Self-adjusting stochastic processes

Let A_t(i, j) be the transition matrix at time t of a process with n states. Such a process may be called self-adjusting if the occurrence of the transition from state h to state k at time t results in a change in the hth row such that A_t+1(h, k) ? A_t(h, k). If the self-adjustment (due to transition h → kx) is A_{t + 1}(h, j) = λA_t(h, j) + (1 ? λ)δ_jk (0 < λ < 1), then with probability 1 the process is eventually periodic. If A₀(i, j) < 1 for all i, j and if the self-adjustment satisfies A_{t + 1}(h, k) = ?(A_t(h, k)) with ?(x) twice differentiable and increasing, x < ?(x) < 1 for 0 ? x < 1,?(1) = ?′(1) = 1, then, with probability 1, lim A_t does not exist.     

On the Instability of Linear Nonautonomous Delay Systems

The unstable properties of the linear nonautonomous delay system x(t) = A(t)x(t) + B(t)x(t – r(t)), with nonconstant delay r(t), are studied. It is assumed that the linear system y(t) = (A(t) + B(t))y(t) is unstable, the instability being characterized by a nonstable manifold defined from a dichotomy to this linear system. The delay r(t) is assumed to be continuous and bounded. Two kinds of results are given, those concerning conditions that do not include the properties of the delay function r(t) and the results depending on the asymptotic properties of the delay function.     

Periodic solutions of semilinear equations of evolution of compact type

The existence of solutions in a weak sense of x′ + (A + B(t, x))x = f(t, x), x(0) = x(T) is established under the conditions that A generates a semigroup of compact type on a Hilbert space H; B(t,x) is a bounded linear operator and f(t, x) a function with values in H; for each square integrable ?(t) the problem with B(t, ?(t)) and f(t, ?(t)) in place of B(t, x) and f(t, x) has a unique solution; and B and f satisfy certain boundedness and continuity conditions.     

Asymptotic stability of a class of integro-differential equations

We examine the asymptotic stability of the zero solution of the first-order linear equation x′(t) = Ax(t) + ∝₀^tB(t ? s) x(s) ds, where B(t) is integrable and does not change sign on [0, ∞). The results are applied to an examination of the stability of equilibrium of some nonlinear population models.     

On the instability of differential systems with varying delay

The unstable properties of the null solution of the nonautonomous delay system x′(t)=A(t)x(t)+B(t)x(t−r₁(t))+f(t,x(t),x(t−r₂(t))) are examined; the nonconstant delays r₁, r₂ are assumed to be continuous bounded functions. The case A=constant is reviewed, where a theorem, recalling the Perron instability theorem for ordinary differential equations, is obtained.     

On some t−(2k, k, λ) designs

t?(2k, k, λ) designs having a property similar to that of Hadamard 3-designs are studied. We consider conditions (i), (ii), or (iii) for t?(2k, k, λ) designs: (i) The complement of each block is a block. (ii) If A and B are a complementary pair of blocks, then ∥ A ∩ C ∥ = ∥ B ∩ C ∥ ± u holds for any block C distinct from A and B, where u is a positive integer. (iii) if A and B are a complementary pair of blocks, then ∥ A ∩ C ∥ = ∥ B ∩ C ∥ or ∥ A ∩ C ∥ = ∥ B ∩ C ∥ ± u holds for any block C distinct from A and B, where u is a positive integer. We show that a t?(2k, k, λ) design with t ? 2 and with properties (i) and (ii) is a 3?(2u(2u + 1), u(2u + 1), u(2u² + u ? 2)) design, and that a t?(2k, k, λ) design with t ? 4 and with properties (i) and (iii) is the 5-(12, 6, 1) design, the 4-(8, 4, 1) design, a $\text{5?(2u}{}^2\text{, u}{}^2\text{,}\text{1}\text{4}\text{(u}{}^2\text{? 3) (u}{}^2\text{? 4))}$ design, or a $\text{5?(}\text{2}\text{3}\text{u(2u + 1),}\text{1}\text{3}\text{u(2u = 1),}\text{1}\text{5 4}\text{u(2u}{}^2\text{+ u ? 9) (2u}{}^2\text{+ u ? 12))}$ design.     

Attainable sets for linear stochastic control systems

Let x_t^u(w) be the solution process of the n-dimensional stochastic differential equation dx_t^u = [A(t)x_t^u + B(t) u(t)] dt + C(t) dW_t, where A(t), B(t), C(t) are matrix functions, W_t is a n-dimensional Brownian motion and u is an admissable control function. For fixed ? ? 0 and 1 ? δ ? 0, we say that x?Rⁿ is (?, δ) attainable if there exists an admissable control u such that P{x_t^u?S_?(x)} ? δ, where S_?(x) is the closed ?-ball in Rⁿ centered at x. The set of all (?, δ) attainable points is denoted by $\text{A}$(t). In this paper, we derive various properties of $\text{A}$(t) in terms of K(t), the attainable set of the deterministic control system $\text{x}\text{?}\text{= A(t)x + B(t)u}$. As well a stochastic bang-bang principle is established and three examples presented.     

