Lawrence Markus

Exact boundary controllability of a hybrid system of elasticity

A hybrid control system is presented: an elastic beam, governed by a partial differential equation, linked to a rigid body which is governed by an ordinary differential equation and to which control forces and torques are applied. The entire system, elastic beam plus rigid body, is proved to be exactly controllable by smooth open-loop controllers applied to the rigid body only, and in arbitrarily short durations. This system is modeled as a two-dimensional space-structure.

Optimal control for nonlinear processes

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Catastrophe Theory and the Study of War

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Nonlinear boundary-value problems arising in chemical reactor theory

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Controllability of [r]-matrix quasi-differential equations

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Control dynamical systems

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Generic Properties of Differential Equations

The methodologies and conceptual frameworks employed by contemporary scholars do not satisfactorily account for the outbreak of international violence. Particularly frustrating to the analyst are the facts: (1) Some wars start and escalate suddenly while others begin and gradually build in intensity; yet in either case assumed causal factors are seen to vary smoothly and continuously through time. (2) The international system may be either at war or at peace at two separate time points even though the causal factors exhibit similar configurations. What is required is a framework that can deal with continuous and discontinuous dependent variables and continuous independent variables. Catastrophe theory provides the necessary classifactory structure in which to construct a systemic-level theory of the occurrence of international violence. Changes in the level of violence in World War I and World War II and the differences between these two conflicts are interpreted in the context of a particular catastrophe model—the butterfly. Catastrophe theory is discussed in light of the problems of description, classification, and empirical generalization in the construction of social science theory.

Dynamical boundary control of elastic plates

Chemical reactions are often housed in long tubes; the reactants are admitted at one end of the tube in a continuous stream, and the products of the reaction are withdrawn at the other end. In many important cases the tube wall is adiabatic so that the heat generated (or absorbed) during the reaction tends to increase (or decrease) the temperature of the reaction mixture. A basic problem in tubular reactor design can be stated: given the flow rate of reactants, the nature of the chemical reaction, and the usual chemical and thermodynamical constants of the process, what is the appropriate length of the reactor tube ? In order to attack this problem effectively it is important to develop a mathematical model of the process, following experience and intuition in the selection and analysis of the significant physical parameters. We shall assume that the problem is one-dimensional along the axis of the tube and that the two significant physical mechanisms are the forced convective flow and a diffusive mechanism superimposed on this flow. The convective flow is characterized by the average axial velocity along the reactor tube and the diffusive mechanism, based on Fick’s Law, usually arises from Taylor diffusion, turbulent diffusion, or molecular diffusion. The first two of these diffusive mechanisms are important in industrial reactors while the third plays a role in flame theory.

Differential Independence of Γ and ζ

Abstract Consider the controllability theory for the linear system x′ = A(t)x + B(t)u, ( l ) where A(t), B(t) are complex-valued matrices in Lloc1 on a real-interval I, and where x(t) is the state and u(t) ϵ Lrmloc∞ is the controller. For an [r]-matrix control system the components x1, x2, …, xnof x, and also u1, u2, …, umof u, as well as the entries of A(t) and B(t), are each r × r matrices, for a fixed integer r ⩾ 1. In the important case of [r]-matrix quasi-differential control systems (and [r]-matrix quasi-differential control equations), where there is a prescribed special format for the matrices A(t) and B(t) (as suggested by the theory of scalar control equations), the authors obtain explicit criteria for the full controllability of ( l ) on I. For r = 1 (the classical case) new and explicit controllability criteria are found for linear control systems with continuous (or merely integrable) time-varying coefficients. The methods are also new and rest only on familiar results of real analysis. Analogous results are further obtained for [r]-matrix linear control systems, for r ⩾ 1. The motivation for considering r > 1 arose from the demand for a level of generality sufficient to deal with the increasingly important matrix linear systems of control theory, such as those associated with matrix Riccati differential equations, and matrix bilinear control systems.

Capture and control in conservative dynamical systems

x(t) = ~([u], Xo, t o ; t). In control theory u(t) is selected to steer x(t) along some prescribed path or to display some desired behavior. Thus the control is introduced for some specific purpose. In this article we take a different viewpoint, that control represents a natural uncertainty in the mathematical description of a physical dynamical system. This uncertainty corresponds to an intrinsic noise or measurement roughness that tends to blur out the microscopic complexities and pathologies of the solution curve family and enables us to discover the basic qualitative features of the dynamical system without the distraction of the fine structure. In this sense the study of control dynamical systems has the same philosophical motivation as the theory of structural stability [2], and has the same technical apparatus as the theory of contingent or multivalued differential equations [1]. In this paper we develop a qualitative theory of control dynamical systems on a compact manifold W, where the free motion (with u(t) -0) is conservative or measure-preserving. We prove a general periodicity theorem, Theorem 1, that every initial point lies on a controlled periodic orbit, which generalizes the Poincard-Carathdodory recurrence theorem. Also we present a general transitivity theorem, Theorem 2, that the set of attainability from any initial state is the entire space W, which can be interpreted as a global controllability theorem.