Operator inclusions and operator-differential inclusions

In Chapter 2, we first introduce a generalized inverse differentiability for set-valued mappings and consider some of its properties. Then, we use this differentiability, Ekeland's Variational Principle and some fixed point theorems to consider constrained implicit function and open mapping theorems and surjectivity problems of set-valued mappings. The mapping considered is of the form F(x, u) + G (x, u). The inverse derivative condition is only imposed on the mapping x F(x, u), and the mapping x G(x, u) is supposed to be Lipschitz. The constraint made to the variable x is a closed convex cone if x F(x, u) is only a closed mapping, and in case x F(x, u) is also Lipschitz, the constraint needs only to be a closed subset. We obtain some constrained implicit function theorems and open mapping theorems. Pseudo-Lipschitz property and surjectivity of the implicit functions are also obtained. As applications of the obtained results, we also consider both local constrained controllability of nonlinear systems and constrained global controllability of semilinear systems. The constraint made to the control is a time-dependent closed convex cone with possibly empty interior. Our results show that the controllability will be realized if some suitable associated linear systems are constrained controllable. In Chapter 3, without defining topological degree for set-valued mappings of monotone type, we consider the solvability of the operator inclusion y0 N1(x) + N2 (x) on bounded subsets in Banach spaces with N1 a demicontinuous set-valued mapping which is either of class (S+) or pseudo-monotone or quasi-monotone, and N2 is a set-valued quasi-monotone mapping. Conclusions similar to the invariance under admissible homotopy of topological degree are obtained. Some concrete existence results and applications to some boundary value problems, integral inclusions and controllability of a nonlinear system are also given. In Chapter 4, we will suppose u A (t,u) is a set-valued pseudo-monotone mapping and consider the evolution inclusions x' (t) + A(t,x((t)) f (t) a.e. and (d)/(dt) (Bx(t)) + A (t,x((t)) f(t) a.e. in an evolution triple (V,H,V*), as well as perturbation problems of those two inclusions

Optimal control problems with time delays: constancy of the Hamiltonian

This paper concerns necessary conditions of optimality for optimal control problems with time delays in the state variable. It is well known that, when there are no time delays and the dynamics are autonomous, the standard necessary conditions in the form of a maximum principle can be supplemented by an extra condition, namely “constancy of the Hamiltonian" along optimal trajectories (and associated costate trajectories). This property, possibly supplemented by other invariance principles, has been used to investigate properties of optimal trajectories, such as solution regularity, without the need to solve the underlying extremal equations. In classical mechanics, for example, the constancy of the Hamiltonian condition can be used to derive a conservation of energy principle from Hamilton's principle of least action. While the maximum principle has been generalized to cover time delays, the validity of constancy of the Hamiltonian-type conditions has not been previously investigated. We provide the first “extra" optimality condition of this nature for autonomous, time delay optimal control problems. The new “constancy of the Hamiltonian" condition involves a correction term, without which the condition is not valid. We illustrate the significance of this condition by applications to minimizer regularity and conservation laws in nonclassical Hamiltonian mechanics

Proceedings of Seminar on Partial Differential Equations in Osaka 2012 : in honor of Professor Hiroki Tanabe’s 80th birthday

Osaka University, August 20‐24, 2012Edited by Atsushi Yagi and Yoshitaka Yamamot