2,414 research outputs found
A Nonsmooth Maximum Principle for Optimal Control Problems with State and Mixed Constraints-Convex Case
Here we derive a nonsmooth maximum principle for optimal control problems
with both state and mixed constraints. Crucial to our development is a
convexity assumption on the "velocity set". The approach consists of applying
known penalization techniques for state constraints together with recent
results for mixed constrained problems.Comment: Published in 'Discrete and Continuous Dynamical Systems, Vol. 2011,
pp. 174-18
Lipschitzian Regularity of the Minimizing Trajectories for Nonlinear Optimal Control Problems
We consider the Lagrange problem of optimal control with unrestricted
controls and address the question: under what conditions we can assure optimal
controls are bounded? This question is related to the one of Lipschitzian
regularity of optimal trajectories, and the answer to it is crucial for closing
the gap between the conditions arising in the existence theory and necessary
optimality conditions. Rewriting the Lagrange problem in a parametric form, we
obtain a relation between the applicability conditions of the Pontryagin
maximum principle to the later problem and the Lipschitzian regularity
conditions for the original problem. Under the standard hypotheses of
coercivity of the existence theory, the conditions imply that the optimal
controls are essentially bounded, assuring the applicability of the classical
necessary optimality conditions like the Pontryagin maximum principle. The
result extends previous Lipschitzian regularity results to cover optimal
control problems with general nonlinear dynamics.Comment: This research was partially presented, as an oral communication, at
the international conference EQUADIFF 10, Prague, August 27-31, 2001.
Accepted for publication in the journal Mathematics of Control, Signals, and
Systems (MCSS). See http://www.mat.ua.pt/delfim for other work
Constrained Nonsmooth Problems of the Calculus of Variations
The paper is devoted to an analysis of optimality conditions for nonsmooth
multidimensional problems of the calculus of variations with various types of
constraints, such as additional constraints at the boundary and isoperimetric
constraints. To derive optimality conditions, we study generalised concepts of
differentiability of nonsmooth functions called codifferentiability and
quasidifferentiability. Under some natural and easily verifiable assumptions we
prove that a nonsmooth integral functional defined on the Sobolev space is
continuously codifferentiable and compute its codifferential and
quasidifferential. Then we apply general optimality conditions for nonsmooth
optimisation problems in Banach spaces to obtain optimality conditions for
nonsmooth problems of the calculus of variations. Through a series of simple
examples we demonstrate that our optimality conditions are sometimes better
than existing ones in terms of various subdifferentials, in the sense that our
optimality conditions can detect the non-optimality of a given point, when
subdifferential-based optimality conditions fail to disqualify this point as
non-optimal.Comment: A number of small mistakes and typos was corrected in the second
version of the paper. Moreover, the paper was significantly shortened.
Extended and improved versions of the deleted sections on nonsmooth Noether
equations and nonsmooth variational problems with nonholonomic constraints
will be published in separate submission
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