Minimum-energy Problem Research Articles

Euclidean space controllability conditions and minimum energy problem for time delay systems with a high gain control

Euclidean space controllability conditions and minimum energy problem for time delay systems with a high gain control

Gradient remediability in linear distributed parabolic systems analysis, approximations and simulations

The aim of this paper is the introduction of a new concept that concerned the analysis of a large class of distributed parabolic systems. It is the general concept of gradient remediability. More precisely, we study with respect to the gradient observation, the existence of an input operator (gradient efficient actuators) ensuring the compensation of known or unknown disturbances acting on the considered system. Then, we introduce and we characterize the notions of exact and weak gradient remediability and their relationship with the notions of exact and weak gradient controllability. Main properties concerning the notion of gradient efficient actuators are considered. The minimum energy problem is studies, and we show how to find the optimal control which compensates the disturbance of the system. Approximations and numerical simulations are also presented.

Minimal energy problems for strongly singular Riesz kernels

AbstractWe study minimal energy problems for strongly singular Riesz kernels , where and , considered for compact ‐dimensional ‐manifolds Γ immersed into . Based on the spatial energy of harmonic double layer potentials, we are motivated to formulate the natural regularization of such minimization problems by switching to Hadamard's partie finie integral operator which defines a strongly elliptic pseudodifferential operator of order on Γ. The measures with finite energy are shown to be elements from the Sobolev space , , and the corresponding minimal energy problem admits a unique solution. We relate our continuous approach also to the discrete one, which has been worked out earlier by D. P. Hardin and E. B. Saff.

Minimum Energy Problem on the Hypersphere

We consider the minimum energy problem on the unit sphere $\mathbb {S}^{d-1}$ in the Euclidean space $\mathbb {R}^{d}$ , d = 3, in the presence of an external field Q, where the charges are assumed to interact according to Newtonian potential 1/r d-2, with r denoting the Euclidean distance. We solve the problem by finding the support of the extremal measure, and obtaining an explicit expression for the density of the extremal measure. We then apply our results to an external field generated by a point charge of positive magnitude, placed at the North Pole of the sphere, and to a quadratic external field.

Energy-efficient joint path-following for attitude-fixed space manipulators with bounds on completion time and motor voltage/current

This paper investigates the minimum-energy joint path-following problem for space manipulators whose base attitude is stabilized by reaction wheels. In the problem, manipulator joint path is specified for rest-to-rest motion and constraints are imposed as the upper bound on both motion completion time and the voltage/current limits of DC motors in manipulator joints and reaction wheels. We suggest a simple two-stage algorithm to address this problem. The algorithm first tries to find a global optimal solution by solving a relaxed convex problem. If the convex relaxation is not successful, then the algorithm solves subproblems iteratively to find a suboptimal solution. Since both problems are formulated as second-order cone programming (SOCP) form, they can be solved efficiently using dedicated SOCP solvers. The effectiveness of the proposed method is verified by numerical experiments.

Minimum-fuel station-change for geostationary satellites using low-thrust considering perturbations

The objective of this paper is to find the minimum-fuel station change for geostationary satellites with low-thrust while considering significant perturbation forces for geostationary Earth orbit (GEO). The effect of Earth's triaxiality, lunisolar perturbations, and solar radiation pressure on the terminal conditions of a long duration GEO transfer is derived and used for establishing the station change model with consideration of significant perturbation forces. A method is presented for analytically evaluating the effect of Earth's triaxiality on the semimajor axis and longitude during a station change. The minimum-fuel problem is solved by the indirect optimization method. The easier and related minimum-energy problem is first addressed and then the energy-to-fuel homotopy is employed to finally obtain the solution of the minimum-fuel problem. Several effective techniques are employed in solving the two-point boundary-value problem with a shooting method to overcome the problem of the small convergence radius and the sensitivity of the initial costate variables. These methods include normalization of the initial costate vector, computation of the analytic Jacobians matrix, and switching detection. The simulation results show that the solution of the minimum-fuel station change with low-thrust considering significant perturbation forces can be obtained by applying these preceding techniques.

Weighted energy problem on the unit sphere

We consider the minimal energy problem on the unit sphere \({\mathbb {S}}^2\) in the Euclidean space \({\mathbb {R}}^3\) immersed in an external field Q, where the charges are assumed to interact via Newtonian potential 1/r, r being the Euclidean distance. The problem is solved by finding the support of the extremal measure, and obtaining an explicit expression for the equilibrium density. We then apply our results to an external field generated by a point charge, and to a quadratic external field.

Energy-Efficient Algorithm for Multicasting in Duty-Cycled Sensor Networks.

