Zvi Artstein

Stabilization with relaxed controls

cannot in general be stabilized using a continuous closed loop control U(X), even if each state separately can be driven asymptotically to the origin. (An example is analyzed in Section 2.) In this paper we examine the possibility of stabilizing such systems with a continuous closed loop relaxed control. We find, indeed, that the family of systems stabilizable with relaxed controls is larger than the family of those stabilizable with ordinary controls. An even larger class is obtained if the continuity of the closed loop at the origin is not required. The latter class includes all one dimensional systems for which states can be driven asymptotically to the origin. This result does not hold in two dimensional systems and we provide a counter-example. It should be pointed that relaxed control-type stabilization is used both in theory and in practice; the method is known as dither. We shall comment on the similarities. Lyapunov functions for the system (1) help us in the construction of the continuous closed loop stabilizers. In fact, we find that the existence of a smooth Lyapunov function is equivalent to the existence of a stabilizing closed loop which is continuous except possibly at the origin; an additional condition on the Lyapunov function implies the continuity at the origin as well. We present these results in Section 4, after a brief introduction of closed loop relaxed controls, notations and terminology in Section 3. Prior to that, in Section 2, we discuss an example illustrating the power of relaxed controls. In the particular case of systems linear in the controls, relaxed controls can be replaced by ordinary controls, this is discussed in Section 5. The role of Lyapunov functions in the stability and stabilization theories is of course well known. Examples of systems with Lyapunov functions are available in the literature. We display some in Section 6, along with general comments on the construction, applications and counterexamples, including one which cannot be continuously stabilized, yet possesses a nonsmooth Lyapunov function. Closed loop stabilization with ordinary controls is analyzed extensively in the literature, see Sontag [8], Sussmann [ll] and references therein. Lyapunov functions techniques in stabil-

Linear systems with delayed controls: A reduction

Linear systems with delayed control action are transformed into systems without delays. Under an absolute continuity condition, the new system is an ordinary differential control equation. In the general case, the new system is a measure-differential control system. It is shown how the controllability, stabilization, and various optimization problems can be analyzed via the reduced systems.

Uniform asymptotic stability via the limiting equations

The construction of Liapunov functions is a most powerful tool for establishing asymptotic and uniform asymptotic stability in ordinary differential equations; see [ 111. One finds, however, that in many cases it is very complicated to construct the appropriate Liapunov function. Another powerful tool was developed by LaSalle: by combining information obtained from simple and natural Liapunov functions with information about geometric and invariance properties of limit sets, one can establish asymptotic properties of solutions including asymptotic stability; see [6, 71. The LaSalle principle enables us to handle a variety of equations for which the geometry of the law of motion is detectable, and then relatively simple Liapunov functions are sufficient. (For the theory of nonautonomous equations see [4], a survey.) The t&-o methods are direct methods; i.e., one should he able to discover the asymptotic stability by looking at the equation and without, for instance, computing the solutions. The purpose of the present paper is to push LaSalle’s ideas further, to include uniform asymptotic stability. i\‘e again insist on direct methods. \Ve are willing to use the geometry and asymptotic properties of the equation, but would like to relax as much as possible the demands on the Liapunov functions. -1 major role in our techniques is played by the limiting equations the equations which describe the limiting behavior of the original nonautonomous law of motion; see 14, 6, 91. 0 ur main result is the characterization (under certain conditions) of uniform asymptotic stability with respect to an equation by mere asymptotic stability, but with respect to all its limiting equations. This abstract characterization becomes practical in those cases where the structure of the limiting equations is relatively clear. \5’e can use the LaSalle principle for handling the limiting equations and therefore obtain the results for the original equation. \Ve will present some examples below. The paper is organized as follows. The main results are prcscnted in Section 3. Secessary preliminaries for applying the results are listed in Section 4. An example is given in Section 5 in which we analyze a damped harmonic oscillator.

A note on fatou's lemma in several dimensions

The theorem answers, in the affirmative, the problem raised by Hildenbrand and Mertens (1971). A partial answer was given in Hildenbrand (1974, lemma 3, p. 69) where it is assumed that {fn(o)},, = r, 2, ,__ is bounded almost everywhere. A need for the full answer arose in the author’s recent work on continuous-time optimal allocation plans (paper forthcoming). We shall employ the theory of integration of correspondences (set-valued functions). Our notations and terminology follow Hildenbrand (1974, ch. I) where all the necessary material can be found; occasionally we shall give a precise reference.

Singularly perturbed ordinary differential equations with dynamic limits

Coupled slow and fast motions generated by ordinary differential equations are examined. The qualitative limit behaviour of the trajectories as the small parameter tends to zero is sought after. Invariant measures of the parametrised fast flow are employed to describe the limit behaviour, rather than algebraic equations which are used in the standard reduced order approach. In the case of a unique invariant measure for each parameter, the limit of the slow motion is governed by a chattering type equation. Without the uniqueness, the limit of the slow motion solves a differential inclusion. The fast flow, in turn, converges in a statistical sense to the direct integral, respectively the set-valued direct integral, of the invariant measures.

Brief paper: Feedback and invariance under uncertainty via set-iterates

We examine discrete-time control systems under non-parametric disturbances. Sets which a given control feedback makes invariant under the disturbance are analyzed via lifting the feedback operation to the space of sets. Properties of being an attractor of the disturbed dynamics and being a minimal invariant set are derived from the corresponding notions of the set-dynamics, yielding, in turn, useful characterizations and error estimates for numerical algorithms which detect the minimal invariant sets. Concrete numerics for some examples of practical feedback rules are offered.

Law of Large Numbers for Random Sets and Allocation Processes

In this paper we establish a strong law of large numbers for unbounded random sets, and then apply it to an optimization problem arising in allocation processes under uncertainty.

Distributions of random sets and random selections

Distributions of selections of a random set are characterized in terms of inequalities, similar to the marriage problem. A consequence is that the ensemble of such distributions is convex compact and depends continuously on the distribution of the random set.

Tracking Fast Trajectories Along a Slow Dynamics: A Singular Perturbations Approach

Controlled coupled slow and fast motions are examined in a singular perturbations setting. The objective is to minimize a cost functional that takes into account both the fast motion, supposing, say, tracking a fast target, and the slow dynamics. A method is offered to cope with the possibility that the fast flow has nonstationary limits. Invariant measures of the fast motion are then the controlled objects on the infinitesimal scale. Optimal amalgamation of them on the slow scale induces the variational limit, whose solutions are near optimal solutions of the perturbed system.

Examples of stabilization with hybrid feedback

Via examples, this paper examines the possibility of stabilizing a continuous control plant, through an interaction with a discrete time controller. Such a hybrid feedback complements the output feedback within the plant when the latter fails to stabilize the system or fails to produce smooth stabilization. The displayed examples point to both the mathematical and the design challenges that the method poses.