Semi-Infinite Programming in Control
Optimal control problems represent a special class of optimization problems which describe dynamical processes. There is a wide range of applications for optimal control problems in engineering and economics. A few of these problems are described and it is shown how they are related and lead to semi-infinite programming problems.
Among the vast literature in the area of semi-infinite programming we want to point out two review articles which include also the aspect of optimal control problems: Polak  reviews engineering applications of semi-inifinite programming and Hettich and Kortanek  give an overview of applications, algorithms and theory in this area.
The dynamical system can be given by a system of difference equations, ordinary differential equations or partial differential equations. With a proper objective function this often leads to optimization problems in function spaces.
A general class of optimal control problems for ordinary differential equations is presented. The controls and the states have to satisfy pointwise bounds. It is shown that a discretization of the control space leads to a semi-infinite programming problem.
A heat conduction process in food industry coupled with the decay of microorganisms is discussed as an important application of optimal control problems. This process, however, also leads to reduction in the vitamines. The optimal control problem consists in finding numerically a control which optimizes the loss of vitamines and satisfies the requirement of achieved sterility for the product. The mathematical model is described by a system of nonlinear partial differential equations. After discretization one obtains a semi-infinite programming problem with nonlinear constraints.
Another application of semi-infinite programming occurs in the control of flutter of an aircraft wing. The avoidance of flutter has to be guaranteed over a certain velocity range at which the aircraft operates. A proper formulation of this problem leads to a semi-infinite programming problem. The dependence of the functions on the indices describing the constraints, however, is not necessarily differentiable and therefore computationally difficult to handle. Therefore, a new approach is presented using the characterization of stability with the Lyapunov equation. This leads to an interesting and novel combination of positive-definite and semi-infinite programming. It is shown how this problem can be solved by a barrier function technique.
KeywordsOptimal Control Problem Optimal Design Problem Lyapunov Equation Aircraft Wing Fixed Design
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