Computation of alternated reachability tubes for linear control systems under uncertainty

The problem of constructing the reachability domain for linear control system in a presence of geometrically bounded unknown disturbance is considered. A high actuality of the problem for engineering applications requires an efficient calculation technique for the reach sets in a class of closed-loop control. The technique suggested in the article is based on the ellipsoidal approximations developed by A.B. Kurzhansky for the alternated reachability domains. In the article these estimates are complemented with an adaptive regularization to guarantee the continuability of the ellipsoidal estimates. The quadratic structure of the regularization combines well with an ellipsoidal nature of the estimates thus making it possible to adjust the existing ellipsoidal estimation schema in a transparent fashion for achieving the continuable and non-singular estimates via adaptive choice of regularization parameters.     

On the convexity of reachability sets of controlled initial-boundary value problems

For a functional-operator equation describing a broad class of controlled initial-boundary value problems, we introduce the notion of abstract reachability set. We obtain sufficient conditions for the convexity and precompactness of that set. The situation of a Nash ?-equilibrium is justified in the sense of program strategies in noncooperative functional-operator games with many players. As an example of reduction of a controlled initial-boundary value problem to the equation under study, we consider the Cauchy problem for a semilinear wave equation with two space variables.     

Sufficient conditions for the controllability of nonlinear distributed systems

For nonlinear distributed systems representable as a Volterra functional operator equation in a Lebesgue space, sufficient conditions for pointwise controllability with respect to a vector of non-linear functionals are proved. The controls are assumed to be piecewise constant vector functions. The reduction of controlled distributed systems to the functional operator equation under study is illustrated by two examples: a Dirichlet boundary value problem for the diffusion equation and a mixed problem for the transport equation.     

State constrained reachability for stochastic hybrid systems

Many control problems can be formulated as driving a system to reach some target states while avoiding some unwanted states. We study this problem for systems with regime change operating in uncertain environments. Nowadays, it is a common practice to model such systems in the framework of stochastic hybrid system models. In this casting, the problem is formalized as a mathematical problem named state constrained stochastic reachability analysis. In the state constrained stochastic reachability analysis, this probability is computed by imposing a constraint on the system to avoid the unwanted states. The scope of this paper is twofold. First we define and investigate the state constrained reachability analysis in an abstract mathematical setting. We define the problem for a general model of stochastic hybrid systems, and we show that the reach probabilities can be computed as solutions of an elliptic integro-differential equation. Moreover, we extend the problem by considering randomized targets. We approach this extension using stochastic dynamic programming. The second scope is to define a developmental setting in which the state constrained reachability analysis becomes more tractable. This framework is based on multilayer modelling of a stochastic system using hierarchical viewpoints. Viewpoints represent a method originated from software engineering, where a system is described by multiple models created from different perspectives. Using viewpoints, the reach probabilities can be easily computed, or even symbolically calculated. The reach probabilities computed in one viewpoint can be used in another viewpoint for improving the system control. We illustrate this technique for trajectory design.     

Optimal stochastic control without convexity conditions in the dynamical equation

Journal of Optimization Theory and Applications - The stochastic control system studied is modelled by then-dimensional stochastic differential equation $$dz_t = f(t, z, u) dt + \sigma (t, z) dW_t...     

Infinite-horizon boundary control of distributed systems

For a boundary controlled dynamic system, algorithms for solving the problem of tracking reference motion and the problem of tracking reference control are described. The algorithms are robust to information noise and computational errors. The solution method is based on the extremal shift method from the theory of positional differential games.     

Mobile control of distributed parameter systems

Book Review

Mobile control of distributed parameter systemsA. G. Butkovskiy and L. M. Pustyl'nikov: translated by L. W. Longdon, Ellis Horwood, Halsted Press, New York, Chichester, Brisbane, Toronto, 1987, 310 pp. US$45.00. ISBN 0-85312-507-4 (Ellis Horwood Limited), ISBN 0-470-20817-1 (Halsted Press).     

