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In this paper we have presented various possible behaviors of two nations in an armaments race. The conceptual aspects and the properties of the Cournot and Stackelberg solutions in terms of the information about the armaments behavior of the other nation have been discussed. The static solution is simplest since it requires a myopic behavior on the part of each nation; i.e. each nation adjusts its armaments level according to the current armament level of the other nation and with no consideration to any future goals. The dynamic solutions on the other hand where future goals are incorporated in the decision making at any instant of time requires more assumptions about the behavior of the other nation. Two cases have been discussed in conjunction with the Cournot solution. In the first, each nation predicts the future armament variations of the other nation, while in the second, each nation predicts the future armaments expenditure policy of the other nation as a function of the armaments levels of both nations. From a practical point of view, such solutions would be useful in a situation where a noncooperative equilibrium is to be negotiated (e.g. Middle East conflict) and each nation is asked (or forced) to commit itself to a certain armaments level or expenditure policy over a certain period of time in the future. The nations are in dynamic Cournot equilibrium if the predictions match the optimum quantities. The dynamic Stackelberg solution on the other hand has been discussed for a situation where armaments decisions are obtained as a function of armaments levels at stages in time by a leader nation first and then by the follower nation., Proceedings of NSF Conference on Control Theory in International Relations Research, Indiana Univ., Bloomington, Ind. Apr 75.

Trajectory sensitivity with respect to small random perturbations in initial state and plant parameters is investigated. An upper bound on the maximal spread around the nominal trajectory at time t for a probability level p is established in the paper. If a sufficiently high degree of exponential stability is attained by additional feedback, the modified system has a lower upper bound compared to that for the system without additional feedback. Moreover, when a deterministic observer is employed, the upper bound is increased but it is still lower than that corresponding to the system without additional feedback and without an observer. (Author), Presented at Proceedings of the 1974 IEEE Conference on Decision and Control, 20-22 Nov 74, Phoenix, Arizona.

Nominally equivalent control realizations consisting of linear combinations of open-loop and state feedback control for a linear time-invariant completely controllable plant are considered. Two controls are said to be nominally equivalent if they are equal for nominal values of plant parameters. The main result of the paper is a development of a procedure for determining a semi-closed loop control which is nominally equivalent to a given semi-closed loop control and which is less sensitive to plant parameter perturbations than the given control. The given semi-closed loop control realization includes open-loop and closed-loop realizations as special cases. Sensitivity reduction is achieved by specifying the change in the feedback matrix in terms of the solution of an algebraic matrix Riccati equation. The corresponding open-loop portion of the control is changed to maintain nominal equivalence. The restructured semi-closed loop system is always stable. When the nominal control is optimal, the sensitivity reduction described in the paper generalizes the one given by Kreindler for optimal closed-loop control compared to the optimal open-loop control. (Author), Sponsored in part by Grant NSF-GK-3893.

It is well-known that the Nash equilibrium solution of a two-person nonzero-sum linear differential game with a quadratic cost function can be expressed in terms of the solution of coupled generalized Riccati-type matrix differential equations. For high order games the numerical determination of the solution of the nonlinear coupled equations may be difficult or even not possible when the application dictates the use of small memory computers. In this paper a series solution is suggested by means of a parameter imbedding method. Instead of solving a high order Riccati matrix equation a lower order matrix Riccati equation corresponding to a zero-sum game is solved. In addition, lower order linear equations have to be solved. These solutions to lower order equations are the coefficients of the series solution for the nonzero-sum game. Cost functions corresponding to truncated solutions are compared with those for exact Nash equilibrium solutions. (Author), Sponsored in part by Grant NSF-GK-3893.

The paper presents a method of control that is near optimum in the presence of small variations of the initial state. The method is demonstrated by considering the problem of fixed-time fuel optimization of a linear system with a bounded control. The control law is implemented using stored data about the nominal control and changes in the optimal switch times due to changes in parameters. Sampling of the state of the system before the first switch time is used to determine the perturbed initial states. A third order example is presented. (Author)

The new approach recently proposed by the authors for treating the parameter variation problem for a class of multivariable systems is extended to a general feedback configuration for multivariable systems. It is shown that the general linear, multivariable, feedback system can be represented by two plant transfer function matrices and three controller transfer function matrices. A formula for the sensitivity matrix of the general structure is derived. Also, sufficient conditions are obtained for assuring a feedback design that is less affected by parameter variations than another feedback design with the same overall transfer matrix. (Author)