On a successive approximation technique in solving some control system optimization problems

AbstractIt has been realized for some time that most realistic optimization problems defy analytical solutions in closed forms and that in most cases it is necessary to resort to judicious combinations of analytical and computational procedures to solve problems. For example, in many optimization problems, one is interested in obtaining structural information on optimal and “good” suboptimal policies. Very often, various analytical as well as computational approximation techniques need be employed to obtain clear understandings of structures of policy spaces.The paper discusses a successive approximation technique to construct minimizing sequences for functionals in extremal problems, and the techniques will be applied, to a class of optimization problems, given by: Minυ F(υ) = Minυ ∫01 g(u,υ)dt where dudt = h(u, v), and where u and v are, in general, elements of Banach spaces.In Section II, B, the minimizing sequences are constructed by approximating g(u, v) by appropriate quadratic expressions with linear constraining differential equations. It is shown that under the stated conditions the functional values converge to the minimal value monotonically.In Section III, the restriction that the differential equation be linear is removed. Differential equation are approximated by appropriate linear equations and the technique of Section II, B is applied to obtain minimizing sequences. The convergence of the minimizing sequence to the solution of the original nonlinear differential equation is shown. An example is also included to illustrate some of the techniques