A. V. Arutyunov

Investigation of the Degeneracy Phenomenon of the Maximum Principle for Optimal Control Problems with State Constraints

In this paper we study the degeneracy phenomenon in optimal control problems with state constraints. It is shown that this phenomenon occurs because of the incompleteness of the standard variants of Pontryagins maximum principle for problems with state constraints. A new maximum principle containing additional information about the behavior of the Hamiltonian at the endtimes is developed. We also obtain some sufficient and necessary conditions for nondegeneracy and pointwise nontriviality of the maximum principle. The results obtained pertain to optimal control problems with systems described by differential inclusions and ordinary differential equations.

Covering mappings in metric spaces and fixed points

We consider metric spaces X and Y with metrics ρ X and ρ Y , respectively. Before stating the main results of this paper, we recall two known assertions. The first of them is the contraction mapping principle, which says that if a space X is complete, then any self-mapping of X satisfying the Lipschitz condition with Lipschitz constant less than 1 (i.e., a contraction) has a (unique) fixed point. The second assertion is Milyutin’s covering mapping theorem. To formulate it, we need the notion of a covering mapping. By B X ( r , x ) we denote the closed ball of radius r centered at x in the space X ; a similar notation is used in the space Y . For any M ⊂ Y , we set B Y ( r , M ) = ( r , y ) (this is the r -neighborhood of the set M ). Definition. Let α > 0. A mapping Ψ : X → Y is said to be α -covering if

Stability of coincidence points and properties of covering mappings

Properties of closed set-valued covering mappings acting from one metric space into another are studied. Under quite general assumptions, it is proved that, if a given α-covering mapping and a mapping satisfying the Lipschitz condition with constant β < α have a coincidence point, then this point is stable under small perturbations (with respect to the Hausdorff metric) of these mappings. This assertion is meaningful for single-valued mappings as well. The structure of the set of coincidence points of an α-covering and a Lipschitzian mapping is studied. Conditions are obtained under which the limit of a sequence of α-covering set-valued mappings is an (α–ɛ)-covering for an arbitrary ɛ > 0.

A Nondegenerate Maximum Principle for the Impulse Control Problem with State Constraints

In this article, a free-time impulsive control problem with state constraints and equality and inequality constraints on the trajectory endpoints is considered. A weakened maximum principle is obtained for problems with data measurable in the time variable, being the time transversality conditions deduced with the help of some extra convexity assumption on the state constraints. In the case of smooth problems a nondegenerate maximum principle is derived by using a penalty function method.

Extremal Problems with Constraints

Let two vector spaces X and Y, a mapping F : X → Y, a scalar-valued function f 0 : X → R 1, and a convex cone C ⋸ Y be given. We consider the problem

Second-order conditions in extremal problems. The abnormal points

{f_0}(x) \to \min |x:F(x) \in C

Covering mappings and their applications to differential equations unsolved for the derivative

(1.1)

Second Order Necessary Conditions for Optimal Impulsive Control Problems

In this paper we study the degeneracy phenomenon arising in optimal control problems with state constraints. It is shown that this phenomenon occurs because of the incompleteness of the standard variants of Pontryagins maximum principle for problems with state constraints. The new maximum principle containing some additional information about the behavior of the Hamiltonian at the endtimes is developed. As application we obtain some sufficient and necessary conditions for nondegeneracy and pointwise nontriviality of the maximum principle. The results obtained envelope the optimal control problems with systems described by differential inclusions and ordinary differential equations.