On an Explicit Solution of the Distributed Algebraic Riccati Equation

An explicit expression for the solution of certain infinite-dimensional Riccati equations is obtained, in the case of self-adjoint operators

Geometric existence theory for the control-affine nonlinear optimal regulator

AbstractFor infinite horizon nonlinear optimal control problems in which the control term enters linearly in the dynamics and quadratically in the cost, well-known conditions on the linearised problem guarantee existence of a smooth globally optimal feedback solution on a certain region of state space containing the equilibrium point. The method of proof is to demonstrate existence of a stable Lagrangian manifold M and then construct the solution from M in the region where M has a well-defined projection onto state space. We show that the same conditions also guarantee existence of a nonsmooth viscosity solution and globally optimal set-valued feedback on a much larger region. The method of proof is to extend the construction of a solution from M into the region where M no-longer has a well-defined projection onto state space

On the Optimal Control of Nonlinear Systems

The lie algebra of tensors on a Hilbert space is used to obtain optimal controls for a class of nonlinear systems

Non Linear Perturbations of Dynamical Systems and Non Linear Controllability

The nonlinear variation of constants formula is generalized to the case where the unperturbed operator has non-elliptic Frechet derivation and is applied to nonlinear controllabilit

The Lie Algebra of a Nonlinear Dynamical System and its Application to Control

Using a recently introduced Lie algebra associated with a nonlinear system and control theory are obtained. In particular, we show that the solutions carry the structure of the associated Lie group. A number of stability and boundedness results are given and a generalisation of classical modal control is developed

Nonautonomous Systems, Lie Algebras and Lyapunov Transformations

An explicit form for the solution of a nonautonomous linear system of differential equations is given by using Duhamels's principle and a generalised Campbell-Haudorff formula. This is applied in the case of a nilpotent generating Lie algebra to Lyapunov transformations

Nonlinear Systems and Kolmogorov's Representation Theorem

A new representation for discrete dynamical systems is presented by applying Kolmogorov's representation theorem to the system functions

Infinite- Dimensional Carleman Linearization, the Lie Series and Optimal Control of Nonlinear PDE's

The Carleman linearization and Lie series techniques are generalized to nonlinear PDE's and applied to nonlinear optimal control theory

Extension of Nonlocal Continuation and Boundedness Theory for Polynomial Systems

In this paper we shall extend a number of results concerning the boundedness and nonlocal continuation of differential equations with sublinear bounds to ones with polynomial vector fields. We shall also show that the solution of many kinds of equations can be obtained as the limit of a sequence of time-varying linear approximations and use this to derive boundedness and stability results. These results directly generalise those of [1]

On Nonlinear Systems and Algebraic Geometry

The theory of linear systems has been developed over many years into a unified collection of results based on the application of linear mathematics. In the state space theory the properties of linear operators have been used to obtain results in controllability, stability etc and in the frequency domain the spectral representation of such operators can be used to generalise classical s-domain methods (see Banks 1983)