The distance to strong stability

It is known that the minimal cone for the constraint system of a conic linear optimization problem is a key component in obtaining strong duality without any constraint qualification. In the particular case of semidefinite optimization, an explicit expression for the dual cone of the minimal cone allows for a dual program of polynomial size that satisfies strong duality. This is achieved due to the fact that we can express the orthogonal complement of a face of the cone of positive semidefinite matrices completely in terms of a system of semidefinite inequalities. In this paper, we extend this result to cones that are either faces of the cone of positive semidefinite matrices or the dual cones of faces of the cone of positive semidefinite matrices. The newly proved result was used in Zhang (4OR 9:403–416, 2011). However, a proof was not given in Zhang (4OR 9:403–416, 2011).

Semidefinite programming lower bounds and branch-and-bound algorithms for the quadratic minimum spanning tree problem

This work shows an approach to reduce the dimensionality of matrix representations of quantum channels. It is achieved by finding a base of the cone of positive semidefinite matrices which represent quantum channels. Next, this is implemented in the Julia programming language as a part of the QuantumInformation.jl package.

Positive semidefinite matrices with a given sparsity pattern

A real-valued continuous function "Equation missing" on an interval "Equation missing" gives rise to a map "Equation missing" via functional calculus from the convex set of "Equation missing" Hermitian matrices all of whose eigenvalues belong to the interval. Since the subpace of Hermitian matrices is provided with the order structure induced by the cone of positive semidefinite matrices, one can consider convexity of this map. We will characterize its convexity by the following trace-inequalities: "Equation missing" for "Equation missing". A related topic will be also discussed.

Aurora and Mehta (J Comb Optim 36(3):965–1006, 2018) show that two graphs $$G_1,G_2$$ , on n vertices each, are isomorphic if and only if the feasible region of a certain linear program, LP-GI, intersects with the Quadratic Assignment Problem (QAP)-polytope in $${\mathbb {R}}^{(n^4+n^2)/2}$$ . The linear program LP-GI in Aurora and Mehta (J Comb Optim 36(3):965–1006, 2018) is obtained by relaxing an integer linear program whose feasible points correspond to the isomorphisms between $$G_1,G_2$$ . In this paper we take an analogous approach with the linear programs replaced with conic programs. A completely positive description of the QAP-polytope was obtained in Povh and Rendl (Discrete Optim 6(3):231–241, 2009). By adding the graph conditions to this description we get a completely positive formulation of the graph isomorphism problem. However, analogous to integer linear programs, it is NP-hard to optimize over the cone of completely positive matrices. So we relax this formulation by replacing the cone of completely positive matrices with the cone of positive semidefinite matrices. We observe that the resulting SDP is the Lovasz Theta function (Lovasz in IEEE Trans Inf Theory 25(1):1–7, 1979) of a graph product of $$G_1,G_2$$ and can be efficiently computed. We provide a natural heuristic that uses the SDP to solve the graph isomorphism problem. We run our heuristic on several pairs of non-isomorphic strongly regular graphs and find the results to be encouraging. Further, by adding the non-negativity constraints to the SDP, we obtain a doubly non-negative formulation, DNN-GI. We show that if the set of optimal points in DNN-GI contains a point of rank at most 3, then the given pair of graphs must be isomorphic.

In this article we give an overview of recent results on hypergroup algebras which are associated with Bessel functions on cones of positive semidefinite matrices and a class of Dunkl-type Bessel functions for root systems of type B. The Bessel hypergroups on matrix cones are obtained from a discrete series with a group-theoretic background by interpolation with respect to a dimension parameter. Passing to the matrix spectra leads to Bessel functions of Dunkl type and three continuous series of hypergroup algebras matching the Dunkl transform in the Weyl-group invariant case.

Throughout this article, a novel control strategy for fractional-order gene regulation networks (FOGRN) of all categories is designed by using the vector Lyapunov function in combination with the M-matrix measure. Firstly, a series of puzzles surrounding the asymptotic stability of two-dimensional FOGRN are studied, and a new asymptotic stability control strategy is formulated based on the vector Lyapunov function in combination with the M-matrix measure, ensuring that the controlled FOGRN has a strong robust stability. In addition, the corresponding asymptotic stability criterion is deduced. On this basis, the problem of asymptotic stability of a three-dimensional FOGRN is studied. Based on the new method, a stabilization control strategy is also formulated with the corresponding asymptotic stability criterion deduced, ensuring that the controlled FOGRN has a strong robust stability as well. Finally, this novel method’s effectiveness and generality are authenticated via simulation experiments.

