horizontal visibility graphs exact results for random time series networks this procedure allows us to apply methods of complex network theory for characterizing time series in this work we present the horizontal visibility algorithm a geometrically simpler and analytically solvable version of our former algorithm focusing on the mapping of random series series of independent identically distributed random variables after presenting some properties of the algorithm we present exact results on the topological properties of graphs associated with random series namely the degree distribution the clustering coefficient and the mean path length we show that the horizontal visibility algorithm stands as a simple method to discriminate randomness in time series since any random series maps to a graph with an exponential degree distribution of the shape pk 1 32 3 k 2 independent of the probability distribution from which the series was generated accordingly visibility graphs with other pk are related to nonrandom series numerical simulations confirm the accuracy of the theorems for finite series in a second part we show that the method is able to distinguish chaotic series from independent and identically distributed i i d theory studying the following situations i noise free low dimensional chaotic series ii low dimensional noisy chaotic series even in the presence of large amounts of noise and iii high dimensional chaotic series coupled map lattice without needs for additional techniques such as surrogate data or noise reduction methods finally heuristic arguments are given to explain the topological properties of chaotic series and several sequences that are conjectured to be random are analyzed

exploring self similarity of complex cellular networks the edge covering method with simulated annealing and log periodic sampling abstract song et al self similarity of complex networks nature 433 2005 392 395 have recently used a version of the box counting method called the node covering method to quantify the self similar properties of 43 cellular networks the minimal number n v of boxes of size l needed to cover all the nodes of a cellular network was found to scale as the power law n v l 1 d v with a fractal dimension d v 3 53 0 26 we implement an alternative box counting method in terms of the minimum number n e of edge covering boxes which is well suited to cellular networks where the search over different covering sets is performed with the simulated annealing algorithm the method also takes into account a possible discrete scale symmetry to optimize the sampling rate and minimize possible biases in the estimation of the fractal dimension with this methodology we find that n e scales with respect to l as a power law n e l d e with d e 2 67 0 15 for the 43 cellular networks previously analyzed by song et al self similarity of complex networks nature 433 2005 392 395 bootstrap tests suggest that the analyzed cellular networks may have a significant log periodicity qualifying a discrete hierarchy with a scaling ratio close to 2

multifractal characterization of protein contact networks abstract the multifractal detrended fluctuation analysis of time series is able to reveal the presence of long range correlations and at the same time to characterize the self similarity of the series the rich information derivable from the characteristic exponents and the multifractal spectrum can be further analyzed to discover important insights into the underlying dynamical process in this paper we employ multifractal analysis techniques in the study of protein contact networks to this end initially a network is mapped to three different time series each of which is generated by a stationary unbiased random walk to capture the peculiarities of the networks at different levels we accordingly consider three observables at each vertex the degree the clustering coefficient and the closeness centrality to compare the results with suitable references we consider also instances of three well known network models and two typical time series with pure monofractal and multifractal properties the first result of notable interest is that time series associated to protein contact networks exhibit long range correlations strong persistence which are consistent with signals in between the typical monofractal and multifractal behavior successively a suitable embedding of the multifractal spectra allows to focus on ensemble properties which in turn gives us the possibility to make further observations regarding the considered networks in particular we highlight the different role that small and large fluctuations of the considered observables play in the characterization of the network topology

multifractal and correlation analyses of protein sequences from complete genomes a measure representation of protein sequences similar to the measure representation of dna sequences proposed in our previous paper yu et al phys rev e 64 031903 2001 and another induced measure are introduced multifractal analysis is then performed on these two kinds of measures of a large number of protein sequences derived from corresponding complete genomes from the values of the d q generalized dimensions spectra and related c q analogous specific heat curves it is concluded that these protein sequences are not completely random sequences for substrings with length k 5 the d q spectra of all organisms studied are multifractal like and sufficiently smooth for the c q curves to be meaningful the c q curves of all bacteria resemble a classical phase transition at a critical point but the analogous phase transitions of higher organisms studied exhibit the shape of double peaked specific heat function but for the classification problem the multifractal property is not sufficient when the measure representations of protein sequences from complete genomes are considered as time series a method based on correlation analysis after removing some memory from the time series is proposed to construct a phylogenetic tree this construction is shown to be reasonably satisfactory

