On a Successive Approximation Technique in Solving Some Control System Optimization Problems

It 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 control optimization problems given by: Minv J(v)=Minv ∫01g(u.v)dt, where du/dt = h(u, v), h(u, v) linear in u and v, and where u and v are, in general, elements of Banach spaces. In Section 2, 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 3, an example is included to illustrate some of the techniques discussed in the paper.