Parameter Dependence of Stable Manifolds for Nonuniform (μ, ν)-dichotomies

We construct stable invariant manifolds for semiflows generated by the nonlinear impulsive differential equation with parameters x'= A(t)x + f(t, x, λ), t≠τi and x(τ+i) = Bix(τi) + gi(x(τi), λ), i ∈ N in Banach spaces, assuming that the linear impulsive differential equation x'= A(t)x, t≠τi and x(τ+i) = Bix(τi), i ∈ N admits a nonuniform (μ, ν)-dichotomy. It is shown that the stable invariant manifolds are Lipschitz continuous in the parameter λ and the initial values provided that the nonlinear perturbations f, g are sufficiently small Lipschitz perturbations.     

Influence of anti-diagonals on the asymptotic stability for linear differential systems

Sufficient conditions are given for asymptotic stability of the linear differential system x′  =  B(t)x with B(t) being a 2  ×  2 matrix. All components of B(t) are not assumed to be positive. The matrix B(t) is naturally divisible into a diagonal matrix D(t) and an anti-diagonal matrix A(t). Our concern is to clarify a positive effect of the anti-diagonal part A(t)x on the asymptotic stability for the system x′  =  B(t)x.      

Influence of anti-diagonals on the asymptotic stability for linear differential systems

Sufficient conditions are given for asymptotic stability of the linear differential system x′  =  B(t)x with B(t) being a 2  ×  2 matrix. All components of B(t) are not assumed to be positive. The matrix B(t) is naturally divisible into a diagonal matrix D(t) and an anti-diagonal matrix A(t). Our concern is to clarify a positive effect of the anti-diagonal part A(t)x on the asymptotic stability for the system x′  =  B(t)x.     

More on Poincaré’s and Perron’s Theorems for Difference Equations∗

Consider the scalar kth order linear difference equation: x(n + k) + pi(n)x(n + k - 1) + … + pk(n)x(n) = 0 where the limits qi=lim_n→∞Pi(n) (i=1,…,k) are finite. In this paper, we confirm the conjecture formulated recently by Elaydi. Namely, every nonzero solution x of (?) satisfies the same asymptotic relation as the fundamental solutions described earlier by Perron, ie., ?= lim sup_n→∞ |x(n)| is equal to the modulus of one of the roots of the characteristics equation χ ^k + q ₁χ ^k?1+…+qk=0. This result is a consequence of a more general theorem concerning the Poincaré difference system x(n+1)=[A+B(n]x(n), where A and B(n) (n=0,1,…) are square matrices such that ‖B(n)‖ →0 as n → ∞. As another corollary, we obtain a new limit relation for the solutions of (?).     

On Heuvers’ Logarithmic Functional Equation

?(x + y) - ?(x) - ?(y) = ?(x ^?1 + y ^?l) are identical to those of the Cauchy equation ?(xy) = ?(x) + ?(y) when ? is a function from the positive real numbers into the reals. In the present article, we prove this equivalence for functions mapping the set of nonzero elements of a field (excluding ?₂) .     

Carleman Estimates for Second-Order Hyperbolic Equations

In the space of variables (x, t) ∈ ? ⁿ⁺¹, we consider a linear second-order hyperbolic equation with coefficients depending only on x. Given a domain D ? ? ⁿ⁺¹ whose projection to the x-space is a compact domain Ω, we consider the question of construction of a stability estimate for a solution to the Cauchy problem with data on the lateral boundary S of D. The well-known method for obtaining such estimates bases on the Carleman estimates with an exponential-type weight function exp(2τ?(x, t)) whose construction faces certain difficulties in case of hyperbolic equations with variable coefficients. We demonstrate that if D is symmetric with respect to the plane t = 0 then we can take ?(x, t) to be the function ?(x, t) = s ²(x, x ⁰) ? pt ², where s(x, x ⁰) is the distance between points x and x ⁰ in the Riemannian metric induced by the differential equation, p is some positive number less than 1, and the fixed point x ⁰ can either belong to the domain Ω or lie beyond it. As for the metric, we suppose that the sectional curvature of the corresponding Riemannian space is bounded above by some number k ₀ ≥ 0. In case of space of nonpositive curvature the parameter p can be taken arbitrarily close to 1; in this case as p → 1 the stability estimates lead to a uniqueness theorem which describes exactly the domain of the solution continuation through S. It turns out that, in case of space of bounded positive curvature, construction of a Carleman estimate is possible only if the product of k ₀ and sup _x∈Ω s ²(x, x ⁰) satisfies some smallness condition.     

