Extremal Norms Research Articles

Optimal stabilizing rates of switched linear control systems under arbitrary known switchings

The problem of stabilizing discrete-time switched linear control systems using continuous control input under arbitrary mode switchings is studied. It is assumed that at each time instance the switching mode can be arbitrarily chosen but is always known by the controller designing the continuous control input; thus the continuous controller is of the general form of an ensemble of mode-dependent state feedback controllers. Under this setting, the fastest (worst-case) stabilizing rate is proposed as a quantitative metric of the systems’ stabilizability. Conditions are derived on when this stabilizing rate can be exactly achieved by an admissible control policy and a counter example is given to show that the stabilizing rate may not always be attained by a mode-dependent linear state feedback control policy. Bounds on the stabilizing rate are derived using (semi)norms. When such bounds are tight, the corresponding extremal norms are characterized geometrically. Numerical algorithms based on ellipsoid and polytope norms are developed for computing bounds of the stabilizing rate and illustrated through examples.

Political Elite and Extremism: A Challenge for the Stability of Pakistan

The extreme behaviours of Pakistani politicians are another dilemma which is prompting different kinds of extreme norms in Pakistani politics. Moreover, political extremism is another factor which is creating a ground for religious extremism in our country. The focal point of this study lies in many factors which are enhancing extremism in different fields of life: such as lack of trustworthy nationwide political parties, multi-party system, switching parties and affiliation for personal interest, fake promises of politicians before elections, and use of Islam for political purposes. The problem of extremism cannot be eliminated until and unless the Pakistani government will not take proper measure to cope with it.

Extremal norms for fiber-bunched cocycles

In traditional Ergodic Optimization, one seeks to maximize Birkhoff averages. The most useful tool in this area is the celebrated Mane Lemma, in its various forms. In this paper, we prove a non-commutative Mane Lemma, suited to the problem of maximization of Lyapunov exponents of linear cocycles or, more generally, vector bundle automorphisms. More precisely, we provide conditions that ensure the existence of an extremal norm, that is, a Finsler norm with respect to which no vector can be expanded in a single iterate by a factor bigger than the maximal asymptotic expansion rate. These conditions are essentially irreducibility and sufficiently strong fiber bunching. Therefore we extend the classic concept of Barabanov norm, which is used in the study of the joint spectral radius. We obtain several consequences, including sufficient conditions for the existence of Lyapunov maximizing sets.

Resilient Stabilization of Switched Linear Control Systems Against Adversarial Switching

This paper studies the problem of stabilizing discrete-time switched linear control systems (SLCSs) using continuous input by a user against adversarial switching by an adversary. It is assumed that at each time the adversary knows the user's decision on the continuous input but not vice versa. A quantitative metric of stabilizability is proposed. Systems at the margin of stabilizability are further classified and studied via the notions of defectiveness and reducibility. Analytical bounds on the stabilizability metric are derived using (semi)norms, with tight bounds provided by extremal norms. Numerical algorithms are also developed for computing this metric. An application example in networked control systems is presented.

Extremal norms of the potentials recovered from inverse Dirichlet problems

Consider the Sturm–Liouville eigenvalue problem , where , and its spectrum is denoted by . For a real number λ, define and . We will set up a formula for explicitly in terms of λ and specify where the infimum can be attained. As an application, we will give the extremal values of the nth eigenvalue of the Dirichlet problem for potentials on a sphere , . The proofs are based on a new Lyapunov-type inequality for Sturm–Liouville equations with potentials.

Canonical Construction of Polytope Barabanov Norms and Antinorms for Sets of Matrices

Barabanov norms have been introduced in Barabanov (Autom. Remote Control, 49 (1988), pp. 152--157) and constitute an important instrument in analyzing the joint spectral radius of a family of matrices and related issues. However, although they have been studied extensively, even in very simple cases it is very difficult to construct them explicitly (see, e.g., Kozyakin (Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), pp. 143--158)). In this paper we give a canonical procedure to construct them exactly, which associates a polytope extremal norm---constructed by using the methodologies described in Guglielmi, Wirth, and Zennaro (SIAM J. Matrix Anal. Appl., 27 (2005), pp. 721--743) and Guglielmi and Protasov (Found. Comput. Math., 13 (2013), pp. 37--97)---to a polytope Barabanov norm. Hence, the existence of a polytope Barabanov norm has the same genericity of an extremal polytope norm. Moreover, we extend the result to polytope antinorms, which have been recently introduced to compute the lower spectral radi...

Extremal norms for positive linear inclusions

For finite-dimensional linear semigroups which leave a proper cone invariant it is shown that irreducibility with respect to the cone implies the existence of an extremal norm. In case the cone is simplicial a similar statement applies to absolute norms. The semigroups under consideration may be generated by discrete-time systems, continuous-time systems or continuous-time systems with jumps. The existence of extremal norms is used to extend results on the Lipschitz continuity of the joint spectral radius beyond the known case of semigroups that are irreducible in the representation theory interpretation of the word.

