Stability and Stabilization of Second-order Systems

A new method of constructing Liapunov functions is applied tosystems described by non-linear, second-order, ordinary differentialequations. General results are presented for which such systemsare asymptotically stable and some consideration is given tothe construction of feedback control loops which guarantee thecontrolled system is asymptotically stable. The general theoryis applied to some simple systems and the results compared withknown results.     

A Study of Two of the Classical Problems of Hydrodynamic Stability by the Liapunov Method

The Liapunov method is extended to a function space with a suitablemetric, and is applied to the problem of the stability of ahorizontal layer of fluid when heated from below (BénardProblem), and also to the problem of the flow between rotating,concentric cylinders (Couette Flow). Criteria for stabilityare obtained by ensuring that certain, positive, bounded functionalsshall be non-increasing. The approach enables the partial differentialequations of the system to be dealt with directly, and alsoallows some account to be taken of the inherent non-linearityof the hydrodynamical system.     

The Tracking Problem for Distributed Systems with Unbounded Control Action

The tracking problem for systems defined by integral equations,given in terms of semigroups, is studied where we impose conditionsthat allow unbounded control action to be considered. It isshown that the optimal control is a combination of open-loopcontrol and the feedback control associated with the solutionto the quadratic cost regulator problem.     

A Lower Bound for the Cost Functional for Control Problems in Hilbert Space

Linear quadratic problems for evolution equations in Hilbertspace are considered. Given any control u₀, a lower bound forthe ratio of the cost of performance of any other control tothe cost of u₀ is obtained. The significance of this lower boundis discussed and the results applied to some simple examplesof both ordinary and partial differential equations.     

The Infinite Time Quadratic Cost Control Problem for Distributed Systems with Unbounded Control Action

This paper studies the quadratic cost control problem, overan infinite time interval, for systems defined by integral equationsgiven in terms of semigroups. Conditions are imposed which allowunbounded control action to be considered. It is shown thatsolution to the problem leads to an integral Riccati equationwith unique solution. The integral Riccati equation may be differentiatedand conditions are given under which the differential Riccatiequation also has unique solution.