Phase Constraints Research Articles

Approximation methods in optimal control problems for nonlinear infinite-dimensional systems

Some notions related to approximate solutions and to the approximation of extremum problems for nonlinear infinite-dimensional systems are proposed. Optimization problems for nonlinear parabolic equations with a fixed terminal state and on an infinite time interval, as well as for singular stationary systems with phase constraints, are illustrated by several examples.

An optimal control problem for a nonlinear elliptic equation with a phase constraint and state variation

We study an optimal control problem for a system described by a nonlinear elliptic equation with a state constraint in the form of an inclusion. We prove the solvability of the problem under consideration and by varying the state of the system obtain necessary optimality conditions.

Optimal therapy protocols in the mathematical model of acute leukemia with several phase constraints

SUMMARYA mathematical model for leukemia therapy based on the Gompertzian law of cell growth is investigated. The effect of the therapy is described in terms of therapy functions. Two types of therapy functions are considered: monotone and non‐monotone. In the first case, the effect of the chemotherapy on the normal cells directly depends on the quantity of the chemotherapeutic agent. In the second case, we consider a therapy function with a threshold effect, which influences the leukemia cells. This effect can describe resistance to the therapy. The optimal control problem has the purpose of minimizing the number of the leukemia cells subject to the constraints of the normal cells, whose number must not fall below a prescribed critical limit and of the amount of the chemotherapeutic agents in the body of the patient (i.e. constraint on the toxicity). Furthermore, a third phase constraint on the cumulative amount of the medicine over the whole therapy process is considered, which can be interpreted as an economic factor of the therapy. This optimization problem is solved analytically with the help of the Pontryagin's Maximum Principle, extension of penalty functions and additional analysis. Copyright © 2013 John Wiley & Sons, Ltd.

On a solution of the Hamilton-Jacobi equation arising in molecular biology

We consider the Cauchy problem for the Hamilton-Jacobi equation with phase constraints that arises in molecular biology. We introduce a continuous generalized solution and propose a method of constructing in with the cost function for the auxiliary optimal control problem.

Wideband Interferometry in Synthetic Aperture Sonar

Synthetic aperture sonar (SAS) interferometry can provide very high resolution images and topographic maps of the seafloor over large swaths. Processing of interferometric data to retrieve reliable depth estimates of the seafloor is, however, nontrivial. Traditional narrowband interferometry processing relies on advanced phase unwrapping techniques, constraints, and prior knowledge to resolve phase ambiguities. These methods all have dependencies throughout the dataset, e.g., a small error in the assumption may cause a global error in the final phase estimate. For wideband systems, there are alternative techniques to estimate the absolute (or ambiguity-free) phase difference directly. We consider four different wideband interferometry techniques for direct phase difference estimation: complex cross correlation, split-spectrum algorithm, and the multichromatic approach. In addition, we develop a weighted split-spectrum algorithm, where the weights minimize the variance in the absolute phase estimate. We benchmark these techniques on simulated data and demonstrate the techniques on real data from the HISAS wideband interferometric SAS on a HUGIN autonomous underwater vehicle. We have found the following: the cross correlation technique always outperforms the other techniques in misregistrated areas caused by severe topographic changes. The split spectrum techniques are substantially faster than the cross correlation technique. The multichromatic approach has similar performance as the multiband split spectrum technique for the same choice of bands. We demonstrate that all the wideband interferometry techniques outperform a standard Goldstein's Branch cut phase unwrapping algorithm on real data from a complicated scene containing an elevated target and severe layover.

