On open-loop and feedback attainability of a closed set for nonlinear control systems

In this note, we investigate the existence of controls which allow to reach a given closed set K through trajectories of a nonlinear control system. In the case where the set is sufficiently regular we give a condition allowing to find a feedback control law which ensures the existence of trajectories to reach the set. We also consider the case where all the trajectories reach K. When K is not necessarily attainable but only viable, we build a set-valued feedback for which the set is invariant. Our approach concerns continuous dynamics, possibly not C¹, so our methods do not come from geometric control theory. Furthermore, we do not require any regularity of the set K in order to obtain our results, except when we want to establish the existence of a feedback control law to achieve our goals.     

Computing the viability kernel in large state dimension

T-viable states in a closed set K under a certain set-valued dynamic are states from which there exists at least one solution remaining in K until a given time horizon T. Minimizing the cost to constraints lets us determine whether a given state is T-viable or not, and this is implementable in large dimension for the state-space. Minimizing on the initial condition itself lets find viable states. Quincampoix's semi-permeability property helps find other states located close to the viability boundary, which is then gradually delineated. The algorithm is particularly suited to the identification of specific trajectories, such as the heavy viable solution, or to the computation of viability kernels associated with delayed dynamics. The volume of the viability kernel and its confidence interval can be estimated by randomly drawing states and checking their viability status. Examples are given.     

Sliding mode control of uncertain systems: a singular perturbation approach

In this paper we consider a nonlinear control system affectedby deterministic uncertainty and described by a system of ordinarydifferential equations. The uncertainty is modelled by a multivaluedmap whose t-measurable and x-Lipschitz selections representthe possible system dynamics of the uncertain system. We proposea dynamical feedback control design, based on the singular perturbationtheory, which allows all the possible system trajectories correspondingto the system dynamics to have the same prescribed behaviour.Specifically, given a manifold K of the state space, definedas the zeros of a smooth map, the proposed control steers andthen holds, during finite or infinite time intervals, any possiblesystem trajectory to any prescribed neighbourhood of K. A resultensuring the exact attainability of K is also provided. Someexamples illustrating the obtained results are presented.     

Null controllability of a class of Newtonian filtration equations

In this paper, we study the null controllability of a class of Newtonian filtration equations. Using the properties of finite extinction time and finite speed of propagation, we construct a control function in feedback form, such that system is exactly null controllable at any time T>0. If we put a nonnegative constraint on the control function, this system is not exactly null controllable any more, but approximately null controllable for a long time.     

An explicit expression for the first return map in the center problem

The classical center-focus problem posed by H. Poincaré in 1880's asks about the characterization of planar polynomial vector fields such that all their integral trajectories are closed curves whose interiors contain a fixed point, a center. In this paper, we present a method allowing for the first time to obtain an explicit expression for the first return map in the center problem.     

Equivalent conditions on periodic feedback stabilization for linear periodic evolution equations

This paper studies the periodic feedback stabilization for a class of linear T  -periodic evolution equations. Several equivalent conditions on the linear periodic feedback stabilization are obtained. These conditions are related to the following subjects: the attainable subspace of the controlled evolution equation under consideration; the unstable subspace (of the evolution equation with the null control) provided by the Kato projection; the Poincaré map associated with the evolution equation with the null control; and two unique continuation properties for the dual equations on different time horizons [0,T]$[0,T]$ and [0,_n0T]$[0,n_{0}T]$ (where _n0$n_0$ is the sum of algebraic multiplicities of distinct unstable eigenvalues of the Poincaré map). It is also proved that a T-periodic controlled evolution equation is linear T-periodic feedback stabilizable if and only if it is linear T-periodic feedback stabilizable with respect to a finite-dimensional subspace. Some applications to heat equations with time-periodic potentials are presented.     

Finding safety in partially controllable chaotic systems

Many discrete-time dynamical systems have a region Q from which all or almost all trajectories leave, or at least they leave in the presence of perturbations that we call disturbances. We partially control systems so that despite disturbances the trajectories of a dynamical system stay in the region Q at least for some initial points in Q. The disturbances can be thought of as either noise or as purposeful, hostile efforts of an enemy to drive the trajectory out of the region. Our goal is to keep trajectories inside Q despite the disturbances and our partial control of chaos method succeeds.Surprisingly this goal can be achieved with a control whose maximum allowable size is smaller than the maximum allowed disturbance. A fundamental step towards this goal is to compute a set called the safe set that had, until now, been found only in certain very special situations.This paper provides a general algorithm for computing safe sets. The algorithm is able to compute the safe sets for a specified region in phase space, the maximum disturbance value, and the maximum allowed control. We call it the Sculpting Algorithm. Its operation is analogous to removing material while sculpting a statue. The algorithm sculpts the safe sets. Our Sculpting Algorithm is independent of the dimension and is fast for one- and two-dimensional dynamical systems. As examples, we apply the algorithm to two paradigmatic nonlinear dynamical systems, namely, the Hénon map and the Duffing oscillator.     

