Peter R. Wolenski

Qualitative properties of trajectories of control systems: A survey

We present a unified approach to a complex of related issues in control theory, one based to a great extent on the methods of nonsmooth analysis. The issues include invariance, stability, equilibria, monotonicity, the Hamilton-Jacobi equation, feedback synthesis, and necessary conditions.

Proximal Analysis and the Minimal Time Function

Under general hypotheses on the target set S and the dynamics of the system, we show that the minimal time function TS(\cdot) is a proximal solution to the Hamilton--Jacobi equation. Uniqueness results are obtained with two different kinds of boundary conditions. A new propagation result is proven, and as an application, we give necessary and sufficient conditions for TS(\cdot) to be Lipschitz continuous near S. A Petrov-type modulus condition is also shown to be sufficient for continuity of TS(\cdot) near S.

The exponential formula for the reachable set of a Lipschitz differential inclusion

The main goal of this paper is to prove a formula for the reachable set of a Lipschitz differential inclusion with convex values. The formula involves a Kuratowski limit of sets that resembles a standard approach of defining the exponential of a matrix—this explains the title. The proof of the main theorem partially relies on a

The subgradient formula for the minimal time function in the case of constant dynamics in Hilbert space

C^1

Some New Regularity Properties for the Minimal Time Function

approximation result due to Filippov, for which a new proof is given. A new approach of characterizing the value function associated with a Mayer optimal control problem is given as an application.

Faster walking speeds increase local instability among people with peripheral neuropathy

The minimal time function with constant dynamics is studied in the context of a Hilbert space. A general formula for the subgradient is proven, and assumptions are identified in which the minimal time function is lower C2.

Convexity in Hamilton--Jacobi Theory II: Envelope Representations

A minimal time problem with linear dynamics and convex target is considered. It is shown, essentially, that the epigraph of the minimal time function

A Sampling Method and Approximation Results for Impulsive Systems

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Control of systems to sets and their interiors

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A Framework for Detecting Glaucomatous Progression in the Optic Nerve Head of an Eye Using Proper Orthogonal Decomposition

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