2,994 research outputs found
Discrete Maximum Principle for Nonsmooth Optimal Control Problems with Delays
We consider optimal control problems for discrete-time systems with delays. The main goal is to derive necessary optimality conditions of the discrete maximum principle type in the case of nonsmooth minimizing functions. We obtain two independent forms of the discrete maximum principle with transversality conditions described in terms of subdifferentials and superdifferentials, respectively. The superdifferential form is new even for non-delayed systems and may be essentially stronger than a more conventional subdifferential form in some situations
Discrete time optimal control with frequency constraints for non-smooth systems
We present a Pontryagin maximum principle for discrete time optimal control
problems with (a) pointwise constraints on the control actions and the states,
(b) frequency constraints on the control and the state trajectories, and (c)
nonsmooth dynamical systems. Pointwise constraints on the states and the
control actions represent desired and/or physical limitations on the states and
the control values; such constraints are important and are widely present in
the optimal control literature. Constraints of the type (b), while less
standard in the literature, effectively serve the purpose of describing
important spectral properties of inertial actuators and systems. The
conjunction of constraints of the type (a) and (b) is a relatively new
phenomenon in optimal control but are important for the synthesis control
trajectories with a high degree of fidelity. The maximum principle established
here provides first order necessary conditions for optimality that serve as a
starting point for the synthesis of control trajectories corresponding to a
large class of constrained motion planning problems that have high accuracy in
a computationally tractable fashion. Moreover, the ability to handle a
reasonably large class of nonsmooth dynamical systems that arise in practice
ensures broad applicability our theory, and we include several illustrations of
our results on standard problems
Variational Analysis of Evolution Inclusions
The paper is devoted to optimization problems of the Bolza and Mayer types for evolution systems governed by nonconvex Lipschitzian differential inclusions in Banach spaces under endpoint constraints described by finitely many equalities and inequalities. with generally nonsmooth functions. We develop a variational analysis of such roblems mainly based on their discrete approximations and the usage of advanced tools of generalized differentiation satisfying comprehensive calculus rules in the framework of Asplund (and hence any reflexive Banach) spaces. In this way we establish extended results on stability of discrete approximations (with the strong W^1,2-convergence of optimal solutions under consistent perturbations of endpoint constraints) and derive necessary optimality conditions for nonconvex discrete-time and continuous-time systems in the refined Euler-Lagrange and Weierstrass-Pontryagin forms accompanied by the appropriate transversality inclusions. In contrast to the case of geometric endpoint constraints in infinite dimensions, the necessary optimality conditions obtained in this paper do not impose any nonempty interiority /finite codimension/normal compactness assumptions. The approach and results developed in the paper make a bridge between optimal control/dynamic optimization and constrained mathematical programming problems in infinite-dimensional spaces
Variational Analysis in Nonsmooth Optimization and Discrete Optimal Control
The paper is devoted to applications of modern methods of variational· analysis to constrained optimization and control problems generally formulated in infinite-dimensional spaces. The main attention is paid to the study of problems with nonsmooth structures, which require the usage of advanced tools of generalized differentiation. In this way we derive new necessary optimality conditions in optimization problems with functional and. operator constraints and then apply them to optimal control problems governed by discrete-time inclusions in infinite dimensions. The principal difference between finite-dimensional and infinite-dimensional frameworks of optimization and control consists of the lack of compactness in infinite dimensions, which leads to imposing certain normal compactness properties and developing their comprehensive calculus, together with appropriate calculus rules of generalized differentiation. On the other hand, one of the most important achievements of the paper consists of relaxing the latter assumptions for certain classes of optimization and control problems. In particular, we fully avoid the requirements of this type imposed on target endpoint sets in infinite-dimensional optimal control for discrete-time inclusions
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 -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
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