are obtained for some of these semilinear equations. The weak solutions are obtained by a measure transformation that verifies absolute continuity with respect to the measure for the solution of the associated linear equation. Some examples of stochastic differential and partial differential equations are given that satisfy the assumptions for the solutions of the semilinear equations.

Adaptive control of linear stochastic evolution systems

An adaptive control problem for some linear stochastic evolution systems in Hilbert spaces is formulated and solved in this paper. The solution includes showing the strong consistency of a family of least squares estimates of the unknown parameters and the convergence of the average quadratic costs with a control based on these estimates to the optimal average cost. The unknown parameters in the model appear affinely in the infinitesimal generator of the C 0 semigroup that defines the evolution system. A recursive equation is given for a family of least squares estimates and the bounded linear operator solution of the stationary Riccati equation is shown to be a continuous function of the unknown parameters in the uniform operator topology

Ergodic Boundary/Point Control of Stochastic Semilinear Systems

A controlled Markov process in a Hilbert space and an ergodic cost functional are given for a control problem that is solved where the process is a solution of a parameter-dependent semilinear stochastic differential equation and the control can occur only on the boundary or at discrete points in the domain. The linear term of the semilinear differential equation is the infinitesimal generator of an analytic semigroup. The noise for the stochastic differential equation can be distributed, boundary and point. Some ergodic properties of the controlled Markov process are shown to be uniform in the control and the parameter. The existence of an optimal control is verified to solve the ergodic control problem. The optimal cost is shown to depend continuously on the system parameter.

Numerical differentiation and parameter estimation in higher-order linear stochastic systems

For a linear time-invariant system of order d/spl ges/2 with a white noise disturbance, the input and the output are assumed to be sampled at regular time intervals. Using only these observations, some approximate values of the first d-1 derivatives are obtained by a numerical differentiation scheme, and the unknown system parameters are estimated by a discretization of the continuous-time least-squares formulas. These parameter estimates have an error which does not approach zero as the sampling interval approaches zero. This asymptotic error is shown to be associated with the inconsistency of the quadratic variation estimate of the white noise local variance based on the sampled observations. The use of an explicit correction term in the least-squares estimates or the use of some special numerical differentiation formulas eliminates the error in the estimates.