Design of non-selective refocusing pulses with phase-free rotation axis by gradient ascent pulse engineering algorithm in parallel transmission at 7 T

At ultra-high magnetic field (⩾7T), B1 and ΔB0 non-uniformities cause undesired inhomogeneities in image signal and contrast. Tailored radiofrequency pulses exploiting parallel transmission have been shown to mitigate these phenomena. However, the design of large flip angle excitations, a prerequisite for many clinical applications, remains challenging due the non-linearity of the Bloch equation. In this work, we explore the potential of gradient ascent pulse engineering to design non-selective spin-echo refocusing pulses that simultaneously mitigate severe B1 and ΔB0 non-uniformities. The originality of the method lays in the optimization of the rotation matrices themselves as opposed to magnetization states. Consequently, the commonly used linear class of large tip angle approximation can be eliminated from the optimization procedure. This approach, combined with optimal control, provides additional degrees of freedom by relaxing the phase constraint on the rotation axis, and allows the derivative of the performance criterion to be found analytically. The method was experimentally validated on an 8-channel transmit array at 7T, using a water phantom with B1 and ΔB0 inhomogeneities similar to those encountered in the human brain. For the first time in MRI, the rotation matrix itself on every voxel was measured by using Quantum Process Tomography. The results are complemented with a series of spin-echo measurements comparing the proposed method against commonly used alternatives. Both experiments confirm very good performance, while simultaneously maintaining a low energy deposition and pulse duration compared to well-known adiabatic solutions.

On question about extension of maximin problem with phase constraints

We study the asymptotic behavior of maximin values of a payoff function, when relaxed constraints are tightened. The payoff function depends on the trajectories of controlled systems of the first and second player. An extension in the class of the Radon measures is used. The asymptotic equivalence between two types of the constraints relaxations is shown.

On a differential game with integral and phase constraints

Consideration is given to a differential game on the plane when the moves of a pursuer and a running away are described by equations of the simple move, and integral constraints are imposed on control functions. In the process of the game the pursuer moves within the prescribed closed convex subset of the plane and the running away moves along its boundary. In the case when the running away has an advantage of resources, the development is made of the runaway strategy that affords the lower bound for the distance between the players. In the case when the pursuer has an advantage of resources, for any initial positions of the players the optimal time of pursuit is found and optimal strategies of the players are set up.

Does the entorhinal cortex use the Fourier transform?

Some neurons in the entorhinal cortex (EC) fire bursts when the animal occupies locations organized in a hexagonal grid pattern in their spatial environment. Place cells have also been observed, firing bursts only when the animal occupies a particular region of the environment. Both of these types of cells exhibit theta-cycle modulation, firing bursts in the 4–12 Hz range. Grid cells fire bursts of action potentials that precess with respect to the theta cycle, a phenomenon dubbed “theta precession.” Various models have been proposed to explain these phenomena, and how they relate to navigation. Among the most promising are the oscillator interference models. The bank-of-oscillators model proposed by Welday et al. (2011) exhibits all these features. However, their simulations are based on theoretical oscillators, and not implemented entirely with spiking neurons. We extend their work in a number of ways. First, we place the oscillators in a frequency domain and reformulate the model in terms of Fourier theory. Second, this perspective suggests a division of labor for implementing spatial maps: position vs. map layout. The animal's position is encoded in the phases of the oscillators, while the spatial map shape is encoded implicitly in the weights of the connections between the oscillators and the read-out nodes. Third, it reveals that the oscillator phases all need to conform to a linear relationship across the frequency domain. Fourth, we implement a partial model of the EC using spiking leaky integrate-and-fire (LIF) neurons. Fifth, we devise new coupling mechanisms, enlightened by the global phase constraint, and show they are capable of keeping spiking neural oscillators in consistent formation. Our model demonstrates place cells, grid cells, and phase precession. The Fourier model also gives direction for future investigations, such as integrating sensory feedback to combat drift, or explaining why grid cells exist at all.

