Multifunctions of bounded variation

Consider control systems described by a differential equation with a control term or, more generally, by a differential inclusion with velocity set F(t,x). Certain properties of state trajectories can be derived when it is assumed that F(t,x) is merely measurable w.r.t. the time variable t . But sometimes a refined analysis requires the imposition of stronger hypotheses regarding the time dependence. Stronger forms of necessary conditions for minimizing state trajectories can be derived, for example, when F(t,x) is Lipschitz continuous w.r.t. time. It has recently become apparent that significant addition properties of state trajectories can still be derived, when the Lipschitz continuity hypothesis is replaced by the weaker requirement that F(t,x) has bounded variation w.r.t. time. This paper introduces a new concept of multifunctions F(t,x) that have bounded variation w.r.t. time near a given state trajectory, of special relevance to control. We provide an application to sensitivity analysis

The Hamiltonian Inclusion for Nonconvex Velocity Sets

Since Clarke's 1973 proof of the Hamiltonian inclusion for optimal control problems with convex velocity sets, there has been speculation (and, more recently, speculation relating to a stronger, partially convexified version of the Hamiltonian inclusion) as to whether these necessary conditions are valid in the absence of the convexity hypothesis. The issue was in part resolved by Clarke himself when, in 2005, he showed that $L^{\infty}$ local minimizers satisfy the Hamiltonian inclusion. In this paper it is shown, by counterexample, that the Hamiltonian inclusion (and so also the stronger partially convexified Hamiltonian inclusion) are not in general valid for nonconvex velocity sets when the local minimizer in question is merely a $W^{1,1}$ local minimizer, not an $L^{\infty}$ local minimizer. The counterexample demonstrates that the need to consider $L^{\infty}$ local minimizers, not $W^{1,1}$ local minimizers, in the proof of the Hamiltonian inclusion for nonconvex velocity sets is fundamental, not just a technical restriction imposed by currently available proof techniques. The paper also establishes the validity of the partially convexified Hamiltonian inclusion for $W^{1,1}$ local minimizers under a normality assumption, thereby correcting earlier assertions in the literature

Decomposition of Differential Games with Multiple Targets

This paper provides a decomposition technique for the purpose of simplifying the solution of certain zero-sum differential games. The games considered terminate when the state reaches a target, which can be expressed as the union of a collection of target subsets considered as ‘multiple targets’; the decomposition consists in replacing the original target by each of the target subsets. The value of the original game is then obtained as the lower envelope of the values of the collection of games, resulting from the decomposition, which can be much easier to solve than the original game. Criteria are given for the validity of the decomposition. The paper includes examples, illustrating the application of the technique to pursuit/evasion games and to flow control

Differential Games Controllers That Confine a System to a Safe Region in the State Space, With Applications to Surge Tank Control

Surge tanks are units employed in chemical processing to regulate the flow of fluids between reactors. A notable feature of surge tank control is the need to constrain the magnitude of the Maximum Rate of Change (MROC) of the surge tank outflow, since excessive fluctuations in the rate of change of outflow can adversely affect down-stream processing (through disturbance of sediments, initiation of turbulence, etc.). Proportional + Integral controllers, traditionally employed in surge tank control, do not take direct account of the MROC. It is therefore of interest to explore alternative approaches. We show that the surge tank controller design problem naturally fits a differential games framework, proposed by Dupuis and McEneaney, for controlling a system to confine the state to a safe region of the state space. We show furthermore that the differential game arising in this way can be solved by decomposing it into a collection of (one player) optimal control problems. We discuss the implications of this decomposition technique, for the solution of other controller design problems possessing some features of the surge tank controller design problem

The application of dynamic programming to optimal inventory control

Published versio

Measurement Placement in Distribution System State Estimation

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The shifted Rayleigh mixture filter for bearings-only tracking of maneuvering targets

Published versio

A New Gaussian Mixture Algorithm for GMTI Tracking Under a Minimum Detectable Velocity Constraint

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The interpretation of discontinuous state feedback control laws as nonanticipative control strategies in differential games

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L∞ estimates on trajectories confined to a closed subset, for control systems with bounded time variation

The term ‘distance estimate’ for state constrained control systems refers to an estimate on the distance of an arbitrary state trajectory from the subset of state trajectories that satisfy a given state constraint. Distance estimates have found widespread application in state constrained optimal control. They have been used to establish regularity properties of the value function, to establish the non-degeneracy of first order conditions of optimality, and to validate the characterization of the value function as a unique solution of the HJB equation. The most extensively applied estimates of this nature are so-called linear L∞L∞ distance estimates. The earliest estimates of this nature were derived under hypotheses that required the multifunctions, or controlled differential equations, describing the dynamic constraint, to be locally Lipschitz continuous w.r.t. the time variable. Recently, it has been shown that the Lipschitz continuity hypothesis can be weakened to a one-sided absolute continuity hypothesis. This paper provides new, less restrictive, hypotheses on the time-dependence of the dynamic constraint, under which linear L∞L∞ estimates are valid. Here, one-sided absolute continuity is replaced by the requirement of one-sided bounded variation. This refinement of hypotheses is significant because it makes possible the application of analytical techniques based on distance estimates to important, new classes of discontinuous systems including some hybrid control systems. A number of examples are investigated showing that, for control systems that do not have bounded variation w.r.t. time, the desired estimates are not in general valid, and thereby illustrating the important role of the bounded variation hypothesis in distance estimate analysis