Related Optimal Control Problems For Generalized Convexity Functional

Convexity plays an important role in mathematical optimization,which is used in management science,economics and finance.Meanwhile,relaxing the convexity is very necessary in many fields and still yet owns some profitable features of convex optimization.In this dissertation,we investigate both theory and application of optimal control problems with generalized convexity functional,our study is relaxing the convexity to reconsider the related optimal control problem.Chapter 1 depicts some properties on the convex analysis and introduce relevant definitions and theorems in the concept of convexity and generalized convexity.Chapter 2 studies the first-order sufficient optimality conditions for the optimal controls under some convexity assumptions.For the Bolza problem,under concavity of the Hamiltonian and convexity of the cost functional,the extreme control must be optimal control.However,for the Mayer problem,the cost functional can be relaxed to pseudo-convex and quasi-convex.In the context of application,some examples illustrate these theoretical results and some counter examples show that the convexity assumptions of these results cannot be further weakened in some sense.Chapter 3 concerns the extension of the previous results on some optimal control problems which is twofold.The first aim is devoted to study the Mayer-type optimal control problem for general finite-dimensional linear systems.In this optimal control problem,the cost functional is quasi-convex and the control domain is a convex compact polyhedron.Under suitable conditions,the uniqueness and stability of optimal systems with respect to perturbations are analyzed,respectively.The second purpose is derived to the regularity and stability property for Mayer problem with non-linear system,where the Hamiltonian and functional are assumed to be convex.However,the error estimates of Euler discretization methods are calculated.Chapter 4 presents the convergence theory of the gradient projection method and proximal point algorithms(briefly GPM and PPA respectively)for some optimal control problems.For the non-convex optimal control problems with affine system,it shows that if the cost functional is quasi-convex,any weakly convergent sequence of controls of GPM converge strongly to the optimal control.Under some mild assumptions,the linear convergence rate of GPM is obtained.For Mayer-type problem with general finite-dimensional linear systems,it proves that if the cost functional is convex,the iterative sequence of controls of PPA converge weakly to the optimal one.Moreover,under some suitable assumptions,the strong convergence is guaranteed and linear convergence rate is constructed based on the controllability index associated with the local form of the maximum principle.In addition,the theoretical results were confirmed by some numerical experiments.Finally,chapter 5 provides brief conclusions for the contributions,and discussions for future works for further consideration.