Generalized iNverse imaging (GIN): Ultrafast fMRI with physiological noise correction

An ultrafast functional magnetic resonance imaging (fMRI) technique, called generalized inverse imaging (GIN), is proposed, which combines inverse imaging with a phase constraint-leading to a less underdetermined reconstruction-and physiological noise correction. A single 3D echo planar imaging (EPI) prescan is sufficient to obtain the necessary coil sensitivity information and reference images that are used to reconstruct standard images, so that standard analysis methods are applicable. A moving dots stimulus paradigm was chosen to assess the performance of GIN. We find that the spatial localization of activation for GIN is comparable to an EPI protocol and that maximum z-scores increase significantly. The high temporal resolution of GIN (50 ms) and the acquisition of the phase information enable unaliased sampling and regression of physiological signals. Using the phase time courses obtained from the 32 channels of the receiver coils as nuisance regressors in a general linear model results in significant improvement of the functional activation, rendering the acquisition of external physiological signals unnecessary. The proposed physiological noise correction can in principle be used for other fMRI protocols, such as simultaneous multislice acquisitions, which acquire the phase information sufficiently fast and sample physiological signals unaliased.

Optimal control in a mathematical model for leukemia therapy with phase constraints

We examine the optimal control problem that arises in the mathematical modeling of leukemia therapy, to solve which the Pontryagin maximum principle and the penalty function method are employed. It is assumed that the drug is capable of killing not only diseased cells, but healthy cells as well. The character of the drug’s interaction with cells is described by appropriate therapy functions.

Existence of optimal control for regularized problem with phase constraint

We establish the existence of a solution to a system of differential equations governing a chemical reactor. We regularize the target functional and prove the existence of an optimal control for the regularized problem. Bibliography: 18 titles.

Простое преследование с фазовыми ограничениями в классе импульсных стратегий

Простое преследование с фазовыми ограничениями в классе импульсных стратегий

Stabilizing and robust fractional order PI controller synthesis for first order plus time delay systems

For all the stable first order plus time delay (FOPTD) systems, a fractional order proportional integral (FOPI) or a traditional integer order proportional integral derivative (IOPID) controller can be designed to fulfill a flat phase constraint and two design specifications simultaneously: gain crossover frequency and phase margin. In this paper, a guideline for choosing two feasible or achievable specifications, and a new FOPI/IOPID controller synthesis are proposed for all the stable FOPTD systems. Using this synthesis scheme, the complete feasible region of two specifications can be obtained and visualized in the plane. With this region as the prior knowledge, all combinations of two specifications can be verified before the controller design. Especially, it is interesting to compare the areas of these two feasible regions for the IOPID and FOPI controllers. This area comparison reveals, for the first time, the potential advantages of one controller over the other in terms of achievable performances. A simulation illustration is presented to show the effectiveness and the performance of the designed FOPI controller compared with the optimized integer order PI controller and the IOPID controller designed following the same synthesis for the FOPI in this paper.

Optimization of the law of moving object evasion from detection under constraints

Consideration was given to optimization of the laws of moving object evasion from detection under the coordinate and phase constraints.

Quantum particle on a Möbius strip, coherent states and projection operators

The coherent states for a quantum particle on a Möbius strip are constructed and their relation to the natural phase space for fermionic fields is shown. The explicit comparison of the obtained states with previous works where the cylinder quantization was used and the spin-1/2 was introduced by hand is given, and the relation between the geometrical phase space, constraints and projection operators is analyzed and discussed.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Coherent states: mathematical and physical aspects’.

Canonical simplicial gravity

A general canonical formalism for discrete systems is developed, which can handle varying phase space dimensions and constraints. The central ingredient is Hamilton's principal function that generates canonical time evolution and ensures that the canonical formalism reproduces the dynamics of the covariant formulation following directly from the action. We apply this formalism to simplicial gravity and (Euclidean) Regge calculus, in particular. A discrete forward/backward evolution is realized by gluing/removing single simplices step by step to/from a bulk triangulation and amounts to Pachner moves in the triangulated hypersurfaces. As a result, the hypersurfaces evolve in a discrete ‘multi-fingered’ time through the full Regge solution. Pachner moves are an elementary and ergodic class of homeomorphisms and generically change the number of variables, but can be implemented as canonical transformations on naturally extended phase spaces. Some moves introduce a priori free data that, however, may become fixed a posteriori by constraints arising in subsequent moves. The end result is a general and fully consistent formulation of canonical Regge calculus, thereby removing a longstanding obstacle in connecting covariant simplicial gravity models to canonical frameworks. The presented scheme is, therefore, interesting in view of many approaches to quantum gravity, but may also prove useful for numerical implementations.

