Necessary conditions for joining optimal singular and nonsingular subarcs

Necessary conditions for optimality of junctions between singular and nonsingular subarcs for singular optimal control problem

A Characterization of all Solutions to the Four Block General Distance Problem

All solutions to the four block general distance problem which arises in H^∞ optimal control are characterized. The procedure is to embed the original problem in an all-pass matrix which is constructed. It is then shown that part of this all-pass matrix acts as a generator of all solutions. Special attention is given to the characterization of all optimal solutions by invoking a new descriptor characterization of all-pass transfer functions. As an application, necessary and sufficient conditions are found for the existence of an H^∞ optimal controller. Following that, a descriptor representation of all solutions is derived

Equilibrium of Heterogeneous Congestion Control: Optimality and Stability

When heterogeneous congestion control protocols that react to different pricing signals share the same network, the current theory based on utility maximization fails to predict the network behavior. The pricing signals can be different types of signals such as packet loss, queueing delay, etc, or different values of the same type of signal such as different ECN marking values based on the same actual link congestion level. Unlike in a homogeneous network, the bandwidth allocation now depends on router parameters and flow arrival patterns. It can be non-unique, suboptimal and unstable. In Tang et al. (“Equilibrium of heterogeneous congestion control: Existence and uniqueness,” IEEE/ACM Trans. Netw., vol. 15, no. 4, pp. 824–837, Aug. 2007), existence and uniqueness of equilibrium of heterogeneous protocols are investigated. This paper extends the study with two objectives: analyzing the optimality and stability of such networks and designing control schemes to improve those properties. First, we demonstrate the intricate behavior of a heterogeneous network through simulations and present a framework to help understand its equilibrium properties. Second, we propose a simple source-based algorithm to decouple bandwidth allocation from router parameters and flow arrival patterns by only updating a linear parameter in the sources’ algorithms on a slow timescale. It steers a network to the unique optimal equilibrium. The scheme can be deployed incrementally as the existing protocol needs no change and only new protocols need to adopt the slow timescale adaptation

Minimum fuel horizontal flightpaths in the terminal area

The problem of minimum fuel airplane trajectories from arbitrary initial states to be fixed final state is considered. There are four state variables (two position coordinates, heading, and constrained velocity) and two constrained controls (thrust and bank angle). The fuel optimality of circular and straight line flightpaths is examined. Representative extremals (trajectories satisfying the necessary conditions of the minimum principle) of various types are computed and used to evaluate trajectories generated by an on line algorithm. Attention is paid to the existence of Darboux points (beyond which an extremal ceases to be globally optimal). One fuel flow rate model includes a term quadratic in thrust; hence, the optimal thrust is continuous and nonsingular. The other fuel flow rate model is linear in thrust, and consequently the optimal thrust is discontinuous and singular

Second-order optimality conditions for the Bolza problem with path constraints

A set of sufficient conditions for a weak minimum is derived for a form of the nonsingular Bolza problem of variational calculus, with interior point constraints and discontinuities in the system equations. Generalized versions of the conjugate point/focal point, normality, convexity and nontangency conditions associated with the ordinary Bolza problem are obtained. The resulting set of sufficient conditions is minimal, in that only minor modifications are required in order to obtain necessary conditions for normal, nonsingular problems of this form. These conditions are relatively easy to implement. Analogous second-order optimality conditions for problems with natural corners or control constraints are also obtained. Previously stated sufficiency conditions for problems with control constraints are shown to be unnecessarily restrictive, in some cases