the structure and complexity of nash equilibria for a selfish routing game in this work we study the combinatorial structure and the computational complexity of nash equilibria for a certain game that models selfish routing over a network consisting of m parallel links we assume a collection of n users each employing a mixed strategy which is a probability distribution over links to control the routing of her own traffic in a nash equilibrium each user selfishly routes her traffic on those links that minimize her expected latency cost given the network congestion caused by the other users the social cost of a nash equilibrium is the expectation over all random choices of the users of the maximum over all links latency through a link we embark on a systematic study of several algorithmic problems related to the computation of nash equilibria for the selfish routing game we consider in a nutshell these problems relate to deciding the existence of a pure nash equilibrium constructing a nash equilibrium constructing the pure nash equilibria of minimum and maximum social cost and computing the social cost of a given mixed nash equilibrium our work provides a comprehensive collection of efficient algorithms hardness results and structural results for these algorithmic problems our results span and contrast a wide range of assumptions on the syntax of the nash equilibria and on the parameters of the system

the price of anarchy for polynomial social cost in this work we consider an interesting variant of the well studied kp model for selfish routing on parallel links which reflects some influence from the much older wardrop model j g wardrop some theoretical aspects of road traffic research proc inst of civil eng part ii 1 1956 325 378 in the new model user traffics are still unsplittable and links are identical social cost is now the expectation of the sum over all links of latency costs each latency cost is modeled as a certain polynomial latency cost function evaluated at the latency incurred by all users choosing the link the resulting social cost is called polynomial social cost or monomial social cost when the latency cost function is a monomial all considered polynomials are of degree d where d 2 and have non negative coefficients we are interested in evaluating nash equilibria in this model and we use the monomial price of anarchy mpoa and the polynomial price of anarchy ppoa as our evaluation measures through establishing some remarkable relations of these costs and measures to some classical combinatorial numbers such as the stirling numbers of the second kind and the bell numbers we obtain a multitude of results for the special case of identical users the fully mixed nash equilibrium where all probabilities are strictly positive maximizes polynomial social cost the mpoa is no more than bd the bell number of order d this immediately implies that the ppoa is no more than 1 t dbt for the special case of two links the mpoa is no more than 2d 2 1 1 n d 1 and this bound is tight for n 2 the mpoa is exactly 2d 1 d d 1 2d 2 d 1 d 1 d d for pure nash equilibria this immediately implies that the ppoa is no more than 2 t d 2t 1 t t 1 2t 2 t 1 t 1 t t

multifractal detrended fluctuation analysis of nonstationary time series abstract we develop a method for the multifractal characterization of nonstationary time series which is based on a generalization of the detrended fluctuation analysis dfa we relate our multifractal dfa method to the standard partition function based multifractal formalism and prove that both approaches are equivalent for stationary signals with compact support by analyzing several examples we show that the new method can reliably determine the multifractal scaling behavior of time series by comparing the multifractal dfa results for original series with those for shuffled series we can distinguish multifractality due to long range correlations from multifractality due to a broad probability density function we also compare our results with the wavelet transform modulus maxima method and show that the results are equivalent

asymptotic formula on average path length of fractal networks modeled on sierpinski gasket abstract in this paper we introduce a new method to construct evolving networks based on the construction of the sierpinski gasket using self similarity and renewal theorem we obtain the asymptotic formula for average path length of our evolving networks

atomic routing games on maximum congestion we study atomic routing games on networks in which players choose a path with the objective of minimizing the maximum congestion along the edges of their path the social cost is the global maximum congestion over all edges in the network we show that the price of stability is 1 the price of anarchy poa is determined by topological properties of the network in particular poa o logn where is the length of the longest path in the player strategy sets and n is the size of the network further 1 poa c 2 log2n where is the length of the longest cycle in the network and c is a constant

topological properties of four networks in protein structures abstract in this paper we investigate the complex networks of interacting amino acids in protein structures the cellular networks and their random controls are treated for the four threshold distances between atoms the numerical simulation and analysis are relevant to the topological properties of the complex networks in the structural classification of proteins and we mainly estimate the network s metrics from the resultant network the cellular network is shown to exhibit a small world feature regardless of their structural class the protein structure presents the positive assortative coefficients when the topological property is described as a tendency for connectivity of high degree nodes we particularly show that both the modularity and the small wordness are significantly followed the increasing function against nodes

