Solution of nonlinear optimal control problems in Hilbert spaces by means of linear programming techniques

Optimal control problems in Hilbert spaces are considered in a measure-theoretical framework. Instead of minimizing a functional defined on a class of admissible trajectory-control pairs, we minimize one defined on a set of measures; this set is defined by the boundary conditions and the differential equation of the problem. The new problem is an infinite-dimensionallinear programming problem; it is shown that it is possible to approximate its solution by that of a finite-dimensional linear program of sufficiently high dimensions, while this solution itself can be approximated by a trajectory-control pair. This pair may not be strictly admissible; if the dimensionality of the finite-dimensional linear program and the accuracy of the computations are high enough, the conditions of admissibility can be said to be satisfied up to any given accuracy. The value given by this pair to the functional measuring the performance criterion can be about equal to theglobal infimum associated with the classical problem, or it may be less than this number. It appears that this method may become a useful technique for the computation of optimal controls, provided the approximations involved are acceptable.     

On the optimum design of piston rings

We study a family of control problems that arise in the design of dynamically loaded bearings which have a minimum power loss; in particular, we study the design of piston rings with minimum parasitic power loss.This research was partially supported by NSF Grant No. MCS-77-05624.     

On Brunovský's example

The example due to Brunovský [Eq. (1)] is analyzed, and further paradoxical properties are exhibited.     

Opportunistic Multi-Access: Multiuser Diversity, Relay-Aided Opportunistic Scheduling, and Traffic-Aided Smooth Admission Control

We study multi-access control in opportunistic communication systems, and propose two new schemes to address channel asymmetry and throughput-guaranteed admission control, respectively. We first devise a relay-aided opportunistic scheduling (RAOS) scheme, in which a user can choose to communicate with the base station either directly or using multiple hops (relay transmissions). We develop relay/direct link construction algorithms using either a channel-capacity-based criterion or a throughput-based criterion, and devise opportunistic scheduling schemes accordingly. Our results show that in the presence of channel asymmetry across users, the RAOS scheme performs significantly better than Qualcomm's HDR scheme. Next, we propose a traffic-aided smooth admission control (SAC) scheme that aims to guarantee throughput provisioning. Simply put, in the SAC scheme, the admission decision is spread over a trial period, by increasing gradually the amount of the time resource allocated to incoming users. Specifically, using the modified weighted proportional fair (WPF) scheduling, we devise a QoS driven weight adaptation algorithm, and the weights assigned to new users are increased in a guarded manner. Then an admission decision is made based on the measured throughput within a time-out window. A key feature is that we exploit explicitly the traffic information and throughput requirements in devising the back-off time. Our results show that the proposed SAC scheme works well in opportunistic communication systems.     

The Number of Neighbors Needed for Connectivity of Wireless Networks

Wireless Networks - Unlike wired networks, wireless networks do not come with links. Rather, links have to be fashioned out of the ether by nodes choosing neighbors to connect to. Moreover the...     

A time-decomposition algorithm for a stiff linear two-point boundary-value problem and its application to a nonlinear optimal control problem

An algorithm is proposed to solve a stiff linear two-point boundary-value problem (TPBVP). In a stiff problem, since some particular solutions of the system equation increase and others decrease rapidly as the independent variable changes, the integration of the system equation suffers from numerical errors. In the proposed algorithm, first, the overall interval of integration is divided into several subintervals; then, in each subinterval a sub-TPBVP with arbitrarily chosen boundary values is solved. Second, the exact boundary values which guarantee the continuity of the solution are determined algebraically. Owing to the division of the integration interval, the numerical error is effectively reduced in spite of the stiffness of the system equation. It is also shown that the algorithm is successfully imbedded into an interaction-coordination algorithm for solving a nonlinear optimal control problem.The authors would like to thank Mr. T. Sera and Mr. H. Miyake for their help with the calculations.     

Lower closure and existence theorems for optimal control problems with infinite horizon

Lower closure theorems are proved for optimal control problems governed by ordinary differential equations for which the interval of definition may be unbounded. One theorem assumes that Cesari's property (Q) holds. Two theorems are proved which do not require property (Q), but assume either a generalized Lipschitz condition or a bound on the controls in an appropriateL _p-space. An example shows that these hypotheses can hold without property (Q) holding.     

Optimal control of coupled spin dynamics: design of NMR pulse sequences by gradient ascent algorithms

In this paper, we introduce optimal control algorithm for the design of pulse sequences in NMR spectroscopy. This methodology is used for designing pulse sequences that maximize the coherence transfer between coupled spins in a given specified time, minimize the relaxation effects in a given coherence transfer step or minimize the time required to produce a given unitary propagator, as desired. The application of these pulse engineering methods to design pulse sequences that are robust to experimentally important parameter variations, such as chemical shift dispersion or radiofrequency (rf) variations due to imperfections such as rf inhomogeneity is also explained.     

Construction of universal rotations from point-to-point transformations

For a desired range of offsets, universal rotations of arbitrary flip angle can be constructed based on point-to-point rotations of I(y) with half the flip angle. This approach allows, for example, creation of broadband or bandselective refocusing pulses from broadband or bandselective excitation pulses. Furthermore, universal rotations about any axis can be obtained from point-to-point transformations that can easily be optimized using optimal control algorithms. The construction procedure is demonstrated on the examples of a broadband refocusing pulse, a broadband 120(x) degrees rotation and a z-rotation with offset pattern.