Set-Valued Analysis, Viability Theory and Partial Differential Inclusions

Systems of first-order partial differential inclusions -- solutions of which are feedbacks governing viable trajectories of control systems -- are derived. A variational principle and an existence theorem of a (single-valued contingent) solution to such partial differential inclusions are stated. To prove such theorems, tools of set-valued analysis and tricks taken from viability theory are surveyed. This paper is the text of a plenary conference to the World Congress on Nonlinear Analysis held at Tampa, Florida, August 19-26, 1992

Local lipschitzness of reachability maps for hybrid systems with applications to safety

Motivated by the safety problem, several definitions of reachability maps, for hybrid dynamical systems, are introduced. It is well established that, under certain conditions, the solutions to continuous-time systems depend continuously with respect to initial conditions. In such setting, the reachability maps considered in this paper are locally Lipschitz (in the Lipschitz sense for set-valued maps) when the right-hand side of the continuous-time system is locally Lipschitz. However, guaranteeing similar properties for reachability maps for hybrid systems is much more challenging. Examples of hybrid systems for which the reachability maps do not depend nicely with respect to their arguments, in the Lipschitz sense, are introduced. With such pathological cases properly identified, sufficient conditions involving the data defining a hybrid system assuring Lipschitzness of the reachability maps are formulated. As an application, the proposed conditions are shown to be useful to significantly improve an existing converse theorem for safety given in terms of barrier functions. Namely, for a class of safe hybrid systems, we show that safety is equivalent to the existence of a locally Lipschitz barrier function. Examples throughout the paper illustrate the results

Hyperbolic Systems of Partial Differential Inclusions

This paper is devoted to the study of first-order hyperbolic systems of partial differential inclusions which are in particular motivated by several problems of control theory, such as tracking problems. The existence of contingent single-valued solutions is proved for a certain class of such systems. Several comparison and localization results (which replace uniqueness results in the case of hyperbolic systems of partial differential equations) allow to derive useful informations on the solutions of these problems

Dynamic Regulation of Controlled Systems, Inertia Principle and Heavy Viable Solutions

Existence of viable (controlled invariant) solutions of a control problem regulated by absolutely continuous open loop controls is proved by using the concept of viability kernels of closed subsets (largest closed controlled invariant subsets). This is needed to provide dynamical feedbacks, i.e., differential equations governing the evolution of viable controls. Among such differential equations, the differential equation providing heavy solutions (in the sense of heavy trends), i.e., governing the evolution of controls with minimal velocity is singled out. Among possible applications, these results are used to define global contingent subsets of the contingent cones which allow to prove the convergence of a modified version of the structure algorithm to a closed viability domain of any closed subset

Partial Differential Inclusions Governing Feedback Controls

The authors derive partial differential inclusions of hyperbolic type, the solutions of which are feedbacks governing the viable (controlled invariant) solutions of a control system. They show that the tracking property, another important control problem, leads to such hyperbolic systems of partial differential inclusions. They begin by proving the existence of the largest solution of such a problem, a stability result and provide an explicit solution in the particular case of decomposable systems. They then state a variational principle and an existence theorem of a. (single-valued contingent) solution to such an inclusion, that they apply to assert the existence of a feedback control

The Viability Kernel Algorithm for Computing Value Functions of Infinite Horizon Optimal Control Problems

We characterize in this paper the epigraph of the value function of a discounted infinite horizon optimal control problem as the viability kernel of an auxiliary differential inclusion. Then the viability kernel algorithm applied to this problem provides the value function of the discretized optimal control problem as the supremum of a nondecreasing sequence of functions iteratively defined. We also use the fact that an upper Painleve-Kuratowski limit of closed viability domains is a viability domain to prove the convergence of the discrete value functions

Observability of Systems under Uncertainty

The authors observe the evolution of a state of a system under uncertainty governed by a differential inclusion through an observation map. The set-valued character due to uncertainty leads them to introduce the "Sharp Input-Output Map", which is a (usual) product, and the "Hazy Input-Output Map", which is a square product. They provide criteria for both sharp and hazy local observability in terms of (global) sharp and hazy observability of a variational inclusion. They reach their conclusions by implementing the following strategy: (1) Provide a general principle of local injectivity and observability of a set-valued map I, which derives these properties from the fact that the kernel of an adequate derivative of I is equal to 0. (2) Supply chain rule formulas which allow to compute the derivatives of the usual product I_{-} and the square product I_{+} from the derivatives of the observation map H and the solution map S. (3) Characterize the various derivatives of the solution map S in terms of the solution maps of the associated variational inclusions. (4) Piece together these results for deriving local sharp and hazy observability of the original system from sharp and hazy observability of the variational inclusions. (5) Study global sharp and hazy observability of the variational inclusions

Controllability and Observability of Control Systems under Uncertainty

This report surveys the results of nonlinear systems theory (controllability and observability) obtained at IIASA during the last three summers. Classical methods based on differential geometry require some regularity and fail as soon as state-dependent constraints are brought to bear on the controls, or uncertainty and disturbances are involved in the system. Since these important features appear in most realistic control problems, new methods had to be devised, which encompass the classical ones, and allow the presence of a priori feedback into the control systems. This is now possible thanks to new tools, in the development of which IIASA played an important role: differential inclusions and set-valued analysis

Set-valued Solutions to the Cauchy Problem for Hyperbolic Systems of Partial Differential Inclusions

We prove the existence of global set-valued solutions to the Cauchy problem for partial differential equations and inclusions, with either single-valued or set-valued initial conditions. The method is based on the equivalence between this problem and problem of finding viability tubes of the associated characteristic system of ordinary differential equations or differential inclusions

On Inverse Function Theorems for 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 to optimization theory and local controllability of differential inclusions