Dynamical systems with time delay

Abstract

In this dissertation, we study necessary conditions and weak invariance properties of dynamical systems with time delay. A number of results have been obtained recently that refine necessary conditions of optimal solutions for nonsmooth dynamical systems without time delay. In this dissertation, we examine the extension of some of these results to problems with time delay. In particular, we study the generalized problem of Bolza with the addition of delay in the state and velocity variables and refer to this problem as the Neutral Problem of Bolza. We consider the relationship between the generalized problem of Bolza with time delay and control systems, establish existence of solutions for the Neutral Problem of Bolza, and use a ``decoupling' technique introduced by Clarke to derive necessary conditions of Hamiltonian and Euler-Lagrange type for this problem. We also apply the same methods to the generalized problem of Bolza with time delay in the state variable only and compare the results obtained in this case with the results obtained in the neutral case. Furthermore, we study the system (S,F) involving a closed set S and a delayed autonomous multifunction F(x(t),x(t-Delta)). Under suitable hypotheses, we provide a characterization of weak invariant properties for F in terms of the lower hamiltonian.