Melkior Ornik

A topological obstruction to reach control by continuous state feedback

The reach control problem (RCP) deals with finding a feedback control which drives the states of an affine control system to leave a simplex through a predetermined facet. In analogy with Brocketts obstruction to continuous feedback stabilization, a topological obstruction to solving the RCP by continuous feedback results in a deep necessary condition for solvability. In this paper, we interpret the problem of topological obstruction as a problem of existence of null-homotopic maps on spheres. This results in a complete and easily implementable characterization of the topological obstruction for the case of systems with two inputs.

Characterization of a topological obstruction to reach control by continuous state feedback

This paper studies a topological obstruction to solving the reach control problem (RCP) by continuous state feedback. Given a simplex and given an affine control system defined on the simplex, the RCP is to find a state feedback to drive closed-loop trajectories initiated in the simplex through an exit facet, without first exiting through other facets. We distill the problem as one of continuously extending a function that maps into a sphere from the boundary of a simplex to its interior. As such, we employ techniques from the extension problem of algebraic topology. Unlike previous work on the same problem, in this paper we remove unnecessary restrictions on the dimension of the simplex, the number of inputs of the system, and the particular geometry of the subset of the state space where the obstruction arises. Thus, the results of this paper represent the culmination of our efforts to characterize the topological obstruction. The conditions obtained in the paper are easily checkable and fully characterize the obstruction.

On a Topological Obstruction in the Reach Control Problem

This paper explores aspects of the Reach Control Problem (RCP) to drive the states of an affine control system to a facet of a simplex without first exiting from other facets. In analogy with the problem of nonlinear feedback stabilization, we investigate a topological obstruction that arises in solving the RCP by continuous state feedback. The problem is fully solved in this paper for the case of two and three dimensions.

An obstruction to solvability of the reach control problem using affine feedback

This paper studies the reach control problem (RCP) using affine feedback on simplices. The contributions of this paper are threefold. First, we identify a new obstruction to solvability of the RCP using affine feedback and provide necessary and sufficient conditions for occurrence of such an obstruction. Second, for two-input systems, these conditions are formulated in terms of scalar linear inequalities. Third, computationally efficient necessary conditions are proposed for checking the obstruction for multi-input systems as feasibility programs in terms of linear inequalities. In contrast to the previous work in the literature, no assumption is imposed on the set of possible equilibria, so the results are applicable to the general RCP.

Pattern identification in distributed systems

A distributed systems interconnection structure emerges as a pattern in the system matrices. This pattern must be preserved through system analysis and control synthesis, and much has been written on these topics. A problem which has not received any attention to date is how to identify a pattern, given the linear system model. This paper proposes a method for identifying a pattern that is mathematically encoded through a commuting relationship with a base matrix. Our method generates the commuting relationship, when it exists. When it does not exist, our method produces the closest approximation to the commuting relationship. Further, it indicates which additional subsystem interconnections would render it achievable. We provide both an exact solution and an almost sure polynomial-time solution in the probabilistic sense. Finally, we give several examples to demonstrate the utility of this method for finding patterns in distributed systems.

Chattering in the Reach Control Problem

Abstract The Reach Control Problem (RCP) is a fundamental problem in hybrid control theory. The goal of the RCP is to find a feedback control that drives the state trajectories of an affine system to leave a polytope through a predetermined exit facet. In the current literature, the notion of leaving a polytope through a facet has an ambiguous definition. There are two different notions. In one, at the last time instance when the trajectory is inside the polytope, it must also be inside the exit facet. In the other, the trajectory is required to cross from the polytope into the outer open half-space bounded by the exit facet. In this paper, we provide a counterexample showing that these definitions are not equivalent for general continuous or smooth state feedback. On the other hand, we prove that analyticity of the feedback control is a sufficient condition for equivalence of these definitions. We generalize this result to several other classes of feedback control previously investigated in the RCP literature, most notably piecewise affine feedback. Additionally, we clarify or complete a number of previous results on the exit behaviour of trajectories in the RCP.

A topological obstruction in a control problem

Abstract One of the important discoveries in control theory is a topological obstruction to continuous feedback stabilization for general nonlinear control systems. In this note we describe another topological obstruction arising from a very different control problem called the reach control problem . Motivated by a classical topological obstruction for extending continuous maps on spheres, we introduce the problem of extending continuous maps on simplices. It is shown that the same condition as in the sphere case gives rise to the obstruction.

MODELING THE ROLE OF MUTATIONS AND DENSITY INDEPENDENT DISPERSAL IN EVOLUTIONARY RESCUE

Faced with a strong and sudden deterioration of environment, a population encounters two possible options — adapt or perish. In general, it is not known which of those outcomes the environmental changes will lead to. Building on experimental research, we introduce a discrete-space, discrete-time model for environmental rescue based on the influence of population dispersal, as well as, potentially beneficial mutations. Numerical results obtained by the model are shown to correspond well to experimentally obtained data.