B. Maslowski

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.

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.

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.

Linear-quadratic Control for Stochastic Equations in a Hilbert Space with Fractional Brownian Motions

A linear-quadratic control problem with a finite time horizon for some infinite-dimensional controlled stochastic differential equations driven by a fractional Gaussian noise is formulated and solved. The feedback form of the optimal control and the optimal cost are given explicitly. The optimal control is the sum of the well-known linear feedback control for the associated deterministic linear-quadratic control problem and a suitable prediction of the adjoint optimal system response to the future noise. The covariance of the noise as well as the control operator in the system equation can in general be unbounded, so the results can also be applied where the noise or the control are on the boundary of the domain or at discrete points in the domain. Some examples of controlled stochastic partial differential equations are given.

Some properties of linear stochastic distributed parameter systems with fractional Brownian motion

A fractional Brownian motion with Hurst parameter in the interval ( 1/2 , 1) is used for the Gaussian noise process in a linear stochastic distributed system or a linear stochastic partial differential equation. These noise processes have properties that have been important for finite dimensional systems. The notion of a mild solution is given and some conditions are given for the existence, the uniqueness and the sample path continuity of the solutions. Limiting distributions are given. Stochastic models with boundary noise instead of distributed noise are also considered. Some examples of a stochastic heat equation and a stochastic wave equation are given that satisfy the conditions for the results.

Adaptive boundary control of stochastic linear distributed parameter systems described by analytic semigroups

A stochastic adaptive control problem is formulated and solved for some unknown linear, stochastic distributed parameter systems that are described by analytic semigroups. The control occurs on the boundary. The “highest-order” operator is assumed to be known but the “lower-order” operators contain unknown parameters. Furthermore, the linear operators of the state and the control on the boundary contain unknown parameters. The noise in the system is a cylindrical white Gaussian noise. The performance measure is an ergodic, quadratic cost functional. For the identification of the unknown parameters a diminishing excitation is used that has no effect on the ergodic cost functional but ensures sufficient excitation for strong consistency. The adaptive control is the certainty equivalence control for the ergodic, quadratic cost functional with switchings to the zero control.

Adaptive Control for Semilinear Stochastic Systems

An adaptive, ergodic cost stochastic control problem for a partially known, semilinear, stochastic system in an infinite dimensional space is formulated and solved. The solutions of the Hamilton--Jacobi--Bellman equations for the discounted cost and the ergodic cost stochastic control problems require some special interpretations because they do not typically exist in the usual sense. The solutions of the parameter dependent ergodic Hamilton--Jacobi--Bellman equations are obtained from some corresponding discounted cost control problems as the discount rate tends to zero. The solutions of the ergodic Hamilton--Jacobi--Bellman equations are shown to depend continuously on the parameter. A certainty equivalence adaptive control is given that is based on the optimal controls from the solutions of the ergodic Hamilton--Jacobi--Bellman equations and a strongly consistent family of estimates of the unknown parameter. This adaptive control is shown to achieve the optimal ergodic cost for the known system.

Some aspects of the adaptive boundary and point control of linear distributed parameter systems

An adaptive control problem for the boundary or the point control of a linear stochastic distributed parameter system (DPS) is formulated and its solution is given. The unknown linear stochastic DPS is described by an evolution equation, in which the unknown parameters appear in the infinitesimal generator of an analytic semigroup and in the unbounded linear transformation for the boundary control. An Ito formula can be verified for smooth functions of the solution of the linear stochastic DPS boundary control considered here. The certainty equivalence adaptive control is shown to be self-tuning by noting 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, i.e., the family of average costs converges to the optimal ergodic cost.<<ETX>>

Adaptive boundary control of linear stochastic distributed parameter systems

An adaptive control problem for the boundary or point control of a linear stochastic distributed parameter system is formulated and its solution is described. Strong consistency is verified for a family of least squares estimates of the unknown parameters. The certainty equivalence adaptive control for an ergodic quadratic cost functional is self-optimizing.<<ETX>>

Some Solutions of Semilinear Stochastic Equations in a Hilbert Space With a Fractional Brownian Motion

Stochastic equations in a Hilbert space with a fractional Brownian motion are used to model stochastic partial differential equations with a space-time noise. Some semilinear stochastic equations are shown to possess one and only one weak solution. These weak solutions are constructed from the solutions of the corresponding linear equations by an absolutely continuous transformation of measures. Some examples of stochastic differential and partial differential equations are given to demonstrate the applicability of the results