horizontal visibility graphs transformed from fractional brownian motions topological properties versus the hurst index abstract nonlinear time series analysis aims at understanding the dynamics of stochastic or chaotic processes in recent years quite a few methods have been proposed to transform a single time series to a complex network so that the dynamics of the process can be understood by investigating the topological properties of the network we study the topological properties of horizontal visibility graphs constructed from fractional brownian motions with different hurst indexes h 0 1 special attention has been paid to the impact of the hurst index on topological properties it is found that the clustering coefficient c decreases when h increases we also found that the mean length l of the shortest paths increases exponentially with h for fixed length n of the original time series in addition l increases linearly with respect to n when h is close to 1 and in a logarithmic form when h is close to 0 although the occurrence of different motifs changes with h the motif rank pattern remains unchanged for different h adopting the node covering box counting method the horizontal visibility graphs are found to be fractals and the fractal dimension d b decreases with h furthermore the pearson coefficients of the networks are positive and the degree degree correlations increase with degree which indicate that the horizontal visibility graphs are assortative with the increase of h the pearson coefficient decreases first and then increases in which the turning point is around h 0 6 the presence of both fractality and assortativity in the horizontal visibility graphs converted from fractional brownian motions is different from many cases where fractal networks are usually disassortative

multifractality levinthal paradox and energy hypersurface instituto de fi sica universidad federal da bahia campus de ondina 40210 340 salvador bahia brazil received 21 june 2000 published 10 january 2001 multifractal properties in the potential energy hypersurface of polypeptides and proteins are investigated characteristic multifractal behavior for different molecular systems is obtained from the f a spectra theanalysis shows that the dimension of the phase space of the problem in uences the accessibility to differentparts of the potential energy hypersurface also we show that it is necessary to take into account the h bondformation between amino acids in the conformational folding search the present ndings indicate that thef a function describes some structural properties of a protein the behavior of the f a spectra gives analternative explanation about the levinthal paradox furthermore the anomalous temperature dependence ofthe raman spin lattice relaxation rates can be related to the perturbations in the secondary structures doi 10 1103 physreve 63 020901 pacs number s 87 14 cc 05 45 df 87 15 aa 05 10 ln

structure and complexity of extreme nash equilibria we study extreme nash equilibria in the context of a selfish routing game specifically we assume a collection of n users each employing a mixed strategy which is a probability distribution over m parallel identical links to control the routing of its own assigned traffic in a nash equilibrium each user selfishly routes its traffic on those links that minimize its expected latency cost the social cost of a nash equilibrium is the expectation over all random choices of the users of the maximum over all links latency through a link we provide substantial evidence for the fully mixed nash equilibrium conjecture which states that the worst nash equilibrium is the fully mixed nash equilibrium where each user chooses each link with positive probability specifically we prove that the fully mixed nash equilibrium conjecture is valid for pure nash equilibria furthermore we show that under a certain condition the social cost of any nash equilibrium is within a factor of 2h 1 e of that of the fully mixed nash equilibrium where h is the factor by which the largest user traffic deviates from the average user traffic considering pure nash equilibria we provide a ptas to approximate the best social cost we give an upper bound on the worst social cost and we show that it is n p hard to approximate the worst social cost within a multiplicative factor better than 2 2 m 1

topological properties and fractal analysis of a recurrence network constructed from fractional brownian motions many studies have shown that we can gain additional information on time series by investigating their accompanying complex networks in this work we investigate the fundamental topological and fractal properties of recurrence networks constructed from fractional brownian motions fbms first our results indicate that the constructed recurrence networks have exponential degree distributions the average degree exponent increases first and then decreases with the increase of hurst index h of the associated fbms the relationship between h and can be represented by a cubic polynomial function we next focus on the motif rank distribution of recurrence networks so that we can better understand networks at the local structure level we find the interesting superfamily phenomenon i e the recurrence networks with the same motif rank pattern being grouped into two superfamilies last we numerically analyze the fractal and multifractal properties of recurrence networks we find that the average fractal dimension db of recurrence networks decreases with the hurst index h of the associated fbms and their dependence approximately satisfies the linear formula db 2 h which means that the fractal dimension of the associated recurrence network is close to that of the graph of the fbm moreover our numerical results of multifractal analysis show that the multifractality exists in these recurrence networks and the multifractality of these networks becomes stronger at first and then weaker when the hurst index of the associated time series becomes larger from 0 4 to 0 95 in particular the recurrence network with the hurst index h 0 5 possesses the strongest multifractality in addition the dependence relationships of the average information dimension d 1 and the average correlation dimension d 2 on the hurst index h can also be fitted well with linear functions our results strongly suggest that the recurrence network inherits the basic characteristic and the fractal nature of the associated fbm series

