Epi-Lipschitzian reachable sets of differential inclusions

The reachable sets of a differential inclusion have nonsmooth topological boundaries in general. The main result of this paper is that under the well-known assumptions of Filippov's existence theorem (about differential inclusions), every epi-Lipschitzian initial compact set (of the Euclidean space) preserves this regularity for a (possibly short) time, i.e. its reachable set is also epi-Lipschitzian for all small times. The proof is based on Rockafellar's geometric characterization of epi-Lipschitzian sets and uses a new result about the "inner semicontinuity" of Clarke tangent cone (to reachable sets) with respect to both time and base point

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

Propagation of singularities for weak KAM solutions and barrier functions

This paper studies the structure of the singular set (points of nondifferentiability) of viscosity solutions to Hamilton-Jacobi equations associated with general mechanical systems on the n-torus. First, using the level set method, we characterize the propagation of singularities along generalized characteristics. Then, we obtain a local propagation result for singularities of weak KAM solutions in the supercritical case. Finally, we apply such a result to study the propagation of singularities for barrier functions

Convexity and Duality in Hamilton-Jacobi Theory

Value functions propagated from initial or terminal costs and constraints by way of a differential or more broadly through a Lagrangian that may take on "alpha," are studied in the case where convexity persists in the state argument. Such value functions, themselves taking on "alpha," are shown to satisfy a subgradient form of the Hamilton-Jacobi equation which strongly supports properties of local Lipschitz continuity, semidifferentibility and Clarke regularity. An extended `method of characteristics' is developed which determines them from Hamiltonian dynamics underlying the given Lagrangian. Close relations with a dual value function are revealed

Approximation of reachable sets using optimal control algorithms

To appearInternational audienceNumerical experiences with a method for the approximation of reachable sets of nonlinear control systems are reported. The method is based on the formulation of suitable optimal control problems with varying objective functions, whose discretization by Euler's method lead to finite dimensional non-convex nonlinear programs. These are solved by a sequential quadratic programming method. An efficient adjoint method for gradient computation is used to reduce the computational costs. The discretization of the state space is more efficiently than by usual sequential realization of Euler's method and allows adaptive calculations or refinements. The method is illustrated for two test examples. Both examples have non-linear dynamics, the first one has a convex reachable set, whereas the second one has a non-convex reachable set

Evolution equations in ostensible metric spaces: First-order evolutions of nonsmooth sets with nonlocal terms

Similarly to funnel equations of Panasyuk, the so-called mutational equations of Aubin provide a generalization of ordinary differential equations to locally compact metric spaces. Here we present their extension to a nonempty set with a possibly nonsymmetric distance. A distribution-like approach leads to so-called right-hand forward solutions. This concept is applied to a type of geometric evolution having motivated the definitions : compact subsets of the Euclidean space evolve according to nonlocal properties of both the set and their limiting normal cones at the boundary. The existence of a solution is based on Euler method using reachable sets of differential inclusions as "elementary deformations" (called forward transitions). Thus, the regularity of these reachable sets at the topological boundaries is studied extensively in the appendix

Evolution equations in ostensible metric spaces. II. Examples in Banach spaces and of free boundaries.

In part I, generalizing mutational equations of Aubin in metric spaces has led to so-called right-hand forward solutions in a nonempty set with a countable family of (possibly nonsymmetric) ostensible metrics. Now this concept is applied to two different types of evolutions that have motivated the definitions : semilinear evolution equations (of parabolic type) in a reflexive Banach space and compact subsets of R^N whose evolution depend on nonlocal properties of both the set and their limiting normal cones at the boundary. For verifying that reachable sets of differential inclusions are appropriate transitions for first-order geometric evolutions, their regularity at the boundary is studied in the appendix

Generalizing evolution equations in ostensible metric spaces: Timed right-hand sleek solutions provide uniqueness of first-order geometric examples.

The mutational equations of Aubin extend ordinary differential equations to metric spaces (with compact balls). In first-order geometric evolutions, however, the topological boundary need not be continuous in the sense of Painleve–Kuratowski. So this paper suggests a generalization of Aubin’s mutational equations that extends classical notions of dynamical systems and functional analysis beyond the traditional border of vector spaces: Distribution– like solutions are introduced in a set just supplied with a countable family of (possibly non-symmetric) distance functions. Moreover their existence is proved by means of Euler approximations and a form of “weak” sequential compactness (although no continuous linear forms are available beyond topological vector spaces). This general framework is applied to a first-order geometric example, i.e. compact subsets of the Euclidean space evolving according to the nonlocal properties of both the current set and its proximal normal cones. Here neither regularity assumptions about the boundaries nor the inclusion principle are required. In particular, we specify sufficient conditions for the uniqueness of these solutions

Dissipative Lipschitz dynamics

In this dissertation we study two related important issues in control theory: invariance of dynamical systems and Hamilton-Jacobi theory associated with optimal control theory. Given a control system modelled as a differential inclusion, we provide necessary and sufficient conditions for the strong invariance property of the system when the dynamic satisfies a dissipative Lipschitz condition. We show that when the dynamic is almost upper semicontinuous and satisfies the dissipative Lipschitz property, these conditions can be expressed in terms of approximate Hamilton-Jacobi inequalities, which subsumes the classic infinitesimal characterization of strongly invariant systems given under the Lipschitz assumtion. In the important case when the dynamic of the system is the sum of a maximal dissipative and a Lipschitz multifunction, the approximate inequalities turn into an exact mixed type inequality that involves the lower and upper Hamiltonian of the dissipative and the Lipschitz piece respectively. We then extend this Hamiltonian characterization to nonautonomous systems by assuming a potentially discontinuous differential inclusion whose right-hand side is the sum of an almost upper semicontinuous dissipative and an almost lower semicontinuous dissipative Lipschitz multifunction. Finally, a Hamilton-Jacobi theory is developed for the minimal time problem of a system with possibly discontinuous monotone Lipschitz dynamic. This is achieved by showing the minimal time function associated to an upper semicontinuous and a monotone Lipschitz data is characterized as the unique proximal semi-solution to an approximate Hamilton-Jacobi equation satisfying an analytical boundary condition