Optimal control of two distributed parameter systems

• Main Author: Williams, Vaughan
• Published: University of Leicester 1972
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.737206

Part I. A heat conduction problem is outlined, where the state of the system under consideration is described by a linear parabolic partial differential equation in one space dimension, with bounded controlling boundary conditions. A cost functional is defined in terms of an integral involving some specified "target function". Integral expressions describing the evolution of the system under the influence of various initial and boundary conditions are given; Butkovskii's Maximum Principle is formally applied to modified forms of these expressions, to give equations determining the optimal control function. Pontryagin's Maximum Principle is applied to a discrete approximation to the distributed system, and a connection is demonstrated between the optimal control thus obtained and that for the distributed system. A further connection is demonstrated with the solution obtained after applying calculus of variations techniques to a notionally unconstrained version of the problem. Difficulties of numerical solution are discussed, and the need for further constraints is pointed out. Solutions to some numerical examples are given. PART II. Equations governing one-dimensional non-turbulent compressible fluid flow in a pipe are given, and the waterhammer problem is described. The method known as "valve stroking" is outlined for a linearized version of this problem. A necessary condition on the optimal solution to the linearized problem is demonstrated, and it is shown that valve stroking does not satisfy this condition, and produces neither a time-optimal nor a mass-flux-optimal solution. A calculus of variations method applied to the non-linear model is shown to produce a non-linear hyperbolic split boundary value problem. A numerical approach, based on iterative gradient methods, is described and some numerical results are presented. Computer programmes for the numerical methods described in parts I and II are included as Appendices.