Design of observers for nonlinear systems with H ∞ performance analysis

This paper investigates the problem of observer design for nonlinear systems. By using differential mean value theorem, which allows transforming a nonlinear error dynamics into a linear parameter varying system, and based on Lyapunov stability theory, an approach of observer design for a class of nonlinear systems with time‐delay is proposed. The sufficient conditions, which guarantee the estimation error to asymptotically converge to zero, are given. Furthermore, an adaptive observer design for a class of nonlinear system with unknown parameter is considered. A method of H _∞ adaptive observer design is presented for this class of nonlinear systems; the sufficient conditions that guarantee the convergence of estimation error and the computing method for observer gain matrix are given. Finally, an example is given to show the effectiveness of our proposed approaches. Copyright © 2013 John Wiley & Sons, Ltd.     

Negative entropy,zero temperature and Markov chains on the interval

We consider ergodic optimization for the shift map on the modified Bernoulli space σ: [0, 1]^? → [0, 1]^?, where [0, 1] is the unit closed interval, and the potential A: [0, 1]^? → ? considered depends on the two first coordinates of [0, 1]^?. We are interested in finding stationary Markov probabilities µ_∞ on [0, 1]^? that maximize the value ∫ Adµ, among all stationary (i.e. σ-invariant) probabilities µ on [0, 1]^?. This problem correspond in Statistical Mechanics to the zero temperature case for the interaction described by the potential A. The main purpose of this paper is to show, under the hypothesis of uniqueness of the maximizing probability, a Large Deviation Principle for a family of absolutely continuous Markov probabilities µ _β which weakly converges to µ_∞. The probabilities µ _β are obtained via an information we get from a Perron operator and they satisfy a variational principle similar to the pressure in Thermodynamic Formalism. As the potential A depends only on the first two coordinates, instead of the probability µ on [0, 1]^?, we can consider its projection ν on [0, 1]². We look at the problem in both ways. If µ_∞ is the maximizing probability on [0, 1]^?, we also have that its projection ν _∞ is maximizing for A. The hypothesis about stationarity on the maximization problem can also be seen as a transhipment problem. Under the hypothesis of A being C ² and the twist condition, that is,

$\frac{{\partial ^2 A}}{{\partial x\partial y}}(x,y) \ne 0, for all (x,y) \in [0,1]^2 ,$

we show the graph property of the maximizing probability ν on [0, 1]². Moreover, the graph is monotonous. An important result we get is: the maximizing probability is unique generically in Mañé’s sense. Finally, we exhibit a separating sub-action for A.

     

Commutators, C0-semigroups and resolvent estimates

We study the existence and the continuity properties of the boundary values on the real axis of the resolvent of a self-adjoint operator H in the framework of the conjugate operator method initiated by Mourre. We allow the conjugate operator A to be the generator of a C₀-semigroup (finer estimates require A to be maximal symmetric) and we consider situations where the first commutator [H,iA] is not comparable to H. The applications include the spectral theory of zero mass quantum field models.     

J-nonexpansive operators

Let J be an involution in the Hilbert space H, i.e., J=J^*, J²=I. A bounded linear operator A, acting in H, is said to be J-nonexpansive if it does not expand the indefinite metric [f, f]=(Jf, f), f H, i.e., [Af, Af][f, f] or, equivalently, A^*JA–J0. In the paper one solves the problem of the determination, for a J-nonexpansive operator A, of a subspace H₀H on which too the operator A^* is J-nonexpansive.Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 45, pp. 63–68, 1986.     

Empirical bayes testing for mean life time of Rayleigh distribution

Consider a Rayleigh distribution withpdfp(x|θ) = 2xθ ^{- 1} exp(- x ²/θ) and mean lifetime μ = √πθ/2. We study the two-action problem of testing the hypothesesH ₀: μ≤ μ₀ againstH ₁: μ > μ₀ using a linear error loss of |μ- μ ₀ | via the empirical Bayes approach. We construct a monotone empirical Bayes test δ _n ^* and study its associated asymptotic optimality. It is shown that the regret of δ _n ^* converges to zero at a rate $\frac{{\ln ^2 n}}{n}$ , wheren is the number of past data available when the present testing problem is considered.     

