Р Э Hourglass Alternative and constructivity of spectral characteristics Hourglass Alternative and constructivity of spectral of matrix products V ICTOR K OZYAKIN characteristics of matrix products Introduction Joint and Lower Spectral Radii Stability vs Stabilizability V ICTOR K OZYAKIN Problems Constructive computability of Kharkevich Institute Kotel’nikov Institute spectral characteristics for Information Transmission Problems of Radio-engineering and Electronics Finiteness Conjecture Russian Academy of Sciences Russian Academy of Sciences Independent Row Uncertainty Hourglass Alternative Р Idea of Proof H -sets of Matrices Э Semiring Theorem Main Result Questions Individual Trajectories One-step Maximization Workshop on switching dynamics & verification Multi-step Maximization Amphithéâtre Darboux, Institut Henry Poincaré (IHP), Paris, France, Minimax Theorem January 28–29, 2016. Acknowledgments
Р Э Hourglass Alternative and constructivity of spectral characteristics of matrix products V ICTOR K OZYAKIN Introduction Joint and Lower Spectral Radii Stability vs Stabilizability Problems Constructive Introduction computability of spectral characteristics Finiteness Conjecture Independent Row Uncertainty Hourglass Alternative Idea of Proof H -sets of Matrices Semiring Theorem Main Result Questions Individual Trajectories One-step Maximization Multi-step Maximization Minimax Theorem Acknowledgments
Р Э Hourglass Alternative Main point of interest: stability/stabilizability of a discrete-time system and constructivity of spectral characteristics of matrix products x in x out V ICTOR K OZYAKIN A + Introduction Joint and Lower Spectral Radii Stability vs Stabilizability Problems Constructive described by a linear ( switching ) equation computability of spectral characteristics Finiteness Conjecture x ( n + 1) = A ( n ) x ( n ), n = 0,1,..., Independent Row Uncertainty Hourglass Alternative Idea of Proof where H -sets of Matrices Semiring Theorem Main Result A i ∈ R d × d , A ( n ) ∈ A = { A 1 , A 2 ,..., A r }, Questions Individual Trajectories x ( n ) ∈ R d . One-step Maximization Multi-step Maximization Minimax Theorem Acknowledgments
Р General problem Э Hourglass Alternative This problem is a special case of the more general problem: and constructivity of spectral characteristics of matrix products When the matrix products A i n ··· A i 2 A i 1 with i n ∈ { 1,..., r } converge V ICTOR K OZYAKIN under different assumptions on the switching sequences { i n } ? Introduction Joint and Lower Spectral Radii “parallel” vs “sequential” computational algorithms: e.g., Gauss-Seidel vs Jacobi method; Stability vs Stabilizability Problems distributed computations; Constructive computability of “asynchronous” vs “synchronous” mode of data exchange in the control theory and data spectral characteristics transmission (large-scale networks); Finiteness Conjecture Independent Row Uncertainty smoothness problems for Daubeshies wavelets (computational mathematics); Hourglass Alternative Idea of Proof one-dimensional discrete Schrödinger equations with quasiperiodic potentials (theory of H -sets of Matrices quasicrystalls, physics); Semiring Theorem Main Result linear or affine iterated function systems (theory of fractals); Questions Individual Trajectories Hopfield-Tank neural networks (biology, mathematics); One-step Maximization “triangular arbitrage” in the models of market economics; Multi-step Maximization Minimax Theorem etc. Acknowledgments
Р Joint and Lower Spectral Radii Э Hourglass Alternative Given a set of ( d × d )-matrices A and a norm �·� on R d , and constructivity of spectral characteristics of matrix products � A i n ··· A i 1 � 1/ n : A i j ∈ A � � V ICTOR K OZYAKIN ρ ( A ) = lim n →∞ sup Introduction Joint and Lower Spectral Radii is called the joint spectral radius (JSR) of A (Rota & Strang, 1960), whereas Stability vs Stabilizability Problems � A i n ··· A i 1 � 1/ n : A i j ∈ A � � Constructive ρ ( A ) = lim ˇ n →∞ inf computability of spectral characteristics Finiteness Conjecture is called the lower spectral radius (LSR) of A (Gurvits, 1995). Independent Row Uncertainty Hourglass Alternative Idea of Proof Remark H -sets of Matrices Semiring Theorem ρ ( A ) and ˇ ρ ( A ) are well defined and independent on the norm �·� ; Main Result Questions �·� in the definitions of JSR and LSR may be replaced by the spectral radius Individual Trajectories One-step Maximization ρ ( · ) of a matrix, see Berger & Wang, 1992 for ρ ( A ) and Gurvits, 1995; Multi-step Maximization Theys, 2005; Czornik, 2005 for ˇ ρ ( A ). Minimax Theorem Acknowledgments
