For the last 30 years the optimization of nonsingular control problems has been an Important part of control engineering, and its mathematical theory is well developed and widely known. On the other hand, singular control problems prove more difficult to analyse and—although necessary conditions for optimality of singular controls have been established over the past decade—It is only recently that sufficient, and necessary and sufficient, conditions have been formulated. The purpose of this book Is to collect together all known results in optimal control theory (as well as appropriate computational methods) which can be applied to the singular problems In optimal control and which up to now have been scattered In numerous journals. Complete and self-contained, the volume begins with an historical survey of singular control problems and leads to the presentation of important, recent results in the field. There are specific real-world applications and the authors discuss those avenues of research which require further Investigation. All those involved In the optimization of dynamical systems will welcome the publication of this book. In addition to advanced students, lecturers and research workers in universities, this will include practising mechanical, chemical and electrical engineers, builders, textile technologists, paper scientists and chemists, and many concerned with non-technical fields such as economics and business management Contents An historical survey of singular control problems Introduction. Singular control in space navigation. Method of Mlele via Green's theorem. Linear systems—quadratic cost Necessary conditions for singular optimal control. Sufficient conditions and necessary and sufficient conditions for optimality. References. Fundamental concepts Introduction. The general optimal control problem. The first variation of J. The second variation of J. A singular control problem. References. Necessary conditions for singular optimal control Introduction. The generalized Legendre-Clebsch condition. Jacobson's necessary condition. References. Sufficient conditions and necessary and sufficient conditions tor non-negativity of nonsingular and singular second variations Introduction. Preliminaries. The nonsingular case. Strong positivlty and the totally singular second variation. A general sufficiency theorem for the second variation. Necessary and sufficient conditions for non-negativity of the totally singular second variation. Necessary conditions for optimality. Other necessary and sufficient conditions. Sufficient conditions for a weak local minimum. Existence conditions for the matrix Rlccati differential equation. Conclusion. References. Computational methods for singular control problems Introduction. Computational application of the sufficiency conditions of theorems in the previous chapter. Computation of optimal singular controls. Joining of optimal singular and non-singular sub-arcs. Conclusion. References. Conclusion The Importance of singular optimal control problems. Necessary conditions. Necessary and sufficient conditions. Computational methods. Switching conditions. Outlook for the future Author index. Sublect index.

Singular optimal control problems

The problem of the decentralized robust control for a class of large-scale interconnected nonlinear dynamical systems with input interconnection and external interconnection perturbations is considered. Based on the stabilizability of each nominal isolated subsystem (i.e., the isolated subsystem in the absence of interconnection perturbations), a class of decentralized local state feedback controllers is proposed, and some sufficient conditions are derived by making use of the Lyapunov stability criterion such that uncertain large-scale interconnected systems can be stabilized asymptotically by these decentralized state feedback controllers. For large-scale systems with only input interconnection perturbations, such decentralized controllers become a class of decentralized stabilizing state feedback controllers. That is, the decentralized stability of such large-scale systems can be guaranteed always by using the decentralized state feedback controllers proposed in the paper. Finally, a numerical example is given to demonstrate the validity of the results.

Decentralized robust control for a class of uncertain large-scale interconnected nonlinear dynamical systems

The decentralized stabilization problem of large-scale systems with structured time-varying uncertainties is considered. A new sufficient condition is established for asymptotically stabilizing the large-scale perturbed systems using decentralized state feedback controllers. We also consider the use of optimal Perron weightings to reduce conservatism and the extension of the results to discrete-time large-scale systems with structured uncertainties. An illustrative example is given to demonstrate that the new sufficient condition is less conservative than the one reported recently. Another example is also given to illustrate the use of the sufficient condition for stability robustness of discrete-time large-scale systems.

Decentralized stabilization of large-scale systems with structured uncertainties

A singular optimal stochastic control problem is studied. A
second-order maximum principle is presented. The second-order
adjoint processes are involved, though the diffusion of the control
system is control independent. The range theorem of vector-valued
measures is used to prove the maximum principle. Examples are given
to illustrate the applications.

https://www.aimsciences.org/article/exportPdf?id=94a442f6-2e16-435c-8974-b644441cff43

A second-order maximum principle for singular optimal stochastic controls

Robustness of disk-stability for perturbed large-scale systems

The design problem of decentralized robust controllers is considered for a class of large-scale systems. The system considered in this paper is composed of interconnected subsystems with time-varying uncertainties. For such large-scale uncertain systems, we give a simple und new method whereby we can derive the sufficient conditions for asymptotically stabilizing the overall system using decentralized feedback robust controllers. An illustrative example is given to demonstrate the design procedure of the decentralized feedback robust controllers, in the light of the method developed in this paper.

Decentralized robust control for a class of large-scale interconnected systems with uncertainties

New necessary conditions are derived for optimality of singular controls (in the sense of Pontryagin's maximum principle) in optimal control problems. We give a method whereby new necessary conditions for optimality of singular controls are derived. As an application of these necessary conditions, an illustrative example is given, which shows that the necessary conditions developed are stronger than the well-known generalized Legendre-Clbesch condition. That is, the singular control in this example satisfies the generalized Legendre-Clebsch condition, but does not satisfy the necessary conditions developed here. The method given here is simple and straightforward. These conditions may be expected to have some applications for singular optimal control problems.

Hansheng Wu

Papers

Decentralized robust control for a class of large-scale interconnected systems with uncertainties

New necessary conditions for optimality of singular controls in optimal control problems