Multicasting is a fundamental network service for one-to-many communications in wireless sensor networks. However, when the sensor nodes work in an asynchronous duty-cycled way, the sender may need to transmit the same message several times to one group of its neighboring nodes, which complicates the minimum energy multicasting problem. Thus, in this paper, we study the problem of minimum energy multicasting with adjusted power (the MEMAP problem) in the duty-cycled sensor networks, and we prove it to be NP-hard. To solve such a problem, the concept of an auxiliary graph is proposed to integrate the scheduling problem of the transmitting power and transmitting time slot and the constructing problem of the minimum multicast tree in MEMAP, and a greedy algorithm is proposed to construct such a graph. Based on the proposed auxiliary graph, an approximate scheduling and constructing algorithm with an approximation ratio of is proposed, where K is the number of destination nodes. Finally, the theoretical analysis and experimental results verify the efficiency of the proposed algorithm in terms of the energy cost and transmission redundancy.

Minimum Riesz Energy Problems for a Condenser with Touching Plates

We consider minimum energy problems in the presence of an external field for a condenser with touching plates A 1 and A 2 in $\mathbb {R}^{n}$ , $n\geqslant 3$ , relative to the α-Riesz kernel |x−y| α−n , $0<\alpha \leqslant 2$ . An intimate relationship between such problems and minimal α-Green energy problems for positive measures on A 1 is shown. We obtain sufficient and/or necessary conditions for the solvability of these problems in both the unconstrained and the constrained settings, investigate the properties of minimizers, and prove their uniqueness. Furthermore, characterization theorems in terms of variational inequalities for the weighted potentials are established. The approach applied is mainly based on the establishment of a perfectness-type property for the α-Green kernel with $0<\alpha \leqslant 2$ which enables us, in particular, to analyze the existence of the α-Green equilibrium measure of a set. The results obtained are illustrated by several examples.

Random normal matrices and Ward identities

Consider the random normal matrix ensemble associated with a potential on the plane which is sufficiently strong near infinity. It is known that, to a first approximation, the eigenvalues obey a certain equilibrium distribution, given by Frostman's solution to the minimum energy problem of weighted logarithmic potential theory. On a finer scale, one can consider fluctuations of eigenvalues about the equilibrium. In the present paper, we give the correction to the expectation of fluctuations, and we prove that the potential field of the corrected fluctuations converge on smooth test functions to a Gaussian free field with free boundary conditions on the droplet associated with the potential.

Performance evaluation of the inverse dynamics method for optimal spacecraft reorientation

This paper investigates the application of the inverse dynamics in the virtual domain method to Euler angles, quaternions, and modified Rodrigues parameters for rapid optimal attitude trajectory generation for spacecraft reorientation maneuvers. The impact of the virtual domain and attitude representation is numerically investigated for both minimum time and minimum energy problems. Owing to the nature of the inverse dynamics method, it yields sub-optimal solutions for minimum time problems. Furthermore, the virtual domain improves the optimality of the solution, but at the cost of more computational time. The attitude representation also affects solution quality and computational speed. For minimum energy problems, the optimal solution can be obtained without the virtual domain with any considered attitude representation.

Riesz External Field Problems on the Hypersphere and Optimal Point Separation

We consider the minimal energy problem on the unit sphere $\mathbb{S}^d$ in the Euclidean space $\mathbb{R}^{d+1}$ in the presence of an external field $Q$, where the energy arises from the Riesz potential $1/r^s$ (where $r$ is the Euclidean distance and $s$ is the Riesz parameter) or the logarithmic potential $\log(1/r)$. Characterization theorems of Frostman-type for the associated extremal measure, previously obtained by the last two authors, are extended to the range $d-2 \leq s < d - 1.$ The proof uses a maximum principle for measures supported on $\mathbb{S}^d$. When $Q$ is the Riesz $s$-potential of a signed measure and $d-2 \leq s <d$, our results lead to explicit point-separation estimates for $(Q,s)$-Fekete points, which are $n$-point configurations minimizing the Riesz $s$-energy on $\mathbb{S}^d$ with external field $Q$. In the hyper-singular case $s > d$, the short-range pair-interaction enforces well-separation even in the presence of more general external fields. As a further application, we determine the extremal and signed equilibria when the external field is due to a negative point charge outside a positively charged isolated sphere. Moreover, we provide a rigorous analysis of the three point external field problem and numerical results for the four point problem.

Modeling of Capillary Burst Valve and Siphon with Hydrophilic Cover for Centrifugal Microfluidic Systems

This paper reports numerical and experimental investigation of capillary burst valve and siphon, which were used in the centrifugal microfluidic systems widely. The simulation model is based on the solution of the minimum energy problem by the Surface Evolver software. In contrast to the solution of meniscus equilibrium equations, it can provide the appropriate boundary conditions for the microvalves with a hydrophilic flat cover. As an experimental verification, a centrifugal microfluidic device based on PDMS and glass was fabricated by soft lithography. The simulated results agree well with experimental data. The technology is useful for the design of microfluidic systems with complex wetting characters and geometries.

On the extremal energy of integral weighted graphs

Let 𝒯(n, m) and ℱ(n, m) denote the classes of weighted trees and forests, respectively, of order n with the positive integral weights and the fixed total weight sum m, respectively. In this article, we determine the minimum energies for both the classes 𝒯(n, m) and ℱ(n, m). We also determine the maximum energy for the class ℱ(n, m). In all cases, we characterize the weighted graphs whose energies reach these extremal values. We also solve the similar maximum energy and minimum energy problems for the classes of (0, 1) weighted trees and forests.