Homogenization of Schrödinger-type equations

We consider a self-adjoint elliptic operator A_ε, ε> 0, on L₂(R^d; Cⁿ) given by the differential expression b(D)*g(x/ε)b(D). Here \(b(D) = \sum\nolimits_{j = 1}^d {b_j D_j }\) is a first-order matrix differential operator such that the symbol b(ξ) has maximal rank. The matrix-valued function g(x) is bounded, positive definite, and periodic with respect to some lattice. We study the operator exponential \({e^{ - i\tau {A_\varepsilon }}}\), where τ ∈ R. It is shown that, as ε → 0, the operator \({e^{ - i\tau {A_\varepsilon }}}\) converges to \({e^{ - i\tau {A^0}}}\) in the norm of operators acting from the Sobolev space H^s(R^d;Cⁿ) (with suitable s) to L₂(R^d;Cⁿ). Here A⁰ is the effective operator with constant coefficients. Order-sharp error estimates are obtained. The question about the sharpness of the result with respect to the type of the operator norm is studied. Similar results are obtained for more general operators. The results are applied to study the behavior of the solution of the Cauchy problem for the Schrödinger-type equation i?_τu_ε(x, τ) = A_εu_ε(x, τ).     

Second order incomplete expiring Cauchy problems

We extend results of Balakrishnan and Dorroh, on 2^nd-order incomplete Cauchy problems, from differentiable to stronglycontinuous semigroups of operators. We show that the Cauchy problem (*) $$\begin{gathered} u''(t) = A(u(t)), t \geqslant 0, u(0) = x, \hfill \\ \mathop {lim}\limits_{t \to \infty } \left\| {u^{(k)} (t)} \right\| = 0, k = 0, 1, 2, \hfill \\ \end{gathered} $$ where A is a linear operator with nonempty resolvent on a Banach space, is well-posed if and only if A has a squares root that generates a C_o semigroup, {T(t)} t>0, that converges to zero, as t goes to infinity, in the strong operator topology. This extension leads to the following application. If A is a linear constant coefficient partial differential operator on L²(?ⁿ), then there exist orthogonal closed subspaces, H₁, H₂, such that H_l⊕H₂=L²(?ⁿ), and (*), on H₁, is well-posed, while the complete Cauchy problem u″(t)=A(u(t)), t??, u(O)=x, u′ (O)=y is well-posed on H₂. We also apply our results to the dying wave equation, on C_o[0, ∞) and L^p(?dv) (1≤p <∞), for a large class of measures v.     

Partial inverse of a monotone operator

ForT a maximal monotone operator on a Hilbert spaceH andA a closed subspace ofH, the “partial inverse”T _A ofT with respect toA is introduced.T _A is maximal monotone. The proximal point algorithm, as it applies toT _A , results in a simple procedure, the “method of partial inverses”, for solving problems in which the object is to findx ∈ A andy ∈ A ^⊥ such thaty ∈ T(x). This method specializes to give new algorithms for solving numerous optimization and equilibrium problems.     

The local determination of Jordan bases for H-self-adjoint operators

Let H be an invertible self-adjoint operator on a finite dimensional Hilbert space $\text{X}$. A linear operator A is said to be H-self-adjoint (or self-adjoint relative to H) if HA = A¹H. Let σ(A) denote, as usual, the spectrum of A. If A is H-self-adjoint, then A is similar to A¹ and λ ∈ σ(A) implies λ&#x0304; ∈ σ (A), so that the spectrum of A issymmetric with respect to the real axis. Given spectral information for A at an eigenvalue λ₀ (≠ λ&#x0304;₀), we investigate the corresponding information at λ&#x0304;₀ and, in particular, the unique pairing of Jordan bases for the root subspaces at λ₀ and λ&#x0304;₀.     

On the convergence region for decomposition solutions

The author's decomposition method using his A_n polynomials for the nonlinearities has been shown to apply to wide classes of nonlinear (or nonlinear stochastic) operator equations providing a computable, accurate solution which converges rapidly. In computation the above is sufficient for a rapid test of convergence region.     