Orthogonal decomposition of the space of algebraic numbers and Lehmerʼs problem

Building on work of Dubickas and Smyth regarding the metric Mahler measure and the authors regarding extremal norms associated to the Mahler measure, the authors introduce a new set of norms associated to the Mahler measure of algebraic numbers which allow for an equivalent reformulation of problems like the Lehmer problem and the Schinzel–Zassenhaus conjecture on a single spectrum. We present several new geometric results regarding the space of algebraic numbers modulo torsion using the L p Weil height introduced by Allcock and Vaaler, including a canonical decomposition of an algebraic number into an orthogonal series with respect to the L 2 height.

A note on extreme norms on $\mathbb R$2

We denote by AN2 the set of all absolute normalized norms on $\mathbb R$2. It is known that the set AN2 and the set of all continuous convex functions ψ on [0,1] with max{1−t,t} ≤ ψ(t) ≤ 1 for t ∈ [0,1] (denoted by Ψ2) are in a one to one correspondence under the equation ψ(t) = ||(1−t,t)||. Recently, we characterized extreme points of AN2 by considering Ψ2. In this paper we give another proof of this result.

Rank-one characterization of joint spectral radius of finite matrix family

In this paper we study the joint/generalized spectral radius of a finite set of matrices in terms of its rank-one approximation by singular value decomposition. In the first part of the paper, we show that any finite set of matrices with at most one element’s rank being greater than one satisfies the finiteness property under the framework of (invariant) extremal norm. Formula for the computation of joint/generalized spectral radius for this class of matrix family is derived. Based on that, in the second part, we further study the joint/generalized spectral radius of finite sets of general matrices through constructing rank-one approximations in terms of singular value decomposition, and some new characterizations of joint/generalized spectral radius are obtained. Several benchmark examples from applications as well as corresponding numerical computations are provided to illustrate the approach.

Exact Computation of Joint Spectral Characteristics of Linear Operators

We address the problem of the exact computation of two joint spectral characteristics of a family of linear operators, the joint spectral radius (JSR) and the lower spectral radius (LSR), which are well-known different generalizations to a set of operators of the usual spectral radius of a linear operator. In this paper we develop a method which—under suitable assumptions—allows us to compute the JSR and the LSR of a finite family of matrices exactly. We remark that so far no algorithm has been available in the literature to compute the LSR exactly. The paper presents necessary theoretical results on extremal norms (and on extremal antinorms) of linear operators, which constitute the basic tools of our procedures, and a detailed description of the corresponding algorithms for the computation of the JSR and LSR (the last one restricted to families sharing an invariant cone). The algorithms are easily implemented, and their descriptions are short. If the algorithms terminate in finite time, then they construct an extremal norm (in the JSR case) or antinorm (in the LSR case) and find their exact values; otherwise, they provide upper and lower bounds that both converge to the exact values. A theoretical criterion for termination in finite time is also derived. According to numerical experiments, the algorithm for the JSR finds the exact value for the vast majority of matrix families in dimensions ≤20. For nonnegative matrices it works faster and finds the JSR in dimensions of order 100 within a few iterations; the same is observed for the algorithm computing the LSR. To illustrate the efficiency of the new method, we apply it to give answers to several conjectures which have been recently stated in combinatorics, number theory, and formal language theory.

Extremal norms of graphs and matrices

In recent years, the trace norm of graphs has been extensively studied under the name graph energy. In this paper, this research is extended to more general matrix norms, for example, the Schatten p-norms and the Ky Fan k-norms. Whenever possible, the results are given both for graphs and general matrices. In various contexts, a puzzling fact was observed: the Schatten p-norms are widely different for 1 ≤ p < 2 and for p ≥ 2.

Finding Extremal Complex Polytope Norms for Families of Real Matrices

In this paper we consider finite families ${\cal F}$ of real $n\!\times\!n$ matrices. In particular, we focus on the computation of the joint spectral radius $\rho({\cal F})$ via the detection of an extremal norm in the class of complex polytope norms, whose unit balls are balanced complex polytopes with a finite essential system of vertices. Such a finiteness property is very useful in view of the construction of efficient computational algorithms. More precisely, we improve the results obtained in our previous paper [N. Guglielmi, F. Wirth, and M. Zennaro, SIAM J. Matrix Anal. Appl., 27 (2005), pp. 721–743], where we gave some conditions on the family ${\cal F}$ which are sufficient to guarantee the existence of an extremal complex polytope norm. Unfortunately, they exclude unnecessarily many interesting cases of real families. Therefore, here we relax the conditions given in our previous paper in order to provide a more satisfactory treatment of the real case.

Asymptotic behavior of extremal solutions and structure of extremal norms of linear differential inclusions of order three

Asymptotic properties of extremal solutions of linear inclusions of order three with zero Lyapunov exponent are investigated. Under certain conditions it is shown that all extremal solutions of such inclusions tend to the same (up to a multiplicative factor) solution, which is central symmetric. The structure of the convex set of extremal norm is studied. A number of extremal points of this set are described.