Equilibrium Problems for Infinite Dimensional Vector Potentials with External Fields

We consider a minimal energy problem with an external field over noncompact classes of infinite dimensional vector measures \((\mu^i)_{i\in I}\) on a locally compact space. The components μi are positive measures normalized by \(\int g_i\,d\mu^i=a_i\) (where ai and gi are given) and supported by closed sets Ai with the sign + 1 or − 1 prescribed such that Ai ∩ Aj = ∅ whenever \({\rm sign}\,A_i\ne{\rm sign}\,A_j\), and the law of interaction between μi, i ∈ I, is determined by the matrix \(\bigl({\rm sign}\,A_i\,{\rm sign}\,A_j\bigr)_{i,j\in I}\). For positive definite kernels satisfying Fuglede’s condition of consistency, sufficient conditions for the existence of equilibrium measures are established and properties of their uniqueness, vague compactness, and strong and vague continuity are studied. Examples illustrating the sharpness of the sufficient conditions are provided. We also obtain variational inequalities for the weighted equilibrium potentials, single out their characteristic properties, and analyze continuity of the equilibrium constants. The results hold, e.g., for classical kernels in \(\mathbb R^n\), \(n\geqslant 2\), which is important in applications.

Throughput and Energy Optimization in Wireless Networks: Joint MAC Scheduling and Network Coding

This paper studies joint scheduling and network coding in wireless multicast networks with independent sources. Designing a network coding solution for wireless networks involves scheduling interference-free transmissions and optimizing a performance measure for the predetermined scheduling sets, followed by constructing network codes for the specific solution. In such a design process, the results of prior steps need to provide enough information to the subsequent steps. In this paper, we formulate a linear optimization problem whose results can be used to design a coding solution. We build our work on previous works and present statistics that show the importance of incorporating unequal timeshares in designing network codes. In particular, our statistics show a throughput improvement of about 35% in maximum flow problems and energy savings between 13% and 30%, depending on the network size, in minimum energy problems. We also present the requirements of code construction algorithms for wireless networks that capture the broadcast property of these networks and can design proper codes when timeshares are unequal. In particular, the adaptation of a centralized network coding scheme for wireless networks is discussed. Overall, the work reported here provides a three-step solution to derive network codes that optimize a performance criterion of interest while also solving the scheduling problem in multihop wireless networks.

On the classical solution to the linear-constrained minimum energy problem

Minimum energy problems involving linear systems with quadratic performance criteria are classical in optimal control theory. The case where controls are constrained is discussed in Athans and Falb (1966) [Athans, M. and Falb, P.L. (1966), Optimal Control: An Introduction to the Theory and Its Applications, New York: McGraw-Hill Book Co.] who obtain a componentwise optimal control expression involving a saturation function expression. We show why the given expression is not generally optimal in the case where the dimension of the control is greater than one and provide a numerical counterexample.

Constrained energy problems with external fields for vector measures

AbstractWe consider a constrained minimal energy problem with an external field over noncompact classes of vector measures (μi)i ∈ I of finite or infinite dimensions on a locally compact space. The components μi are nonnegative Radon measures satisfying normalizing conditions, supported by given Ai and such that σi − μi ≥ 0, the constraints σi, i ∈ I, being given. For a particular matrix of interaction between μi, i ∈ I, and a rather general class of positive definite kernels, sufficient conditions for the existence of minimizers are established and their uniqueness and vague compactness are studied. Examples illustrating the sharpness of the sufficient conditions are provided. We also analyze continuity properties of minimizers in the vague and strong topologies when Ai and σi are varied. Almost all results are valid also for classical kernels in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb R^n$\end{document}, which is important in applications.

On power sums of convex functions in local minimum energy problem

In this paper, new inequalities on the power sums of a convex function are derived and the monotonically decreasing nature of the Riemann sum of a function including a certain strong convexity is shown. It is also shown that a derived inequality has a direct implication in a local minimum energy problem in two-dimensional hexagonal packing.

Time-optimal trajectory planning of robot manipulators in point-to-point motion using an indirect method

In this study, an iterative method for computing the time-optimal point-to-point control of robot manipulators is studied, subject to limits on the actuator torques and actuator jerks. This method uses the indirect solution of an open-loop optimal control problem so that the optimal control problem is translated to a non-linear two-point boundary value problem (TPBVP). Since there are many difficulties in finding the switching points and solving the TPBVP, a simple iterative method is proposed. This method is based on solving the minimum-energy problem without the necessity of finding the switching point. So, the resultant TPBVP can be solved using the usual algorithms. In addition, since the solution of the TPBVPs is sensitive to the initial guess, a simple procedure for making proper initial guesses is introduced. Furthermore, since the inverse kinematic of redundant manipulators is an ill-posed problem, the general form for boundary conditions corresponding to the final time is considered. In addition, it is shown that the proposed method has no sensitivity for specifying the time-optimal trajectory of manipulators that may contain singular arcs.