Analytic-numerical solutions with a priori error bounds for a class of strongly coupled mixed partial differential systems

This paper deals with the construction of analytic-numerical solutions with a priori error bounds for systems of the type u_t = Au_xx, u(0,t) + u_x(0,t) = 0, Bu(1,t) + Cu_x(1,t) = 0, 0 < x < 1, t > 0, u(x,0) = f(x). Here A, B, C are matrices for which no diagonalizable hypothesis is assumed. First an exact series solution is obtained after solving appropriate vector Sturm-Liouville-type problems. Given an admissible error ε and a bounded subdomain D, after appropriate truncation an approximate solution constructed in terms of data and approximate eigenvalues is given so that the error is less than the prefixed accuracy ε, uniformly in D.     

On the Cauchy problem for some abstract nonlinear differential equations

In the present paper, we study the Cauchy problem in a Banach spaceE for an abstract nonlinear differential equation of form $$\frac{{d^2 u}}{{dt^2 }} = - A\frac{{du}}{{dt}} + B(t)u + f(t,W)$$ whereW = (A ₁(t)u,A ₂(t)u,?,A _?(t)u), (A _i (t),i = 1, 2, ?,?), (B(t),t ∈I = [0,b]) are families of closed operators defined on dense sets inE intoE, f is a given abstract nonlinear function onI ×E ^? intoE and ?A is a closed linear operator defined on dense set inE intoE, which generates a semi-group. Further, the existence and uniqueness of the solution of the considered Cauchy problem is studied for a wide class of the families (A _i(t),i = 1, 2, ?,?), (B(t),t ∈I). An application and some properties are also given for the theory of partial diferential equations.     

Algebraic properties of compatible Poisson brackets

We discuss algebraic properties of a pencil generated by two compatible Poisson tensors A(x) and B(x). From the algebraic viewpoint this amounts to studying the properties of a pair of skew-symmetric bilinear forms A and B defined on a finite-dimensional vector space. We describe the Lie group G _P of linear automorphisms of the pencil P = {A + λB}. In particular, we obtain an explicit formula for the dimension of G _P and discuss some other algebraic properties such as solvability and Levi-Malcev decomposition.     

Applications of the Fourier—Wiener transform to differential equations on infinite dimensional spaces,II

In this paper, we investigate the existence and regularity of solutions for Cauchy problems associated with the following two equations: (1) u_t(x, t) = (?1)^{k + 1}Δ^ku(x, t), and (2) u_tt(x, t) = (?1)^{k + 1}Δ^ku on infinite dimensional spaces. The nonhomogeneous cases associated with (1) and (2) are also considered.     

Accurate approximation of time-varying matrices of linear differential systems

We consider the problem of the identification of the time-varying matrix A(t) of a linear m-dimensional differential system y′ = A(t)y. We develop an approximation A_n,k = ∑ⁿ_{j ? 1}c_j{Y(t_k + τ_j) Y^?1(t_k) ? I} to A(t_k) for grid points t_k = a + kh, k = 0,…, N using specified τ_j = θ_jh, 0 < θ_j < 1, j = 1, …, n, and show that for each t_k, the L₁ norm of the error matrix is $\text{O}$(hⁿ). We demonstrate an efficient scheme for the evaluation of A_n,k and treat sample problems.     

On the null-controllability of nonlinear delay systems with restrained controls

Sufficient conditions are developed for the null-controllability of the nonlinear delay process (1) $\text{x}\text{?}\text{(t) = L(t, x}{}_{\mathrm{t}}\text{) + B(t) u(t) + f(t, x}{}_{\mathrm{t}}\text{, u(t))}$ when the values of the control functions u lie in an m-dimensional unit cube C^m of E^m. Conditions are placed on f which guarantee that if the uncontrolled system $\text{x}\text{?}\text{(t) = L(t, x}{}_{\mathrm{t}}\text{)}$ is uniformly asymptotically stable and if the linear control system x(t) = L(t, x_t) + B(t) u(t) is proper, then (1) is null-controllable.     

C0-semigroups on a locally convex space

A necessary and sufficient condition that a densely defined linear operator A in a sequentially complete locally convex space X be the infinitesimal generator of a quasi-equicontinuous C₀-semigroup on X is that there exist a real number β ? 0 such that, for each λ > β, the resolvent (λI ? A)^?1 exists and the family {(λ ? β)^k(λI ? A)^?k; λ > β, k = 0, 1, 2,…} is equicontinuous. In this case all resolvents (λI ? A)^?1, λ > β, of the given operator A and all exponentials exp(tA), t ? 0, of the operator A belong to a Banach algebra $\text{B}{}_{\mathrm{г}}\text{(X)}$ which is a subspace of the space L(X) of all continuous linear operators on X, and, for each t ? 0 and for each x?X, one has lim_{k → z} (I ? k^?1tA)^?kx = exp(tA) x. A perturbation theorem for the infinitesimal generator of a quasi-equicontinuous C₀-semigroup by an operator which is an element of $\text{B}{}_{\mathrm{г}}\text{(X)}$ is obtained.