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

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

On the Lyapunov Second Method for Data Measurable in Time

In this paper the authors study time dependent Lyapunov functions for nonautonomous systems described by differential inclusions. In particular it is shown that Lyapunov functions are viscosity supersolutions of a Hamilton-Jacobi equation. For this aim a new viability theorem for differential inclusions with time dependent state constraints is proved. The viability conditions are formulated both using contingent cones and in a dual way, using subnormal cones (negative polar of contingent cone)

Viability Kernels of Differential Inclusions with Constraints: Algorithms and Applications

The authors investigate a differential inclusion whose solutions have to remain in a given closed set. The viability kernel is the set of the initial conditions starting at which, there exist solutions to the differential inclusion remaining in this closed set. In this paper, the authors provide an algorithm which determine this set and they apply it to some concrete examples

Value Functions and Optimality Conditions for Semilinear Control Problems. II: Parabolic Case

In this paper the authors study properties of the value function and of optimal solutions of a semilinear Mayer problem in infinite dimensions. Applications concern systems governed by a state equation of parabolic type. In particular, the issues of the joint Lipschitz continuity and semiconcavity of the value function are treated in order to investigate the differentiability of the value function along optimal trajectories

Isaacs' Equations for Value-Functions of Differential Games

The authors study value functions of a differential game with payoff which depends on the state at a given end time. They consider differential games with feedback strategies and with nonanticipating strategies. They prove that value-functions are solutions to some Hamilton-Jacobi-Isaacs equations in the viscosity and contingent sense. For these two notions of strategies, with some regularity assumptions, The authors prove that value-functions are the unique solution of Isaacs' equations