The rate of approximation by means of polynomials with restricted coefficients

For a sequenceA = {A_k} _k?1 ^∞ of positive constants letP _A = {p(x): p(x) = Σ _k?1 ⁿ a _k x _k ,n = 1,2, …, ¦a _k ≦ A _k ^k }. We consider the rate of approximation by elements ofP _A , of continuous functions in [0, 1] which vanish at x = 0. Also a classP _A is called “efficient” if globally it guarantees the Jackson rate of approximation. Some necessary conditions for efficiency and some sufficient ones are derived.     

On the essential commutant of analytic Toeplitz operators associated with spherical isometries

Let T∈Bn(H) be an essentially normal spherical isometry with empty point spectrum on a separable complex Hilbert space H, and let AT⊂B(H) be the unital dual operator algebra generated by T. In this note we show that every operator S∈B(H) in the essential commutant of AT has the form S=X+K with a T-Toeplitz operator X and a compact operator K. Our proof actually covers a larger class of subnormal operator tuples, called A-isometries, which includes for example the tuple T=(Mz₁,…,Mzn)∈B(H²n(σ)) consisting of the multiplication operators with the coordinate functions on the Hardy space H²(σ) associated with the normalized surface measure σ on the boundary ∂D of a strictly pseudoconvex domain D⊂Cn. As an application we determine the essential commutant of the set of all analytic Toeplitz operators on H²(σ) and thus extend results proved by Davidson (1977) [6] for the unit disc and Ding and Sun (1997) [11] for the unit ball.     

Homogenization of hyperbolic equations

We consider a self-adjoint matrix elliptic operator A _{ε, ε} > 0, on L ₂(R ^d ;C ⁿ ) given by the differential expression b(D)*g(x/ε)b(D). The matrix-valued function g(x) is bounded, positive definite, and periodic with respect to some lattice; b(D) is an (m × n)-matrix first order differential operator such that m ≥ n and the symbol b(ξ) has maximal rank. We study the operator cosine cos(τA _ε ^1/2 ), where τ ∈ R. It is shown that, as ε → 0, the operator cos(τA _ε ^1/2 ) converges to cos(τ(A ⁰)^1/2) in the norm of operators acting from the Sobolev space H ^s (R ^d ;C ⁿ ) (with a suitable s) to L ₂(R ^d ;C ⁿ ). Here A ₀ is the effective operator with constant coefficients. Sharp-order error estimates are obtained. The question about the sharpness of the result with respect to the type of the operator norm is studied. Similar results are obtained for more general operators. The results are applied to study the behavior of the solution of the Cauchy problem for the hyperbolic equation ? _τ ² u _ε (x, τ) = ?A _ε u _ε (x, τ).     

Monotone empirical Bayes tests for some discrete nonexponential families

This paper deals with the empirical Bayes two-action problem of testingH ₀ : θ ≤ θ_o versusH ₁ : θ > θ_o using a linear error loss for some discrete nonexponential families having probability function either $\begin{gathered} f_1 (x|\theta ) = (x\alpha + 1 - \theta )\theta ^x /\prod\limits_{j = 0}^x {(j\alpha + 1)} \\ or \\ f_1 (x|\theta ) = \left[ {\theta \prod\limits_{j = 0}^{x - 1} {(j\alpha + 1 - \theta )} } \right]/\left[ {\prod\limits_{j = 0}^x {(j\alpha + 1)} } \right] \\ \end{gathered} $ Two empirical Bayes tests δ_n* and δ_n** are constructed. We have shown that both δ_n* and δ_n** are asymptotically optimal, and their regrets converge to zero at an exponential decay rate O(exp( -cn)) for some c > 0, wheren is the number of historical data available when the present decision problem is considered.     

