An investigation of two semi-linear problems of optimum control
Kennedy, William Joseph, Jr.
Faulkner, Frank D.
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Two problems are presented which are linear on two adjacent intervals but not on their union. These problems are associated with the differential equation X = AX + BF, 0 t < t/cx + DF, t(1) < t < T where X is the matrix (x/y), where F is a 2 x 1 matrix, and where A,B,C, and D are 2 x 2 matrices of functions of t. t(1) is a variable, hence the differential equation is non-linear. Problems associated with this differential equation are called semi-linear. In the first problem, a condition is found on t1 and F which must be satisfied whenever x(T) is to be a maximum with y(T) fixed. In the second problem, conditions on F and t1 are found which must be satisfied for x(T) to be a maximum for a fixed y(T) and for a fixed x(t1). A numerical routine is developed which yields successive approximations to the maximum value of x(T). The basic theory of the methods is presented, and the problems are developed in the context of optimum control.
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