Р Another Formulae for JSR Э Hourglass Alternative and constructivity of spectral characteristics of matrix products V ICTOR K OZYAKIN Elsner, 1995; Shih, 1999 — via infimum of norms; Introduction Joint and Lower Spectral Radii Protasov, 1996; Barabanov, 1988 — via special kind of norms with Stability vs Stabilizability Problems additional properties; Constructive computability of Chen & Zhou, 2000 — via trace of matrix products; spectral characteristics Finiteness Conjecture Blondel & Nesterov, 2005 — via Kronecker (tensor) products of matrices; Independent Row Uncertainty Hourglass Alternative Parrilo & Jadbabaie, 2008 — via homogeneous polynomials instead of Idea of Proof norms; H -sets of Matrices Semiring Theorem etc. Main Result Questions Individual Trajectories One-step Maximization Multi-step Maximization Minimax Theorem Acknowledgments
Р Stability vs Stabilizability Э Hourglass Alternative and constructivity of spectral characteristics of matrix products Difference between the joint and lower spectral radii: V ICTOR K OZYAKIN Introduction Joint and Lower Spectral Radii Stability vs Stabilizability The inequality ρ ( A ) < 1 characterizes the Schur stability of A : Problems Constructive computability of ρ ( A ) < 1 = ⇒ ∀ { i n } : � A i n ··· A i 2 A i 1 � → 0. spectral characteristics Finiteness Conjecture Independent Row Uncertainty Hourglass Alternative Idea of Proof H -sets of Matrices The inequality ˇ ρ ( A ) < 1 characterizes the Schur stabilizability of A : Semiring Theorem Main Result Questions ρ ( A ) < 1 ˇ ∃ { i n } : � A i n ··· A i 2 A i 1 � → 0. = ⇒ Individual Trajectories One-step Maximization Multi-step Maximization Minimax Theorem Acknowledgments
Р JSR vs LSR Э Hourglass Alternative and constructivity of spectral characteristics of matrix products The LSR has ‘less stable’ continuity properties than the JSR, see V ICTOR K OZYAKIN Bousch & Mairesse, 2002; Introduction Joint and Lower Spectral Radii Until recently, ‘good’ properties for the LSR, including numerical Stability vs Stabilizability Problems algorithms of computation, were obtained only for matrix sets A having Constructive an invariant cone, see Protasov, Jungers & Blondel, 2009/10; Jungers, 2012; computability of spectral characteristics Guglielmi & Protasov, 2013; Finiteness Conjecture Independent Row Uncertainty Bochi & Morris, 2015 started a systematic investigation of the continuity Hourglass Alternative Idea of Proof properties of the LSR. H -sets of Matrices Semiring Theorem Main Result Their investigation is based on the concepts of dominated splitting and Questions k-multicones from the theory of hyperbolic linear cocycles. In particular, Individual Trajectories One-step Maximization they gave a sufficient condition for the Lipschitz continuity of the LSR Multi-step Maximization Minimax Theorem Acknowledgments
Р First Problems Э Hourglass Alternative Inequalities and constructivity of spectral characteristics ρ ( A ) < 1, ρ ( A ) < 1 ˇ of matrix products V ICTOR K OZYAKIN might seem to give an exhaustive answer to the questions on stability or Introduction stabilizability of a switching system. Joint and Lower Spectral Radii Stability vs Stabilizability Theoretically: Problems Constructive this is indeed the case. computability of spectral characteristics Finiteness Conjecture Independent Row Uncertainty In practice: Hourglass Alternative the computation of ρ ( A ) and ˇ ρ ( A ) is generally impossible in a closed Idea of Proof H -sets of Matrices formula form = ⇒ need in approximate computational methods; Semiring Theorem Main Result there are no a priory estimates for the rate of convergence of the related Questions Individual Trajectories limits in the definitions of ρ ( A ) and ˇ ρ ( A ); One-step Maximization the required amount of computations rapidly increases in n and Multi-step Maximization Minimax Theorem dimension of a system. Acknowledgments
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