On system gains for linear and nonlinear systems

System gains, and bounds for system gains, are determined for stable linear and nonlinear systems in different signal setups which include ℓ_p signal setups and certain persistent signal generalizations of the ℓ₂ and ℓ₁ signal setups. These results show that robust H_∞ and ℓ₁ control generalize to very versatile persistent signal settings. Relationships between different system gains are also derived. Finally, an application of nonlinear system gain bounds is given by establishing induced ℓ_∞ modelling error bounds for a class of (generalized) piecewise linear systems approximated with simpler linear time-invariant models.     

Spectral properties of infinite-dimensional closed-loop systems

We consider feedback systems obtained from infinite-dimensional well-posed linear systems by output feedback. Thus, our framework allows for unbounded control and observation operators. Our aim is to investigate the relationship between the open-loop system, the feedback operator K and the spectrum (in particular, the eigenvalues and eigenvectors) of the closed-loop generator A^K. We give a useful characterization of that part of the spectrum σ(A^K) which lies in the resolvent set of A, the open-loop generator, via the “characteristic equation” involving the open-loop transfer function. We show that certain points of σ(A) cannot be eigenvalues of A^K if the input and output are scalar (so that K is a number) and K≠0. We devote special attention to the case when the output feedback operator K is compact. It is relatively easy to prove that in this case, σ_e(A), the essential spectrum of A, is invariant, that is, it is equal to σ_e(A^K). A related but much harder problem is to determine the largest subset of σ(A) which remains invariant under compact output feedback. This set, which we call the immovable spectrum of A, strictly contains σ_e(A). We give an explicit characterization of the immovable spectrum and we investigate the consequences of our results for a certain class of well-posed systems obtained by interconnecting an infinite chain of identical systems.     

Maximal versus strong solution to algebraic Riccati equations arising in infinite Markov jump linear systems

We deal with a perturbed algebraic Riccati equation in an infinite dimensional Banach space which appears, for instance, in the optimal control problem for infinite Markov jump linear systems (from now on iMJLS). Infinite or finite here has to do with the state space of the Markov chain being infinite countable or finite (see, e.g., [M.D. Fragoso, J. Baczynski, Optimal control for continuous time LQ—problems with infinite Markov jump parameters, SIAM J. Control Optim. 40(1) (2001) 270–297]). By using a certain concept of stochastic stability (a sort of L₂-stability), we have proved in [J. Baczynski, M.D. Fragoso, Maximal solution to algebraic Riccati equations linked to infinite Markov jump linear systems, Internal Report LNCC, no. 6, 2006] existence (and uniqueness) of maximal solution for this class of equations. As it is noticed in this paper, unlike the finite case (including the linear case), we cannot guarantee anymore that maximal solution is a strong solution in this setting. Via a discussion on the main mathematical hindrance behind this issue, we devise some mild conditions for this implication to hold. Specifically, our main result here is that, under stochastic stability, along with a condition related with convergence in the infinite dimensional scenario, and another one related to spectrum—weaker than spectral continuity—we ensure the maximal solution to be also a strong solution. These conditions hold trivially in the finite case, allowing us to recover the result of strong solution of [C.E. de Souza, M.D. Fragoso, On the existence of maximal solution for generalized algebraic Riccati equations arising in stochastic control, Systems Control Lett. 14 (1990) 233–239] set for MJLS. The issue of whether the convergence condition is restrictive or not is brought to light and, together with some counterexamples, unveil further differences between the finite and the infinite countable case.     

A note on a system theoretic approach to a conjecture by Peller-Khrushchev

Based on the construction of infinite dimensional balanced realizations an alternative solution to the following inverse spectral problem is presented: Given a monotonically decreasing sequence of positive numbers (σ_n)_{n 1}, does there exist a Hankel operator whose sequence of singular values is (σ_n)_{n 1}?     

H∞ filtering for linear periodic systems with parameter uncertainty

This paper focuses on H_∞ filtering for a class of linear periodic systems with a certain type of norm-bounded time-varying parameter uncertainty which appears in both the state and output matrices. The problem addressed is the design of a linear periodic estimator that guarantees both the quadratic stability and and prescribed H_∞ performance on infinite horizon for the estimation error for all admissible parameter uncertainties. A solution to this problem is obtained via a Riccati equation approach.     