On the neighborhood in stabilizing period-T orbits for chaotic m degree polynomial dynamical system

In this paper, a problem of stabilizing a period-T orbit in discrete chaotic m degree polynomial dynamical systems is studied. The aim is to present a new method for determining the neighborhood of a period-T point in which the system remains stable when subjected to a linear feedback control. A theorem on the existence of neighborhood is rigorously proved using idea from functional analysis and polar coordinate transformation. The ways of implementing the obtained theorem in the Hénon map are proposed. The validity of this method is shown by numerical simulation.     

On the closure in W1q of the attainable subspace of linear time lag systems

This paper presents the necessary and sufficient condition for the closedness in the topology of W₁^q(T ? 1, T; Rⁿ) of the set of all complete final states x_{[T ? 1,T]} of a linear retarded system, which can be attained by L^p controls. The criterion is fully computable. The analyticity of the matrix coefficients of the system equation is assumed. A number of related results are presented, including a criterion for the closedness of the set of all trajectories of a nondelayed linear system with measurable coefficients.     

Transitive partially hyperbolic diffeomorphisms on 3-manifolds

The known examples of transitive partially hyperbolic diffeomorphisms on 3-manifolds belong to 3 basic classes: perturbations of skew products over an Anosov map of T², perturbations of the time one map of a transitive Anosov flow, and certain derived from Anosov diffeomorphisms of the torus T³. In this work we characterize the two first types by a local hypothesis associated to one closed periodic curve.     

Dependence on the parameters of the set of trajectories of the control system described by a nonlinear Volterra integral equation

In this paper the control system with limited control resources is studied, where the behavior of the system is described by a nonlinear Volterra integral equation. The admissible control functions are chosen from the closed ball centered at the origin with radius µ in L _p (p > 1). It is proved that the set of trajectories generated by all admissible control functions is Lipschitz continuous with respect to µ for each fixed p, and is continuous with respect to p for each fixed µ. An upper estimate for the diameter of the set of trajectories is given.     

Sufficient conditions for viability under imperfect measurement

Viability theory gives a necessary and sufficient condition for the existence of a (set-valued) state feedback control such that all trajectories of the closed-loop system starting from the graph of a given tube in the state space remain in the tube. Here we investigate the same problem in the case where only incomplete and inexact measurement of the state is available. In the time-invariant case, we give a sufficient condition for the existence of anoutput feedback regulation map. The condition is shown to be equivalent to Haddad's viability condition if the measurement is perfect.     

Structure of approximate solutions for discrete-time control systems arising in economic dynamics

In this work we study the structure of approximate solutions of an autonomous discrete-time control system with a compact metric space of states X which is a subset of a finite-dimensional Euclidean space. This control system is described by a nonempty closed set Ω⊂X×X which determines a class of admissible trajectories (programs) and by a bounded upper semicontinuous function v:Ω→R¹ which determines an optimality criterion. We are interested in turnpike properties of the approximate solutions which are independent of the length of the interval, for all sufficiently large intervals. Usually, in economic dynamics, the turnpike properties have been studied for systems such that all their good programs converge to a turnpike which was an interior point of Ω. In this paper we establish turnpike results for a large class of control systems for which the turnpike is not necessarily an interior point of Ω.     

Comparison of two randomized policy M/G/1 queues with second optional service, server breakdown and startup

The problem addressed in this paper is to compare the minimum cost of the two randomized control policies in the M/G/1 queueing system with an unreliable server, a second optional service, and general startup times. All arrived customers demand the first required service, and only some of the arrived customers demand a second optional service. The server needs a startup time before providing the first required service until the system becomes empty. After all customers are served in the queue, the server immediately takes a vacation and the system operates the (T, p)-policy or (p, N)-policy. For those two policies, the expected cost functions are established to determine the joint optimal threshold values of (T, p) and (p, N), respectively. In addition, we obtain the explicit closed form of the joint optimal solutions for those two policies. Based on the minimal cost, we show that the optimal (p, N)-policy indeed outperforms the optimal (T, p)-policy. Numerical examples are also presented for illustrative purposes.     

