Peter Falb

Invariants and Canonical Forms under Dynamic Compensation

This paper is concerned with the development of a complete abstract invariant as well as a set of canonical forms under dynamic compensation for linear systems characterized by proper, rational transfer matrices. More specifically, it is shown that one can always associate with any proper rational transfer matrix,

Stochastic differential equations in Hilbert space

T(s)

Time-, fuel-, and energy-optimal control of nonlinear norm-invariant systems

, a special lower left triangular matrix,

A multipoint method of third order

\xi _T (s)

Applications of Algebraic Geometry in System Theory.

, called the interactor. This matrix is then shown to represent an abstract invariant under dynamic compensation which, together with the rank of

Time-Optimal Velocity Control of a Spinning Space Body

T(s)

On cross-correlation bounds and the positivity of certain nonlinear operators

, represents a complete abstract invariant. A set of canonical forms under dynamic compensation is also developed along with appropriate dynamic compensation.

Remarks on the infinite dimensional Riccati equation

Abstract Existence and uniqueness theorems for stochastic evolution equations are developed in a Hilbert space context. The results are based on a blending of the theorems for evolution equations with stochastic integration for Hilbert space valued random processes. The results are applied to stochastically forced parabolic partial differential equations such as have arisen in heat transfer problems.

On a theorem of Bochner

Nonlinear systems of the form \dot{X}(t)=g[x(t);t]+u(t) , where x(t), u(t) , and g[x(t); t] are n vectors, are examined in this paper. It is shown that if \parellelx(t)\parellel = \sqrt{x_{1}^{2}(t) + ... + x_{n}^{2}(t)} is constant along trajectories of the homogeneous system \dot{X}(t)=g[x(t); t] and if the control u(t) is constrained to lie within a sphere of radius M , i.e., \parellelu(t}\parellel \leq M , for all t , then the control u^{\ast}(t)= - Mx(t} /\parellelx(t)\parellel drives any initial state \xi to 0 in minimum time and with minimum fuel, where the consumed fuel is measured by \int \liminf{0} \limsup{T}\parellel u(t) \parelleldt . Moreover, for a given response time T , the control \utilde(t) = -\parellel\xi\parellel x(t)/T \parellel x(t) \parellel drives \xi to 0 and minimizes the energy measured by \frac{1}{2}\int \liminf{0} \limsup{T}\parellelu(t)\parellel^{2}dt . The theory is applied to the problem of reducing the angular velocities of a tumbling asymmetrical space body to zero.

Time optimal control for plants with numerator dynamics

AbstractLetF be a mapping of the Banach spaceX into itself. A convergence theorem for the iterative solution ofF(x)=0 is proved for the multipoint algorithmxn+1=xn−ø(xn), where