Some exact solutions for the helical flow of a generalized Oldroyd-B fluid in a circular cylinder

The helical flow of an Oldroyd-B fluid with fractional derivatives, also named generalized Oldroyd-B fluid, in an infinite circular cylinder is studied using Hankel and Laplace transforms. The motion is due to the cylinder that, at time t=0⁺ begins to rotate around its axis with an angular velocity Ωt, and to slide along the same axis with linear velocity Vt. The components of the velocity field and the resulting shear stresses are presented under integral and series form in terms of the generalized G and R functions. The solutions that have been obtained satisfy all imposed initial and boundary conditions, and are presented as sums of two terms, one of them being a similar solution for a Newtonian fluid. Similar solutions for generalized Maxwell fluids, as well as those for ordinary Oldroyd-B and Maxwell fluids are obtained as limiting cases of our general solutions. Furthermore, the solutions for Newtonian fluids performing the same motion, are also obtained as special cases of our solutions for α=β=1 and λ_r→λ.     

State-feedback implementation of cascade compensators

For general dynamical system Σ and cascade compensator conditions are developed for the existence of a state-feedback law F which when applied to Σ causes the resulting closed-loop system Σ_F to exhibit the same input-output behavior as the cascade connection of Σ with . For a controllable, observable linear system Σ with transfer matrix T_Σ, it is shown that an invertible linear cascade compensator with transfer matrix can be implemented with state feedback if and only if the McMillan degree of T_Σ equal the McMillan degree of .     

Homomorphic characterizations of recursively enumerable languages with very small language classes

In this paper, we attempt to characterize the class of recursively enumerable languages with much smaller language classes than that of linear languages. Language classes, and , of (i,j) linear languages and (i,j) minimal linear languages are defined by posing restrictions on the form of production rules and the number of nonterminals. Then the homomorphic characterizations of the class of recursively enumerable languages are obtained using these classes and a class, , of minimal linear languages. That is, for any recursively enumerable language L over Σ, an alphabet Δ, a homomorphism h : Δ^*→Σ^* and two languages L₁ and L₂ over Δ in some classes mentioned above can be found such that L = h(L₁∩L₂). The membership relations of L₁ and L₂ of the main results are as follows:(I) For posing restrictions on the forms of production rules, the following result is obtained:(1) and .This result is the best one and cannot be improved using . However, with posing more restriction on L₂, this result can be improved and the follwing statement is obtained.(2) and .(II) For posing restrictions on the numbers of nonterminals, the follwing result is obtained.(3) and .We believe this result is also the best.     

Adaptive policies for discrete-time stochastic control systems with unknown disturbance distribution

We introduce adaptive policies for discrete-time, infinite horizon, stochastic control systems x₁₊₁ = F(x₁, a₁, ξ₁, T =0, 1, …, with discounted reward criterion, where the disturbance process ξ₁ is a sequence of i.i.d. random elements with unknown distribution. These policies are shown to be asymptotically optimal and for each of them we obtain (almost surely) uniform approximations of the optimal reward function.     

Maximal solution to algebraic Riccati equations linked to infinite Markov jump linear systems

In this paper, we deal with a perturbed algebraic Riccati equation in an infinite dimensional Banach space. Besides the interest in its own right, this class of equations appears, for instance, in the optimal control problem for infinite Markov jump linear systems (from now on iMJLS). Here, infinite or finite has to do with the state space of the Markov chain being infinite countable or finite (see Fragoso and Baczynski in SIAM J Control Optim 40(1):270–297, 2001). By using a certain concept of stochastic stability (a sort of L ₂-stability), we prove the existence (and uniqueness) of maximal solution for this class of equation and provide a tool to compute this solution recursively, based on an initial stabilizing controller. When we recast the problem in the finite setting (finite state space of the Markov chain), we recover the result of de Souza and Fragoso (Syst Control Lett 14:233–239, 1999) set to the Markovian jump scenario, now free from an inconvenient technical hypothesis used there, originally introduced in Wonham in (SIAM J Control 6(4):681–697). Research supported by grants CNPq 520367-97-9, 300662/2003-3 and 474653/2003-0, FAPERJ 171384/2002, PRONEX and IM-AGIMB.