Riccati equations for generalized singular estimate control systems

We consider the Bolza problem associated with boundary/point control systems governed by strongly continuous semigroups. In continuation of our work in Lasiecka and Tuffaha [I. Lasiecka and A. Tuffaha, Riccati equations for the Bolza problem arising in boundary/point control problems governed by C ₀–semigroups satisfying a singular estimate, J. Optim. Theory Appl. 136 (2008), pp. 229–246; I. Lasiecka and A. Tuffaha, A Bolza optimal synthesis problem for singular estimate control systems, Control Cybernet 38(4B) (2009), pp. 1429–1460], we yet extend the theory to a more general class of control problems that are not analytic providing sharp blow-up rates for the regularity. Solvability of the associated Riccati equations and an optimal feedback synthesis are established. The presence of unbounded control actions, such as boundary/point controls, naturally lead to a singularity at the terminal point t?=?T of the optimal control and of the corresponding feedback operator as before. The class of control systems considered in this article is a generalization to the class usually referred to in the literature as ‘Singular Estimate Control Systems’. The prototype is still that of a PDE system consisting of coupled hyperbolic parabolic dynamics interacting on an interface with point/boundary control. The distinct feature of the class considered in this article is that the degree of unboundedness in the control is stronger than that allowed in the usual singular estimate control system configuration, giving rise to less regular optimal state trajectories.     

Controllability cost of conservative systems: resolvent condition and transmutation

This article concerns the exact controllability of unitary groups on Hilbert spaces with unbounded control operator. It provides a necessary and sufficient condition not involving time which blends a resolvent estimate and an observability inequality. By the transmutation of controls in some time L for the corresponding second-order conservative system, it is proved that the cost of controls in time T for the unitary group grows at most like exp(αL²/T) as T tends to 0. In the application to the cost of fast controls for the Schrödinger equation, L is the length of the longest ray of geometric optics which does not intersect the control region. This article also provides observability resolvent estimates implying fast smoothing effect controllability at low cost, and underscores that the controllability cost of a system is not changed by taking its tensor product with a conservative system.     

A deteriorating cold standby repairable system with priority in use

In this paper, a cold standby repairable system consisting of two dissimilar components and one repairman is studied. In this system, it is assumed that the working time distributions and the repair time distributions of the two components are both exponential and component 1 is given priority in use. After repair, component 2 is “as good as new” while component 1 follows a geometric process repair. Under these assumptions, using the geometric process and a supplementary variable technique, some important reliability indices such as the system availability, reliability, mean time to first failure (MTTFF), rate of occurrence of failure (ROCOF) and the idle probability of the repairman are derived. A numerical example for the system reliability R(t) is given. And it is considered that a repair-replacement policy based on the working age T of component 1 under which the system is replaced when the working age of component 1 reaches T. Our problem is to determine an optimal policy T^∗ such that the long-run average cost per unit time of the system is minimized. The explicit expression for the long-run average cost per unit time of the system is evaluated, and the corresponding optimal replacement policy T^∗ can be found analytically or numerically. Another numerical example for replacement model is also given.     

Adaptive PID-tracking control of muscle-like actuated compliant robotic systems with input constraints

In this paper, adaptive control strategies are analyzed to control a compliant robot arm in form of a double pendulum structure. Powered by antagonistic, muscle-like actuators, the controller shall enable the system to track prescribed trajectories for a desired movement of the arm, as quickly and precisely as possible. But, due to the natural muscle behavior, the control variables underly prescribed bounds: the limited control inputs are generated by muscle intensities. Therefore, feedback strategies are sought which have to be limited/saturated from the very beginning. For this, at first, simulation studies of adaptive λ-tracking controllers are performed to apply the best-rated ones to the compliant robotic system. After some model adjustments concerning the mechanics of a prototype (for experimental verification), an intensity control is determined, to divide the controller variables to the respective muscle pairs. Various numerical simulations of a system with DoF=2 show the effectiveness of the saturated controllers in set-point and tracking control. The efficiency of these controllers is verified and proven in experiments of a prototype.     

A generalization of Caffarelli’s contraction theorem via (reverse) heat flow

A theorem of L. Caffarelli implies the existence of a map, pushing forward a source Gaussian measure to a target measure which is more log-concave than the source one, which contracts Euclidean distance (in fact, Caffarelli showed that the optimal-transport Brenier map T _opt is a contraction in this case). We generalize this result to more general source and target measures, using a condition on the third derivative of the potential, by providing two different proofs. The first uses a map T, whose inverse is constructed as a flow along an advection field associated to an appropriate heat-diffusion process. The contraction property is then reduced to showing that log-concavity is preserved along the corresponding diffusion semi-group, by using a maximum principle for parabolic PDE. In particular, Caffarelli??s original result immediately follows by using the Ornstein?CUhlenbeck process and the Prékopa?CLeindler Theorem. The second uses the map T _opt by generalizing Caffarelli??s argument, employing in addition further results of Caffarelli. As applications, we obtain new correlation and isoperimetric inequalities.