It is shown that three modes of control, viz., bang-bang, bang-bang with zero and continuous action which may have saturation, cover a wide variety of performance criteria to be minimized in optimal controls of linear systems. For a pure inertial system which starts from an equilibrium state and heads towards the origin in state space, optimal controls for various performance criteria and terminal conditions are determined.

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This paper treats the synthesis of optimum control systems by using Pontryagin's maximum principle. When the magnitude of manipulated variable is limited by saturation, the optimum control system which completes the transient response in minimum time behaves like a relay-servo. Therefore, the important task of the controller is to reverse the sign of the manipulated variable at the proper instants according to the operating condition of the system. A set of points in the phase space at which the sign of the manipulated variable is to be reversed forms a switching surface.
Discussed in this paper are the optimum switching of a linear third-order oscillatory system and the optimum switching curve of a nonlinear control system governed by Van der Pol's equation with additional forcing term.

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Described in this paper is an optimizing control of the generalized process whose equations of motion are represented as the simultaneous difference differential equations. For this purpose, there is given a unified approach to maximize the given functional under many constraint conditions, using the calculus of variation as a powerful tool. An emphasis is laid on the discontinuity of the functional and difference equations, and three types of continuity conditions of corner points on the stationary curve are obtained. Given as an example of the processes including both dead time and discontinuity are sampling control systems and general scheme of optimizing the sampling systems is discussed.
Being difficult to get an analytic solution of the boundary value problem of Euler equation which gives necessary conditions of the process maximum profits, two approximating methods using digital computers are shown.

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An optimizing control system with multi control variables may be considered as a conventional feedback system in which the partial derivatives of the criterion value to control variables are kept at zero. According to the method of measuring the derivatives, optimizing schemes are classified into two groups, one is a perturbation signal method and another trial method. Both methods are essentially identical in the manner of feeding back the derivatives. However, minor differences come from the measurement process. In the former method, the measurement continuously feeds signals so that the system response is quick. However, filters are generally needed to eliminate the mutual interference of multi loops and determine the system response. In the latter method, on the other hand, the measurement is discrete and a sampler takes the place of the filter in the former method. Generally, the derivatives are measured after the criterion value reaches a steady state, so sampling period of the sampler is long. However, both the time constant of the filter and the sampling period can be chosen at any value, depending upon how accurate information is needed.

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Learntrol (abbreviation of learning control) is a new control system which has been developed in the authors' laboratory. Proposed in this paper are Learntrol I & II. The basic theory, result of simulation experiment with a digital computer and discussion on the result are presented.
The phenomenal characteristic of Learntrol II is that the searching time can be amazingly shortened or only two-step search is required by the aid of the jumping search. The theoretical characteristic is that Learntrol II has an ability of insight learning or it consists mainly of two concepts of map and frequency table. Stored in the map is the summarized behaviors in the step-by-step search and in the frequency table (Learntrol I lacks only this table) the frequency of each behavior.
The step-by- step search is divided into such three parts as the direction-sensing search, monotonously hill-climbing search and convergently peak-confirming search, and the insight search is added to each of them. The rest part remained after the step-by-step search is removed from the learning search, is divided into such two parts as the direction-sensing search and jumping search which corresponds to the last two parts of the step-by-step search.
The theory of Learntrol II is widely applicable to all the systems which contain the repeating decisions

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