a new model for selfish routing in this work we introduce and study a new potentially rich model for selfish routing over non cooperative networks as an interesting hybridization of the two prevailing such models namely the kpmodel e koutsoupias c h papadimitriou worst case equilibria in g meinel s tison eds proceedings of the 16th annual symposium on theoretical aspects of computer science in lecture notes in computer science vol 1563 springer verlag 1999 pp 404 413 and the wmodel j g wardrop some theoretical aspects of road traffic research proceedings of the of the institute of civil engineers 1 pt ii 1952 325 378 in the hybrid model each of nusers is using a mixed strategy to ship its unsplittable traffic over a network consisting of m parallel links in a nash equilibrium no user can unilaterally improve its expected individual cost to evaluate nash equilibria we introduce quadratic social cost as the sum of the expectations of the latencies incurred by the squares of the accumulated traffic this modeling is unlike the kp model where social cost e koutsoupias c h papadimitriou worst case equilibria in g meinel s tison eds proceedings of the 16th annual symposium on theoretical aspects of computer science in lecture notes in computer science vol 1563 springer verlag 1999 pp 404 413 is the expectation of the maximum latency incurred by the accumulated traffic but it is like the w model since the quadratic social cost can be expressed as a weighted sum of expected individual costs we use the quadratic social cost to define quadratic coordination ratio here are our main findings quadratic social cost can be computed in polynomial time this is unlike the p completeness d fotakis s kontogiannis e koutsoupias m mavronicolas p spirakis the structure and complexity of nash equilibria for a selfish routing game in p widmayer f triguero r morales m hennessy s eidenbenz r conejo eds proceedings of the 29th international colloquium on automata languages and programming in lecture notes in computer science vol 2380 springer verlag 2002 pp 123 134 of computing social cost for the kp model for the case of identical users and identical links the fully mixed nash equilibrium m mavronicolas p spirakis the price of selfish routing algorithmica 48 1 2007 91 126 where each user assigns positive probability to every link maximizes quadratic social cost as our main result we present a comprehensive collection of tight constant that is independent of m and n strictly less than 2 lower and upper bounds on the quadratic coordination ratio for several interesting special cases some of the bounds stand in contrast to corresponding super constant bounds on the coordination ratio previously shown in a czumaj b vocking tight bounds for worst case equilibria acm transactions on algorithms 3 1 2007 e koutsoupias m mavronicolas p spirakis approximate equilibria and ball fusion theory of computing systems 36 6 2003 683 693 e koutsoupias c h papadimitriou worst case equilibria in g meinel s tison eds proceedings of the 16th annual symposium on theoretical aspects of computer science in lecture notes in computer science vol 1563 springer verlag 1999 pp 404 413 m mavronicolas p spirakis the price of selfish routing algorithmica 48 1 2007 91 126 for the kp model

ambiguities in recurrence based complex network representations of time series recently different approaches have been proposed for studying basic properties of time series from a complex network perspective in this work the corresponding potentials and limitations of networks based on recurrences in phase space are investigated in some detail we discuss the main requirements that permit a feasible system theoretic interpretation of network topology in terms of dynamically invariant phase space properties possible artifacts induced by disregarding these requirements are pointed out and systematically studied finally a rigorous interpretation of the clustering coefficient and the betweenness centrality in terms of invariant objects is proposed

measure representation and multifractal analysis of complete genomes this paper introduces the notion of measure representation of dna sequences spectral analysis and multifractal analysis are then performed on the measure representations of a large number of complete genomes the main aim of this paper is to discuss the multifractal property of the measure representation and the classification of bacteria from the measure representations and the values of the dq spectra and related cq curves it is concluded that these complete genomes are not random sequences in fact spectral analyses performed indicate that these measure representations considered as time series exhibit strong long range correlation here the long range correlation is for the k strings with dictionary ordering and it is different from the base pair correlations introduced by other people for substrings with length k58 the dq spectra of all organisms studied are multifractal like and sufficiently smooth for the cq curves to be meaningful with the decreasing value of k the multifractality lessens the cq curves of all bacteria resemble a classical phase transition at a critical point but the analogous phase transitions of chromosomes of nonbacteria organisms are different apart from chromosome 1 of c elegans they exhibit the shape of double peaked specific heat function a classification of genomes of bacteria by assigning to each sequence a point in two dimensional space d21 d1 and in three dimensional space d21 d1 d22 was given bacteria that are close phylogenetically are almost close in the spaces d21 d1 and d21 d1 d22

