Dynamical systems with time delay

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\u27 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

Second-order optimality conditions for the Bolza problem with path constraints

A set of sufficient conditions for a weak minimum is derived for a form of the nonsingular Bolza problem of variational calculus, with interior point constraints and discontinuities in the system equations. Generalized versions of the conjugate point/focal point, normality, convexity and nontangency conditions associated with the ordinary Bolza problem are obtained. The resulting set of sufficient conditions is minimal, in that only minor modifications are required in order to obtain necessary conditions for normal, nonsingular problems of this form. These conditions are relatively easy to implement. Analogous second-order optimality conditions for problems with natural corners or control constraints are also obtained. Previously stated sufficiency conditions for problems with control constraints are shown to be unnecessarily restrictive, in some cases

Finite-Difference Approximations And Optimal Control Of Differential Inclusions

This dissertation concerns the study of the generalized Bolza type problem for dynamic systems governed by constrained differential inclusions. We develop finite-discrete approximations of differential inclusions by using the implicit Euler scheme and the Runge-Kutta scheme for approximating time derivatives, while an appropriate well-posedness of such approximations is justified. Our principal result establishes the uniform approximation of strong local minimizers for the continuous-time Bolza problem by optimal solutions to the corresponding discretized finite-difference systems by the strengthen $W^{1,2}$-norm approximation of this type in the case ``intermediate (between strong and weak minimizers) local minimizers under additional assumptions. Especially the implicitly discrete approximation is under the general ROSL setting. Finally, we derive necessary optimality conditions for each scheme for the discretized Bolza problems via suitable generalized differential constructions of variational analysis

Optimality and Characteristics of Hamilton-Jacobi-Bellman Equations

In this paper the authors study the Bolza problem arising in nonlinear optimal control and investigate under what circumstances the necessary conditions for optimality of Pontryagin's type are also sufficient. This leads to the question when shocks do not occur in the method of characteristics applied to the associated Hamilton-Jacobi-Bellman equation. In this case the value function is its (unique) continuously differentiable solution and can be obtained from the canonical equations. In optimal control this corresponds to the case when the optimal trajectory of the Bolza problem is unique for every initial state and the optimal feedback is an upper semicontinuous set-valued map with convex, compact images

A Higher-order Maximum Principle for Impulsive Optimal Control Problems

We consider a nonlinear system, affine with respect to an unbounded control $u$ which is allowed to range in a closed cone. To this system we associate a Bolza type minimum problem, with a Lagrangian having sublinear growth with respect to $u$. This lack of coercivity gives the problem an {\it impulsive} character, meaning that minimizing sequences of trajectories happen to converge towards discontinuous paths. As is known, a distributional approach does not make sense in such a nonlinear setting, where, instead, a suitable embedding in the graph-space is needed. We provide higher order necessary optimality conditions for properly defined impulsive minima, in the form of equalities and inequalities involving iterated Lie brackets of the dynamical vector fields. These conditions are derived under very weak regularity assumptions and without any constant rank conditions

Optimal control of the sweeping process over polyhedral controlled sets

The paper addresses a new class of optimal control problems governed by the dissipative and discontinuous differential inclusion of the sweeping/Moreau process while using controls to determine the best shape of moving convex polyhedra in order to optimize the given Bolza-type functional, which depends on control and state variables as well as their velocities. Besides the highly non-Lipschitzian nature of the unbounded differential inclusion of the controlled sweeping process, the optimal control problems under consideration contain intrinsic state constraints of the inequality and equality types. All of this creates serious challenges for deriving necessary optimality conditions. We develop here the method of discrete approximations and combine it with advanced tools of first-order and second-order variational analysis and generalized differentiation. This approach allows us to establish constructive necessary optimality conditions for local minimizers of the controlled sweeping process expressed entirely in terms of the problem data under fairly unrestrictive assumptions. As a by-product of the developed approach, we prove the strong $W^{1,2}$-convergence of optimal solutions of discrete approximations to a given local minimizer of the continuous-time system and derive necessary optimality conditions for the discrete counterparts. The established necessary optimality conditions for the sweeping process are illustrated by several examples

Conjugate Points and Shocks in Nonlinear Optimal Control

In this paper the authors use the method of characteristics to extend the Jacobi conjugate points theory to the Bolza problem arising in nonlinear optimal control. This yields necessary and sufficient optimality conditions for weak and strong local minima stated in terms of the existence of a solution to a corresponding matrix Riccati differential equation. The same approach allows to investigate as well smoothness of the value function