Chattering regimes and Lagrangian manifolds in problems with phase constraints

Given a class of optimal control problems with a phase constraint, we construct a Lagrangian manifold containing extremals with countably many tangencies to the boundary of that constraint. The construction proceeds by resolving the singularities of the Hamiltonian system of the problem at the accumulation point of the points of contact of extremals with that boundary.

Optimal synthesis in the time minimization problem under phase constraints

The time minimization problem in a linear system with a scalar control under phase constraints is considered. This problem is reduced to the problem of maximum deviations. Necessary conditions of control optimality are obtained. The results are applied to the problem of uniaxial stabilization of a satellite with consideration of a gravitational torque. The stabilization is performed by a flywheel engine with a limited angular momentum.

Differential games with many pursuers when evader moves on the surface of a cylinder

We study a pursuit differential game with many Pursuers when the Evader moves on the surface of a given cylinder. Maximal speeds of all players are equal. We consider two cases: in the first case, the Pursuers move arbitrarily without phase constraints; and in the second case, the Pursuers move on the surface of the cylinder. In both cases, we give necessary and sufficient conditions to complete the pursuit. In addition, in the second case, we show that pursuit differential game on a cylinder are equivalent to a differential game on the plane with many groups of Pursuers where each group consists of infinite number of pursuers having the same control parameter. References Isaacs, R. Differential Games. A Mathematical Theory with Applications to Warfare and Pursuit, Control, and Optimization. New York: Wiley, 1963. Petrov, N. N. A problem of group pursuit with phase constraints. J. Appl. Math. Mech. 1988, 52, No 6, 1030--1033. Petrosyan, L. A. Survival differential game with many participants. Dokl. Akad. Nauk USSR. 1965, 161, No 2, 285--287. Petrosyan, L. S. Differentsial'nye igry presledovaniya (Pursuit Differential Games). Leningrad (SPb): Leningr.State Univ(SPbSU), 1977. Pshennichnyi, B. N. Simple pursuit of several targets. Kibernetika. 1976, No 3, 145--146. Chernous'ko, F. L. A problem of evasion of several pursuers. J. Appl. Maths Mechs.1976, 40, No 1, 14--24. Ivanov R. P. Simple pursuit-evasion on a compact. Dokl. Akad. Nauk SSSR. 1980, 254, No 6, 1318--1321. Melikyan, A. A. and Ovakimyan, N. V. Singular trajectories in the problem of simple pursuit on a manifold. J. Appl. Math. Mech. 1991, 55, No 1, 42--48. Melikyan, A. A. and Ovakimyan, N. V. Differential Games of Simple Pursuit and Approach on Manifolds. Institute of Mechanics, National Academy of Sciences of Armenia. Yerevan. Preprint, 1993. Kuchkarov, A. Sh. The problem of optimal approach in locally euclidean spaces. Automation and Remote Control. 2007, 68, No 6, 974-978. Kuchkarov, A. Sh. A simple pursuit--evasion problem on a ball of a Riemannian manifold. Mathematical Notes. 2009, 85, No 2, 190--197. Azamov, A. On a problem of escape along a prescribed curve. J. Appl. Math. Mech. 1982, 46, No 4, 553--555. Kuchkarov, A. Sh. and Rikhsiev, B. B. on the solution of a pursuit problem with phase constraints. Automation and Remote Control. 2001, 62, No 8, 1259--1262. Ibragimov G. I. A game problem on a closed convex set. Siberian advances in mathematics., 2002, 12, No 3, 16--31. Nikulin, V. V. and Shafarevich, I. R. Geometriya i Gruppy (Geometric and Groups). Moscow: Nauka, 1983.