Infinite-variate wide-sense Markov processes and functional analysis for bounded operator-forming vectors

Let p, q be arbitrary parameter sets, and let $\text{H}$ be a Hilbert space. We say that x = (xⁱ)_i?q, xⁱ ? $\text{H}$, is a bounded operator-forming vector (?$\text{H}$_F^q) if the Gram matrix 〈x, x〉 = [(xⁱ, x^j)]_i?q,j?q is the matrix of a bounded (necessarily ≥ 0) operator on $\text{l}{}_{\mathrm{q}}{}^2$, the Hilbert space of square-summable complex-valued functions on q. Let A be p × q, i.e., let A be a linear operator from $\text{l}{}_{\mathrm{q}}{}^2$ to $\text{l}{}_{\mathrm{p}}{}^2$. Then exists a linear operator ǎ from (the Banach space) $\text{H}$_F^q to $\text{H}$_F^p on $\text{D}$(A) = {x:x ? $\text{H}$_F^q, $\text{A〈x, x〉}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ is p × q bounded on $\text{l}{}_{\mathrm{q}}{}^2$} such that y = ǎx satisfies y^j?σ(x) = {space spanned by the xⁱ}, 〈y, x〉 = A〈x, x〉 and $\text{〈y, y〉 = A〈x, x〉}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{(A〈x, x〉}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}{}^1$. This is a generalization of our earlier [J. Multivariate Anal.4 (1974), 166–209; 6 (1976), 538–571] results for the case of a spectral measure concentrated on one point. We apply these tools to investigate q-variate wide-sense Markov processes.     

Control subspaces of minimal dimension. Elementary introduction. Discotheca

In this paper there is introduced and studied the following characteristic of a linear operator A acting on a Banach space Χ: , where Cyc A=R∶R is a subspace of Χ, dim R<+∞. Spqn (AⁿR∶n?0)=χ. Always disc A ?μ_A=(the multiplicity of the spectrum of the operator (dim R∶R∈Cyc A), where (by definition) in each A-cyclic subspace there is contained a cyclic subspace of dimension ? disc A. For a linear dynamical system x(t)=Ax(t)+Bu,(t) which is controllable, the characteristic disc A of the evolution operator A shows how much the control space can be diminished without losing controllability. In this paper there are established some general properties ofdisc (for example, conditions are given under which disc(A⊕B))=max(discA, disc B); disc is computed for the following operators: S (S is the shift in the Hardy space H²); disc S=2, (but μ_S=i); disc S _n ^* =n (butμ=1), where S_n=S⊕. ⊕S; disc S=2, (but μ_S=1), where S is the bilateral shift. It is proved that for a normal operator N with simple spectrum, disc N=μ_N=1 ? (the operator N is reductive). There are other results also, and also a list of unsolved problems.     

A non-steady flow of liquid in a porous pipe with variable permeability

LetB denote the infinitesimal operator of a strongly continuous semigroup S(t), with resolvent R_λ, on Banach space L. We define related operators P and V so that λR_λf = Pf + λVf + o(λ), as λ → 0⁺. For α, η > 0 and possibly unbounded, linear operator A, we let U_{α, η}(t) represent a strongly continuous semigroup generated by αA + ηB. We show that under appropriate simultaneous convergence of α and η, U_{α, η}(t) converges strongly to a strongly continous semigroup U(t), having infinitesimal operator characterized through PA(VA)^rf where r =min{j ? 0, PA(VA)^j ≠ 0}. We apply the abstract perturbation theorem to a singular perturbation initial-value problem, of Tihonov-type, for a non-linear system of ordinary differential equations.     

The structurally stable linear systems on the half-line are those with exponential dichotomies

Let A be a selft-adjoint operator on the Hilbert space L²Ω, ?) = {u ε L_loc²(Ω)|∫_Ω|² ?(x)dx < + ∞} defined by means of a closed, semibounded, sesquilinear form a(·, ·). We obtain a necessary and sufficuents condition for the spectrum of A to be discrete. We apply this result to a Sturm-Liouville problem for an elliptic operator with discontinuous coefficients defined on an unbounded domain and to the study of the spectrum of a Hamiltonian defined by a pseudodifferential operator.     

Dirac Operator with a Potential of Special Form and with the Periodic Boundary Conditions

We consider the Dirac operator on the interval [0, 1] with the periodic boundary conditions and with a continuous potential Q(x) whose diagonal is zero and which satisfies the condition Q(x) = Q^T(1?x), x ∈ [0, 1]. We establish a relationship between the spectrum of this operator and the spectra of related functional-differential operators with involution. We prove that the system of eigenfunctions of this Dirac operator has the Riesz basis property in the space L ₂ ² [0, 1].