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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]

This paper provides a probabilistic proof of the comparison result for viscosity solutions of path-dependent semilinear PDEs. We consider the notion of viscosity solutions introduced in [I. Ekren, et al., Ann. Probab., 42 (2014), pp. 204-236], which considers as test functions all those smooth processes which are tangent in mean. When restricted to the Markovian case, this definition induces a larger set of test functions and reduces to the notion of stochastic viscosity solutions analyzed in [E. Bayraktar and M. Sirbu, Proc. Amer. Math. Soc., 140 (2012), pp. 3645-3654; SIAM J. Control Optim., 51 (2013), pp. 4274-4294]. Our main result takes advantage of this enlargement of the test functions and provides an easier proof of comparison. This is most remarkable in the context of the linear path-dependent heat equation. As a key ingredient for our methodology, we introduce a notion of punctual differentiation, similar to the corresponding concept in the standard viscosity solutions [L. A. Caffarelli and X. Cabre, Amer. Math. Soc. Colloq. Publ., 43, AMS, Providence, RI, 1995], and we prove that semimartingales are almost everywhere punctually differentiable. This smoothness result can be viewed as the counterpart of the Aleksandroff smoothness result for convex functions. A similar comparison result was established earlier in [I. Ekren et al., Ann. Probab., 42 (2014), pp. 204-236]. The result of this paper is more general and, more importantly, the arguments that we develop do not rely on any representation of the solution. [ABSTRACT FROM AUTHOR]

This paper provides a probabilistic proof of the comparison result for viscosity solutions of path-dependent semilinear PDEs. We consider the notion of viscosity solutions introduced in [I. Ekren, et al., Ann. Probab., 42 (2014), pp. 204-236], which considers as test functions all those smooth processes which are tangent in mean. When restricted to the Markovian case, this definition induces a larger set of test functions and reduces to the notion of stochastic viscosity solutions analyzed in [E. Bayraktar and M. Sirbu, Proc. Amer. Math. Soc., 140 (2012), pp. 3645-3654; SIAM J. Control Optim., 51 (2013), pp. 4274-4294]. Our main result takes advantage of this enlargement of the test functions and provides an easier proof of comparison. This is most remarkable in the context of the linear path-dependent heat equation. As a key ingredient for our methodology, we introduce a notion of punctual differentiation, similar to the corresponding concept in the standard viscosity solutions [L. A. Caffarelli and X. Cabre, Amer. Math. Soc. Colloq. Publ., 43, AMS, Providence, RI, 1995], and we prove that semimartingales are almost everywhere punctually differentiable. This smoothness result can be viewed as the counterpart of the Aleksandroff smoothness result for convex functions. A similar comparison result was established earlier in [I. Ekren et al., Ann. Probab., 42 (2014), pp. 204-236]. The result of this paper is more general and, more importantly, the arguments that we develop do not rely on any representation of the solution. [ABSTRACT FROM AUTHOR]

The joint spectral radius (JSR) of a set of matrices characterizes the maximal asymptotic growth rate of an infinite product of matrices of the set. This quantity appears in a number of applications including the stability of switched and hybrid systems. A popular method used for the stability analysis of these systems searches for a Lyapunov function with convex optimization tools. We investigate dual formulations for this approach and leverage these dual programs for developing new analysis tools for the JSR. We show that the dual of this convex problem searches for the occupations measures of trajectories with high asymptotic growth rate. We both show how to generate a sequence of guaranteed high asymptotic growth rate and how to detect cases where we can provide lower bounds on the JSR. All results of this paper are presented for the general case of constrained switched systems, that is, systems for which the switching signal is constrained by an automaton. [ABSTRACT FROM AUTHOR]

We study the following optimization problem over a dynamical system that consists of several linear subsystems: Given a finite set of nÃ--n matrices and an n-dimensional vector, find a sequence of K matrices, each chosen from the given set of matrices, to maximize a convex function over the product of the K matrices and the given vector. This simple problem has many applications in operations research and control, yet a moderate-sized instance is challenging to solve to optimality for state-of-the-art optimization software. We propose a simple exact algorithm for this problem. Our algorithm runs in polynomial time when the given set of matrices has the oligo-vertex property, a concept we introduce in this paper for a finite set of matrices. We derive several sufficient conditions for a set of matrices to have the oligo-vertex property. Numerical results demonstrate the clear advantage of our algorithm in solving large-sized instances of the problem over one state-of-the-art global optimization solver. We also propose several open questions on the oligo-vertex property and discuss its potential connection with the finiteness property of a set of matrices, which may be of independent interest. [ABSTRACT FROM AUTHOR]

In this paper we consider a distributed stochastic optimization problem without gradient/subgradient information for local objective functions and subject to local convex constraints. Objective functions may be nonsmooth and observed with stochastic noises, and the network for the distributed design is time-varying. By adding stochastic dithers to local objective functions and constructing randomized differences motivated by the Kiefer--Wolfowitz algorithm, we propose a distributed subgradient-free algorithm for finding the global minimizer with local observations. Moreover, we prove that the consensus of estimates and global minimization can be achieved with probability one over the time-varying network, and we obtain the convergence rate of the mean average of estimates as well. Finally, we give numerical examples to illustrate the performance of the proposed algorithms. [ABSTRACT FROM AUTHOR]

This paper studies distributed algorithms for the nonsmooth extended monotropic optimization problem, which is a general convex optimization problem with a certain separable structure. The considered nonsmooth objective function is the sum of local objective functions as-signed to agents in a multiagent network, with local set constraints and affine equality constraints. Each agent only knows its local objective function, local set constraint, and the information ex-changed between neighbors. To solve the constrained convex optimization problem, we propose two novel distributed continuous-time subgradient-based algorithms, with projected output feedback and derivative feedback, respectively. Moreover, we prove the convergence of proposed algorithms to the optimal solutions under some mild conditions and analyze convergence rates, with the help of the techniques of variational inequalities, decomposition methods, and differential inclusions. Finally, we give an example to illustrate the efficacy of the proposed algorithms. [ABSTRACT FROM AUTHOR]

This paper is focused on a cooperative control design which guarantees robust connectivity and invariance of a multiagent network inside a bounded domain, under the presence of additional bounded input terms in each agent's dynamics. In particular, under the assumptions that the domain is convex and has a smooth boundary, we can design a repulsion vector field near its boundary, which ensures invariance of the agents' trajectories and does not affect the robustness properties of the control part that is exploited for connectivity maintenance. [ABSTRACT FROM AUTHOR]

In this paper we provide a solution to the gain-scheduling synthesis problem (GSP) for structured parametric and dynamic passive uncertainties, which encompasses an approach for norm-bounded uncertainties as considered in classical structured singular value theory. If compared to known approaches, the distinguishing feature is the use of frequency-dependent multipliers to guarantee stability and performance of the controlled system. On the basis of suitable parameterizations of these multipliers, we develop necessary and sufficient conditions for the existence of gain-scheduling controllers and discuss the benefits of this new design methodology. [ABSTRACT FROM AUTHOR]