extending the notion of rationality of selfish agents second order nash equilibria motivated by the increasing interest of the computer science community in the study and understanding of non cooperative systems we present a novel model for formalizing the rational behavior of agents with a more farsighted view of the consequences of their actions this approach yields a framework creating new equilibria which we call second order equilibria starting from a ground set of traditional ones by applying our approach to pure nash equilibria we define the set of second order pure nash equilibria and present their applications to the prisoner s dilemma game to an instance of braess s paradox in the wardrop model and to the kp model with identical machines

competitive routing in networks with polynomial costs we study a class of noncooperative general topology networks shared by n users each user has a given flow which it has to ship from a source to a destination we consider a class of polynomial link cost functions adopted originally in the context of road traffic modeling and show that these costs have appealing properties that lead to predictable and efficient network flows in particular we show that the nash equilibrium is unique and is moreover efficient these properties make the polynomial cost structure attractive for traffic regulation and link pricing in telecommunication networks we finally discuss the computation of the equilibrium in the special case of the affine cost structure for a topology of parallel links

fractality in complex networks critical and supercritical skeletons fractal scaling char22 a power law behavior of the number of boxes needed to tile a given network with respect to the lateral size of the box char22 is studied we introduce a box covering algorithm that is a modified version of the original algorithm introduced by song et al nature london 433 392 2005 this algorithm enables easy implementation fractal networks are viewed as comprising a skeleton and shortcuts the skeleton embedded underneath the original network is a special type of spanning tree based on the edge betweenness centrality it provides a scaffold for the fractality of the network when the skeleton is regarded as a branching tree it exhibits a plateau in the mean branching number as a function of the distance from a root for nonfractal networks on the other hand the mean branching number decays to zero without forming a plateau based on these observations we construct a fractal network model by combining a random branching tree and local shortcuts the scaffold branching tree can be either critical or supercritical depending on the small worldness of a given network for the network constructed from the critical supercritical branching tree the average number of vertices within a given box grows with the lateral size of the box according to a power law an exponential form in the cluster growing method the critical and supercritical skeletons are observed in protein interaction networks and the world wide web respectively the distribution of box masses i e the number of vertices within each box follows a power law p m m ensuremath sim m ensuremath ensuremath eta the exponent ensuremath eta depends on the box lateral size ensuremath ell b for small values of ensuremath ell b ensuremath eta is equal to the degree exponent ensuremath gamma of a given scale free network whereas ensuremath eta approaches the exponent ensuremath tau ensuremath gamma ensuremath gamma ensuremath 1 as ensuremath ell b increases which is the exponent of the cluster size distribution of the random branching tree finally we study the perimeter h ensuremath alpha of a given box ensuremath alpha i e the number of edges connected to different boxes from a given box ensuremath alpha as a function of the box mass m b ensuremath alpha it is obtained that the average perimeter over the boxes with box mass m b is likely to scale as h m b ensuremath sim m b irrespective of the box size ensuremath ell b

analytical framework for recurrence network analysis of time series recurrence networks are a powerful nonlinear tool for time series analysis of complex dynamical systems while there are already many successful applications ranging from medicine to paleoclimatology a solid theoretical foundation of the method has still been missing so far here we interpret an ensuremath varepsilon recurrence network as a discrete subnetwork of a continuous graph with uncountably many vertices and edges corresponding to the system s attractor this step allows us to show that various statistical measures commonly used in complex network analysis can be seen as discrete estimators of newly defined continuous measures of certain complex geometric properties of the attractor on the scale given by ensuremath varepsilon in particular we introduce local measures such as the ensuremath varepsilon clustering coefficient mesoscopic measures such as ensuremath varepsilon motif density path based measures such as ensuremath varepsilon betweennesses and global measures such as ensuremath varepsilon efficiency this new analytical basis for the so far heuristically motivated network measures also provides an objective criterion for the choice of ensuremath varepsilon via a percolation threshold and it shows that estimation can be improved by so called node splitting invariant versions of the measures we finally illustrate the framework for a number of archetypical chaotic attractors such as those of the bernoulli and logistic maps periodic and two dimensional quasiperiodic motions and for hyperballs and hypercubes by deriving analytical expressions for the novel measures and comparing them with data from numerical experiments more generally the theoretical framework put forward in this work describes random geometric graphs and other networks with spatial constraints which appear frequently in disciplines ranging from biology to climate science

