The relevance of mathematics in control theory is described. The role of mathematics in the early development of control theory is reviewed with the emphasis on the models of controlled objects. The relation between the operational calculus and the classical control theory is discussed. The contact between control theory and mathematics became more and more wider and deeper especially after the emergence of modern control theory in 1960. Some specific aspects of the mathematical problems posed by control theory are reviewed. Possible future directions of the mathematics/control interface is described in conjunction with robustness issues of control systems design.

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A practical and effective technique called "Fast Automatic Differentiation (FAD) " computes the partial derivatives with respect to all the variables of a function as well as an estimate of the rounding error incurred in the computed value of the function. It has a time complexity at most a (universal) constant times as large as that of evaluating the function alone. In large-scale numerical analysis and optimization, we can use FAD instead of the conventional techniques of which the complexity depends upon the number of variables. Moreover, with this technique, we can define a numerically meaningful "norm" to yield a convergence criterion for an iterative method. In the last decade, several researchers have proposed essentially the same techniques as FAD in different contexts so that there are several formalisms by which to get to FAD, e. g., "computational graph", "matrix multiplication" and "Lagrange multipliers". Thus, FAD will become one of the fundamental techniques in numerical computation.

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Recently, a remarkable progress has been made in the field of the global minimization of nonconvex functions over a polytope. The purpose of this article is to survey one of the most successful approaches in this field, namely parametric programming approaches to quasilinear nonconvex minimization problems. The problems to be discussed are: linear multiplicative programming problems, i. e., the minimization of the product of two affine functions; minimization of the sum of two linear fractional functions; minimization of concave quadratic functions and bilinear programming problems. It will be shown that a global minimum of a fairly large scale problems can be obtained efficiently by applying parametric simplex algorithms. Further, it will be shown that a convex multiplicative programming problems, i. e., the minimization of the product of two convex functions, can be solved by parametrizatioh and branch and bound techniques.

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This paper presents a method for control of flood propagating through a reservoir or river channel using the combination of the finite element method and optimal control theory. The numerical model of the flood propagation can be expressed by the linear two-dimensional shallow water equation and the equation can be solved by the two-step explicit finite element method. As the boundary conditions, inflow discharge as the external force of optimal control problem is imposed on upstream site. The control vector is chosen as the outflow discharge, and is imposed on downstream site for the gate operation. Discretized shallow water equation by FEM is used for the state equation of the optimum problem. The performance function is formulated' by water elevation and control discharge. Then introducing a vector of Lagrange multiplier, We define the H function which is so called Hamiltonian. After that, using Euler equation and transversality condition as the necessary conditions for such optimum problem, eventually the two point boundary-value problem for state vector and the multiplier is introduced. To solve this problem, the conjugate gradient method and the Sakawa-Shindo method which minimizes the Hamiltonian directly, are mainly used in this paper. It is shown that the water elevation can be controlled as flat as possible by adjusting the control discharge of the dam gate in numerical examples.

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