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We analyze the maximal growth of trajectories of discrete-time linear switching system, i.e., controlled linear systems with the control set being an arbitrary compact set of matrices. This is done by applying the optimal convex Lyapunov function called the Barabanov norm, which provides a very refined analysis of trajectories. Until recently that notion remained rather theoretical apart from special cases. In 2015 N. Guglielmi and M. Zennaro [SIAM J. Matrix Anal. Appl., 36 (2015), pp. 634-655] showed that many systems possess at least one efficiently computed Barabanov norm. In this paper we classify all possible Barabanov norms and prove that, under mild assumptions, which can be verified algorithmically, those norms are unique and are either piecewise-linear or piecewise-quadratic. For some narrow classes of systems, there are more complicated Barabanov norms, but they can still be classified and constructed. Using those results we find all trajectories of the fastest growth. Examples and numerical results are provided. [ABSTRACT FROM AUTHOR]

The relationship of the theory of port-Hamiltonian systems with the mathematical concept of monotonicity is explored. The earlier introduced notion of incrementally port-Hamiltonian systems is extended to systems defined with respect to maximal cyclically monotone relations, together with their generating convex functions. This gives rise to interesting subclasses of incrementally port-Hamiltonian systems, with examples stemming from physical systems modeling as well as from convex optimization. Furthermore, it is shown how cyclical monotonicity for Dirac structures is equivalent to separability. An in-depth treatment is given of the composition of maximal monotone and maximal cyclically monotone relations, where in the latter case the resulting maximal cyclically monotone relation is shown to be computable through the use of generating functions. The results on compositionality are employed for steady-state analysis and for a convex optimization approach to the computation of the equilibria of interconnected incrementally port-Hamiltonian systems. Finally, the relation to incremental and differential passivity is discussed, and it is shown how incrementally port-Hamiltonian systems with strictly convex Hamiltonians are equilibrium independent passive. [ABSTRACT FROM AUTHOR]

We present a convex solution for the design of generalized accelerated gradient algorithms for strongly convex objective functions with Lipschitz continuous gradients. We utilize integral quadratic constraints and the Youla parameterization from robust control theory to formulate a solution of the algorithm design problem as a convex semidefinite program. We establish explicit formulas for the optimal convergence rates and extend the proposed synthesis solution to extremum control problems. [ABSTRACT FROM AUTHOR]

In a real Hilbert space setting, we investigate the asymptotic properties of the solutions of a second-order differential system in view of linearly constrained convex minimization. The inertial dynamics are governed by a Hessian-driven damping term associated with the convex function to be minimized and potential effects induced by the linear constraints. We provide conditions on both the damping and the potential for which the solutions converge towards some feasible point of the convex minimization problem; this convergence is towards some minimizer provided that the solutions' initial data is specifically preselected. In addition, we present asymptotic estimates on the convergence rate of the solutions depending on the interaction between damping and potential effects. Our analysis is mainly based on energy-like arguments that capture the dissipative nature of the inertial dynamics by means of a Bregman distance. We complement our study with the fact that the second-order dynamics admit a first-order representation in terms of the Arrow--Hurwicz differential system. Numerical experiments further illustrate our theoretical findings. [ABSTRACT FROM AUTHOR]