Optimal norms and the computation of joint spectral radius of matrices

The notion of spectral radius of a set of matrices is a natural extension of spectral radius of a single matrix. The finiteness conjecture (FC) claims that among the infinite products made from the elements of a given finite set of matrices, there is a certain periodic product, made from the repetition of the optimal product, whose rate of growth is maximal. FC has been disproved. In this paper it is conjectured that FC is almost always true, and an algorithm is presented to verify the optimality of a given product. The algorithm uses optimal norms, as a special subset of extremal norms. Several conjectures related to optimal norms and non-decomposable sets of matrices are presented. The algorithm has successfully calculated the spectral radius of several parametric families of pairs of matrices associated with compactly supported multi-resolution analyses and wavelets. The results of related numerical experiments are presented.

Extremal [formula omitted]-norms of linear operators and self-similar functions

We prove that for any p ∈ [ 1 , + ∞ ] a finite irreducible family of linear operators possesses an extremal norm corresponding to the p-radius of these operators. As a corollary, we derive a criterion for the L p -contractibility property of linear operators and estimate the asymptotic growth of orbits for any point. These results are applied to the study of functional difference equations with linear contractions of the argument (self-similarity equations). We obtain a sharp criterion for the existence and uniqueness of solutions in various functional spaces, compute the exponents of regularity, and estimate moduli of continuity. This, in particular, gives a geometric interpretation of the p-radius in terms of spectral radii of certain operators in the space L p [ 0 , 1 ] .

An algorithm for finding extremal polytope norms of matrix families

In this paper the problem of the computation of the joint spectral radius of a finite set of matrices is considered. We present an algorithm which, under some suitable assumptions, is able to check if a certain product in the multiplicative semigroup is spectrum maximizing. The algorithm proceeds by attempting to construct a suitable extremal norm for the family, namely a complex polytope norm. As examples for testing our technique, we first consider the set of two 2-dimensional matrices recently analyzed by Blondel, Nesterov and Theys to disprove the finiteness conjecture, and then a set of 3-dimensional matrices arising in the zero-stability analysis of the 4-step BDF formula for ordinary differential equations.

On the Complexity of Computing the Capacity of Codes That Avoid Forbidden Difference Patterns

We consider questions related to the computation of the capacity of codes that avoid forbidden difference patterns. The maximal number of $n$-bit sequences whose pairwise differences do not contain some given forbidden difference patterns increases exponentially with $n$. The exponent is the capacity of the forbidden patterns, which is given by the logarithm of the joint spectral radius of a set of matrices constructed from the forbidden difference patterns. We provide a new family of bounds that allows for the approximation, in exponential time, of the capacity with arbitrary high degree of accuracy. We also provide a polynomial time algorithm for the problem of determining if the capacity of a set is positive, but we prove that the same problem becomes NP-hard when the sets of forbidden patterns are defined over an extended set of symbols. Finally, we prove the existence of extremal norms for the sets of matrices arising in the capacity computation. This result makes it possible to apply a specific (even though non polynomial) approximation algorithm. We illustrate this fact by computing exactly the capacity of codes that were only known approximately.

Complex Polytope Extremality Results for Families of Matrices

In this paper we consider finite families of complex $n\times n$-matrices. In particular, we focus on those families that satisfy the so-called finiteness conjecture, which was recently disproved in its more general formulation. We conjecture that the validity of the finiteness conjecture for a finite family of nondefective type is equivalent to the existence of an extremal norm in the class of complex polytope norms. However, we have not been able to prove this complex polytope extremality conjecture, but we are able to prove the small complex polytope extremality theorem under some more restrictive hypotheses on the underlying family of matrices. In addition, our theorem assures a certain finiteness property on the number of vertices of the unit ball of the extremal complex polytope norm, which could be very useful for the construction of suitable algorithms aimed at the actual computation of the spectral radius of the family.

Stability of one-leg -methods for the variable coefficient pantograph equation on the quasi-geometric mesh

 where a(t) and b(t) are continuous complex-valued functions and q ∈ (0, 1) is a fixed constant. We also extend our analysis to the neutral equation y � (t) = a(t) y(t) + b(t) y( qt ) + c(t) y � ( qt ), where c(t) is also a continuous complex-valued function. In recent years, stability properties of numerical methods for this kind of equation have been studied by numerous authors who have mainly considered the constant coefficient case and integration over meshes with constant stepsize. In general, the developed techniques give rise to non-standard recurrence relations. Instead, in the present paper we study constrained variable stepsize schemes. In Bellen et al. (1997, Appl. Numer. Math.,24, 279–293) the behaviour of the class of Θ-methods was analysed with relevance for the constant coefficient version of the above equation. As with the constant coefficient case, we show here that the methods in this class are stable if and only if 1 Θ > 1/2. The basic tool we use in our analysis is the spectral radius of a family of matrices, computed by means of polytope extremal norms. An extension of the stability results to a more general class of equations is also discussed in the last section.