approximating congestion dilation in networks via quality of routing games a classic optimization problem in network routing is to minimize c d where c is the maximum edge congestion and d is the maximum path length also known as dilation the problem of computing the optimal c d is np complete even when either c or d is a small constant we study routing games in general networks where each player i selfishly selects a path that minimizes ci di the sum of congestion and dilation of the player s path we first show that there are instances of this game without nash equilibria we then turn to the related quality of routing qor games which always have nash equilibria qor games represent networks with a small number of service classes where paths in different classes do not interfere with each other with frequency or time division multiplexing qor games have o log4 n price of anarchy when either c or d is a constant thus nash equilibria of qor games give poly log approximations to hard optimization problems

clustering of casablanca stock market based on hurst exponent estimates abstract this paper deals with the problem of casablanca stock exchange cse topology modeling as a complex network during three different market regimes general trend characterized by ups and downs increasing trend and decreasing trend in particular a set of seven different hurst exponent estimates are used to characterize long range dependence in each industrial sector generating process they are employed in conjunction with hierarchical clustering approach to examine the co movements of the casablanca stock exchange industrial sectors the purpose is to investigate whether cluster structures are similar across variable increasing and decreasing regimes it is observed that the general structure of the cse topology has been considerably changed over 2009 variable regime 2010 increasing regime and 2011 decreasing regime time periods the most important findings follow first in general a high value of hurst exponent is associated to a variable regime and a small one to a decreasing regime in addition hurst estimates during increasing regime are higher than those of a decreasing regime second correlations between estimated hurst exponent vectors of industrial sectors increase when casablanca stock exchange follows an upward regime whilst they decrease when the overall market follows a downward regime

fractal and complex network analyses of protein molecular dynamics based on protein molecular dynamics we investigate the fractal properties of energy pressure and volume time series using the multifractal detrended fluctuation analysis mf dfa and the topological and fractal properties of their converted horizontal visibility graphs hvgs the energy parameters of protein dynamics we considered are bonded potential angle potential dihedral potential improper potential kinetic energy van der waals potential electrostatic potential total energy and potential energy the shape of the h q h q curves from mf dfa indicates that these time series are multifractal the numerical values of the exponent h 2 h 2 of mf dfa show that the series of total energy and potential energy are non stationary and anti persistent the other time series are stationary and persistent apart from series of pressure with h 0 5h 0 5 indicating the absence of long range correlation the degree distributions of their converted hvgs show that these networks are exponential the results of fractal analysis show that fractality exists in these converted hvgs for each energy pressure or volume parameter it is found that the values of h 2 h 2 of mf dfa on the time series exponent of the exponential degree distribution and fractal dimension dbdb of their converted hvgs do not change much for different proteins indicating some universality we also found that after taking average over all proteins there is a linear relationship between h 2 h 2 from mf dfa on time series and db db of the converted hvgs for different energy pressure and volume

stochastic optimal lqr control with integral quadratic constraints and indefinite control weights a standard assumption in traditional deterministic and stochastic optimal minimizing linear quadratic regulator lqr theory is that the control weighting matrix in the cost functional is strictly positive definite in the deterministic case this assumption is in fact necessary for the problem to be well posed because positive definiteness is required to make it a convex optimization problem however it has recently been shown that in the stochastic case when the diffusion term is dependent on the control the control weighting matrix may have negative eigenvalues but the problem remains well posed in this paper the completely observed stochastic lqr problem with integral quadratic constraints is studied sufficient conditions for the well posedness of this problem are given indeed we show that in certain cases these conditions may be satisfied even when the control weighting matrices in the cost and all of the constraint functionals have negative eigenvalues it is revealed that the seemingly nonconvex problem with indefinite control weights can actually be a convex one by virtue of the uncertainty in the system finally when these conditions are satisfied the optimal control is explicitly derived using results from duality theory