789 research outputs found
Value Functions for Bolza Problems with Discontinuous Lagrangians and Hamilton-Jacobi Inequalities
We investigate the value function of the Bolza problem of the Calculus of
Variations with a lower semicontinuous Lagrangian and a
final cost , and show that it is locally Lipschitz for whenever
is locally bounded. It also satisfies Hamilton-Jacobi inequalities in a
generalized sense.
When the Lagrangian is continuous, then the value function is the unique
lower semicontinuous solution to the corresponding Hamilton-Jacobi equation,
while for discontinuous Lagrangian we characterize the value function by using
the so called contingent inequalities.Comment: 33 pages. Control, Optimization and Calculus of Variations, to appea
Steven H. Gale, Sharp Cut: Harold Pinter's Screenplays and the Artistic Process. Lexington: University Press of Kentucky, 2003
reviewZadanie pt. „Digitalizacja i udostępnienie w Cyfrowym Repozytorium Uniwersytetu Łódzkiego kolekcji czasopism naukowych wydawanych przez Uniwersytet Łódzki” nr 885/P-DUN/2014 dofinansowane zostało ze środków MNiSW w ramach działalności upowszechniającej nauk
Autonomous Integral Functionals with Discontinuous Nonconvex Integrands: Lipschitz Regularity of Minimizers, DuBois-Reymond Necessary Conditions, and Hamilton-Jacobi Equations
This paper is devoted to the autonomous Lagrange problem of the calculus of
variations with a discontinuous Lagrangian. We prove that every minimizer is
Lipschitz continuous if the Lagrangian is coercive and locally bounded. The
main difference with respect to the previous works in the literature is that we
do not assume that the Lagrangian is convex in the velocity. We also show that,
under some additional assumptions, the DuBois-Reymond necessary condition still
holds in the discontinuous case. Finally, we apply these results to deduce that
the value function of the Bolza problem is locally Lipschitz and satisfies (in
a generalized sense) a Hamilton-Jacobi equation.Comment: 29 page
Uniqueness of solutions to Hamilton-Jacobi equations arising in the Calculus of Variations
We prove the uniqueness of the viscosity solution to the Hamilton-Jacobi
equation associated with a Bolza problem of the Calculus of Variations,
assuming that the Lagrangian is autonomous, continuous, superlinear, and
satisfies the usual convexity hypothesis. Under the same assumptions we prove
also the uniqueness, in a class of lower semicontinuous functions, of a
slightly different notion of solution, where classical derivatives are replaced
only by subdifferentials. These results follow from a new comparison theorem
for lower semicontinuous viscosity supersolutions of the Hamilton-Jacobi
equation, that is proved in the general case of lower semicontinuous
Lagrangians.Comment: 14 page
Second-order sensitivity relations and regularity of the value function for Mayer's problem in optimal control
This paper investigates the value function, , of a Mayer optimal control
problem with the state equation given by a differential inclusion. First, we
obtain an invariance property for the proximal and Fr\'echet subdifferentials
of along optimal trajectories. Then, we extend the analysis to the
sub/superjets of , obtaining new sensitivity relations of second order. By
applying sensitivity analysis to exclude the presence of conjugate points, we
deduce that the value function is twice differentiable along any optimal
trajectory starting at a point at which is proximally subdifferentiable. We
also provide sufficient conditions for the local regularity of on
tubular neighborhoods of optimal trajectories
On the Linearization of Nonlinear Control Systems and Exact Reachability
The author studies the problem of exact local reachability of infinite dimensional nonlinear control systems. The main result shows that the exact local reachability of a linearized system implies that of the original system. The main tool is an inverse map ping theorem for a map from a complete metric space to a reflexive Banach space
The Maximum Principle for a Differential Inclusion Problem
In this report, the Pontryagin principle is extended to optimal control problems with feedbacks (i.e., in which the controls depend upon the state). New techniques of non-smooth analysis (asymptotic derivatives of set-valued maps and functions) are used to prove this principle for problems with finite and infinite horizons
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