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

A Priori Estimates for Operational Differential Inclusions

The author proves a set-valued Gronwall lemma and a relaxation theorem for the semilinear differential inclusion x' E Ax + F(t,x), x(0)=x_0 where A is the infinitesimal generator of a C_0-semigroup on a separable Banach space X and F : [0,T] x X --> X is a set-valued map. This result is important for investigation of many futures of semilinear inclusions, for instance, infinitesimal generators of reachable sets, variational inclusions, etc

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

A Viability Approach to the Skorohod Problem

The theory of stochastic differential equations with reflecting boundary conditions leads to the "Skorohod" problem. This report proposes a solution to this problem using techniques from viability theory and nonsmooth analysis, allowing very general situations to occur

Optimal Trajectories Associated to a Solution of Contingent Hamilton-Jacobi Equation

In this paper we study the existence of optimal trajectories associated with a generalized solution to Hamilton-Jacobi-Bellman equation arising in optimal control. In general, we cannot expect such solutions to be differentiable. But, in a way analogous to the use of distributions in PDE, we replace the usual derivatives with "contingent epiderivatives" and the Hamilton-Jacobi equation by two "contingent Hamilton-Jacobi inequalities". We show that the value function of an optimal control problem verifies these "contingent inequalities". Our approach allows the following three results: (1) The upper semicontinuous solutions to contingent inequalities are monotone along the trajectories of the dynamical system. (2) With every continuous solution V of the contingent inequalities, we can associate an optimal trajectory along which V is constant. (3) For such solutions, we can construct optimal trajectories through the corresponding optimal feedback. They are also "viscosity solutions" of a Hamilton-Jacobi equation. Finally we prove a relationship between super-differentials of solutions introduced in Crandall-Evans-Lions and the Pontryagin principle and discuss the link of viscosity solutions with Clarke's approach to the Hamilton-Jacobi equation

Adjoint Differential Inclusions in Necessary Conditions for the Minimal Trajectories of Differential Inclusions

This paper extends Pontryagin's maximum principle to differential inclusions and nonsmooth criterion functions, relying on a checkable "surjectivity property" of a "linearized set-valued system" around the optimal trajectory. As an example, Pontryagin's principle is obtained for optimal control problems with constraints on both the initial and the final states. The research described here was undertaken within the framework of the Dynamics of Macrosystems Feasibility Study in the System and Decision Sciences Program

Local Invertibility of Set-Valued Maps

We prove several equivalent versions of the inverse function theorem: an inverse function theorem for smooth maps on closed subsets, one for set-valued maps, a generalized implicit function theorem for set-valued maps. We provide applications of the above results to the problem of local controllability of differential inclusions

High Order Inverse Function Theorems

We prove several first order and high order inverse mapping theorems for maps defined on a complete metric space and provide a number of applications