This article introduces a new notion of stability for internal autonomous system descriptions that is referred to as ‘strong stability’ and characterises the case of avoiding overshoots for all initial conditions within a given sphere. This is a stronger notion of stability compared with alternative definitions (asymptotic, Lyapunov), which allows the analysis and design of control systems described by natural coordinates to have no overshooting response for arbitrary initial conditions. For the case of linear time-invariant systems necessary and sufficient conditions for strong stability are established in terms of the negative definiteness of the symmetric part of the state matrix. The invariance of strong stability under orthogonal transformations is established and this enables the characterisation of the property in terms of the invariants of the Schur form of the state matrix. Some interesting relations between strong stability and special forms of coordinate frames are examined and links between the skewness of the eigenframe and the violation of strong stability are derived. The characterisation of properties of strong stability given here provides the basis for the development of feedback theory for strong stabilisation.

The cone of distance matrices

We investigate stability of a linear autonomous functional differential equation of neutral type. The basis of the study is the well-known explicit solution representation formula including an integral operator, the kernel of which is called the Cauchy function of the equation under study. It is shown that the definitions of Lyapunov, asymptotic and exponential stabilities can be formulated without loss of generality in terms of the corresponding properties of the Cauchy function. The conclusion is drawn that stability with respect to initial data depends on the functional space which the initial function belongs to, and, as a consequence, that there is the need to indicate this space in the definition of stability. It is shown that, along with the concept of asymptotic stability, a certain stronger property should be introduced, which we call strong asymptotic stability. The main study is devoted to stability with respect to initial function from spaces of integrable functions. Special attention is paid to the study of asymptotic and exponential stability. We use the following known properties of the Cauchy function of an equation of neutral type: this function is piecewise continuous, and its jumps are determined by a Cauchy problem for a linear difference equation. We obtain that the strong asymptotic stability of the equation under consideration for initial data from the space L1 is equivalent to an exponential estimate of the Cauchy function and; moreover, we show that these properties are equivalent to the exponential stability with respect to initial data from the spaces Lp for all p from 1 to infinity inclusive. However, we show that strong asymptotic stability with respect to the initial data from the space Lp for p greater than one may not coincide with exponential stability.

If V is a Lyapunov function of an equation du/dt = u′ = Zu in a Banach space then asymptotic stability of an equilibrium point may be easily proved if it is known that sup(V′) < 0 on sufficiently small spheres centered at the equilibrium point. In this paper weak asymptotic stability is proved for a bounded infinitesimal generator Z under a weaker assumption V′ ≤ 0 (which alone implies ordinary stability only) if some observability condition, involving Z and the Frechet derivative of V′, is satisfied. The proof is based on an extension of LaSalle′s invariance principle, which yields convergence in a weak topology and uses a strongly continuous Lyapunov function. The theory is illustrated with an example of an integro‐differential equation of interest in the theory of chemical processes. In this case strong asymptotic stability is deduced from the weak one and explicit sufficient conditions for stability are given. In the case of a normal infinitesimal generator Z in a Hilbert space, strong asymptotic stability is proved under the following assumptions Z*+Z is weakly negative definite and Ker Z = {0}. The proof is based on spectral theory.

(in the sense of Lyapunov) should be considered in the physical state space, not in the state space of another system which is a lower dimensional realization of some input-output relation. The dimension of the stat.e space depends on t,he stability property one is interested in. It. is, hence, not correct to state that the dimension of the st.ate space is (an - 1) for a nonuniformly damped power system, and only (2n - 2) for a uniformly damped or undamped power system; eveh this is a priori' expect,ed for the power system model, since t.he physical variables are t.he speeds and the load angle differences, but not the load angles themselves. However, I usel Anderson’s t.heorem only to derive a Lyapunov funct.ion for the power syst,em stability problem and not, to conclude its sign dehiteness; the validity and the usefulness of the Lyapunov function are checked a posteriori. Because of t.he reasons pointed out by Sastry and Murthy, it would indeed be incorrect to conclude that for the power syst,em case a positive-definite mat.rix P is obtained from Anderson’s technique. My purpose is to develop a systematic procedure to derive a useful Lyapunov function. Actually, once the Lyapunov function is obtained, one can forget about Anderson’s theorem and investigate the sign definiteness of the tentative Lyapunov function. This function should be considered in the (2n - l>dimensional state space with

On conditions for asymptotic stability of dissipative infinite-dimensional systems with intermittent damping

Modeling and Lyapunov controller design for transverse vibration of a flexible cable transporter system with arbitrarily varying lengths are presented using Hamilton's principle and Lyapunov theory. The axial velocity of the system is assumed to be unknown in the model. This is different from existing literature where the axial velocity is assumed either to be constant or prescribed. The governing equations include two coupled non-linear Partial Differential Equations (PDEs) and boundary conditions. The interactions between the cables and the slider, pulleys, and motors are included in the model. Numerical solution of the derived governing equations is obtained using Galerkin's method. Based on the Lyapunov theory, we propose two boundary controllers and one domain point-wise controller, which suppress the transverse vibration of the cables effectively while assuring the attainment of the slider goal. The proposed controllers dissipate the vibration energy and guarantee the stability of the closed-loop system in Lyapunov sense. Simulation results demonstrate the effectiveness of the proposed controllers.