UBC Theses and Dissertations
The sensitivity of optimal value functions in differential inclusion problems Loewen, Philip Daniel
In many practical situations, the parameters of a control system are not known exactly. For this reason, much recent literature is concerned with the sensitivity of dynamic optimization problems to small perturbations of their basic character. This thesis discusses the sensitivity of a general differential inclusion problem to perturbations of its objective function, dynamic constraint, and endpoint conditions. Such perturbations, here introduced by a finite-dimensional vector u, define a Value function V(u) by inducing changes in the problem's optimal value. After Chapter I reviews the calculus of generalized gradients, Chapter II provides a formula for a certain quantitative index of the problem's sensitivity to changes in u: the generalized gradient of V. The basic approach is that used by Clarke in "Optimization and Nonsmooth Analysis" (New York: Wiley-Interscience, 1983)., Theorem 3.4.3. The results presented in Chapter II compare well with Clarke's in the appropriate special case. Chapter III extends the sensitivity analysis of Chapter II to treat free time problems, in which the planning period is a further choice variable. A brief discussion of controllability and an example from mathematical economics constitute Chapter IV.