Dynamic convexity for natural thermostatted systems

Natural thermostatted systems are mechanical systems whose Lagrangian is the difference of a kinetic and a potential energy, subjected to the nonholonomic constraint of a constant kinetic energy. When any two points of the configuration space are joined by a thermostatted motion, we say that the system is dynamically convex. A thermostatted charged particle on the plane with a constant electric field is not a dynamically convex system. We prove a general sufficient condition for dynamic convexity, from which whole classes of examples are easily constructed.     

Optimality conditions for age-structured control systems

We consider a fairly general model (extension of the Gurtin-MacCamy model of population dynamics) of an age structured control system with nonlocal dynamics and nonlocal boundary conditions. A necessary optimality condition is obtained in the form of Pontryagin's maximum principle, which is applicable to a number of practically meaningful models where the previously known results fail. We discuss such models (an epidemic control, and a capital accumulation model) as illustrations.     

Existence results without convexity conditions for general problems of optimal control with singular components

This note presents a new, quick approach to existence results without convexity conditions for optimal control problems with singular components in the sense of E. J. McShane (SIAM J. Control5 (1967), 438–485). Starting from the resolvent kernel representation of the solutions of a linear integral equation, a version of Fatou's lemma in several dimensions is shown to lead directly to a compactness result for the attainable set and an existence result for a Mayer problem. These results subsume those of L. W. Neustadt (J. Math. Anal. Appl.7 (1963), 110–117), C. Olech (J. Differential Equations2 (1966), 74–101), M. Q. Jacobs (“Mathematical Theory of Control,” pp. 46–53, Academic Press, 1967), L. Cesari (SIAM J. Control12 (1974), 319–331) and T. S. Angell (J. Optim. Theory Appl.19 (1976), 63–79).     

New technique of sensitivity reduction for distributed control systems

Control systems governed by linear parabolic partial differential equations are considered. A new technique for the reduction of the state sensitivity to small parameter variations is presented. A new cost function, which does not depend on the parameter, is defined. The corresponding optimal control is obtained in an open-loop form, then it is transformed into a feedback form so that a better reduction of the sensitivity is guaranteed. A simple example illustrates the application of the technique.     

Solvability of mixed-type optimal control problems for distributed systems

In this paper we consider optimal control problems for distributed systems which are not solved with respect to the time derivative, where performance functionals take a quadratic form. We concurrently use the distributed and starting controls.     

Global reachability results for systems with constrained controllers

The global reachability problem is to determine if, given a specified initial state, the state of a system can be steered via an admissible control to every point in the state space. This paper addresses the problem of global reachability when there are magnitude constraints on the controls. Necessary conditions and sufficient conditions for a system to be globally reachable are presented. The results are compared with those available for the global controllability problem.Dedicated to G. LeitmannThe research of the first author was supported by the National Science Foundation under Grant No. ECS-82-10284.     

Dissipativity for Lur’e distributed parameter control systems

In this work, dissipativity of Lur’e distributed parameter control systems has been addressed. Delay-dependent sufficient conditions for the dissipativity with respect to the infinite-dimensional version of energy supply rate (Q₁,S₁,R₁) characterized exclusively by unbounded operator Q₁ are established in terms of linear operator inequalities (LOIs). Finally, the heat equation illustrates our result.     

Local approximation of control systems

Given two control systems where the control enters linearly, a sufficient condition is derived that one system locally approximates the other, i.e., there exists a map between the state spaces which carries the trajectory of the first system for any control into the trajectory of the second system for the same control with an error that grows like a power of t.     

Sharp conditions for global stability of Lotka-Volterra systems with distributed delays

We give a criterion for the global attractivity of a positive equilibrium of n-dimensional non-autonomous Lotka-Volterra systems with distributed delays. For a class of autonomous Lotka-Volterra systems, we show that such a criterion is sharp, in the sense that it provides necessary and sufficient conditions for the global asymptotic stability independently of the choice of the delay functions. The global attractivity of positive equilibria is established by imposing a diagonal dominance of the instantaneous negative feedback terms, and relies on auxiliary results showing the boundedness of all positive solutions. The paper improves and generalizes known results in the literature, namely by considering systems with distributed delays rather than discrete delays.