Optimal controls for stochastic systems with singular noise

Stochastic control systems of the form dx₁ = f(t, x, u_t)dt + g(t, x)db_t (0 t 1), with g singular and general cost, are discussed. It is shown that there is an optimal relaxed control u that depends on the past of x and the driving process b. Nonstandard methods are used.     

The formula of the solution for some classes of initial boundary value problems for the hyperbolic equation with two independent variables

A new representation is proved of the solutions of initial boundary value problems for the equation of the form u _xx (x, t) + r(x)u _x (x, t) ? q(x)u(x, t) = u _tt (x, t) + μ(x)u _t (x, t) in the section (under boundary conditions of the 1st, 2nd, or 3rd type in any combination). This representation has the form of the Riemann integral dependent on the x and t over the given section.     

A sixth-order extrapolation method for special nonlinear fourth-order boundary value problems

A sixth-order convergent finite difference method is developed for the numerical solution of the special nonlinear fourth-order boundary value problem y^(iv)(x) = f(x, y), a < x < b, y(a) = A₀, y″(a) = B₀, y(b) = A₁ y′(b) = B₁, the simple-simple beam problem.The method is based on a second-order convergent method which is used on three grids, sixth-order convergence being obtained by taking a linear combination of the (second-order) numerical results calculated using the three individual grids.Special formulas are proposed for application to points of the discretization adjacent to the boundaries x = a and x= b, the first two terms of the local truncation errors of these formulas being the same as those of the second-order method used at the other points of each grid.Modifications to these two formulas are obtained for problems with boundary conditions of the form y(a) = A₀, y′(a) = C₀, y(b) = A₁, y′(b) = C₁, the clamped-clamped beam problem.The general boundary value problem, for which the differential equation is y^(iv)(x) = f(x, y, y′, y″, y‴), is also considered.     

Conservative difference schemes for diffusion problems with boundary and interface conditions

For 0≤x≤1, 0≤t≤T we consider the diffusion equation $$\gamma (x)u_t (x, t) - (B u)_x (x, t) = f(x, t)$$ with (alternatively)B u:=(a(x)u) _x +b(x)u ora(x)u _x +β(x)u. There are given initial valuesu(x,0), influx rates?(B u) (0,t) and (B u) (1,t) across the lateral boundaries and an influx rate (B u) (ζ?0,t)?(B u) (ζ+0,t) at an interface ζ∈(0, 1) where the elsewhere smooth functions γ,a, b, β are allowed to have jump discontinuities.a and γ are assumed to be positive. Interpretingu(x, t) as temperature and γ(x) u (x, t) as energy density we can easily express the total energy \(E(t) = \int\limits_0^1 {\gamma (x) u (x, t)} \) in terms of integrals of the given data. We describe and analyse explicit and implicit one-step difference schemes which possess a discrete quadrature analogue exactly matchingE(t) at the time grid points. These schemes also imitate the isotonic dependence of the solution on the data. Hence stability can be proved by Gerschgorin's method and, under appropriate smoothness assumptions, convergence is 0 ((Δx)²+Δt).     

Asymptotic behavior and blow up of solutions for semilinear parabolic equations at critical energy level

In this paper we study the initial boundary value problem of semilinear parabolic equations with semilinear term f(u). By using the family of potential wells method we prove that if f(u) satisfies some conditions, J(u₀) ≤ d and I(u₀) > 0, then the solution decays to zero exponentially as t → ∞. On the other hand, if J(u₀) ≤ d, I(u₀) < 0, then the solution blows up in finite time.     

Adaptive robust control of nonholonomic systems with stochastic disturbances

Based on the Brockett’s necessary condition for feedback asymptotic stabilization[1], nonholonomic systems fail to be stabilized at the origin by any static continuous state feedback though they are open loop controllable. There are two novel approaches …     

Modified nonlinearly dispersive mK(m,n,k) equations: I. New compacton solutions and solitary pattern solutions

In this paper exact solutions of the modified nonlinearly dispersive KdV equations (simply called mK(m,n,k) equations), u^m−1u_t+a(uⁿ)_x+b(u^k)_xxx=0, are investigated by using some direct ansatze. As a result, abundant new compacton solutions: solitons with the absence of infinite wings, solitary pattern solutions having infinite slopes or cups, solitary wave solutions and periodic wave solutions are obtained.     

Parabolic equations with double variable nonlinearities

The paper is devoted to the study of the homogeneous Dirichlet problem for the doubly nonlinear parabolic equation with nonstandard growth conditions:

ut=div(a(x,t,u)|u^|α(x,t)|∇u^|p(x,t)−2∇u)+f(x,t)     

High-accuracy cubic spline alternating group explicit methods for 1D quasi-linear parabolic equations†

In this article, we study the application of the alternating group explicit (AGE) and Newton-AGE iterative methods to a two-level implicit cubic spline formula of O(k ²+kh ²+h ⁴) for the solution of 1D quasi-linear parabolic equation u _xx =φ (x, t, u, u _x , u _t ), 0<x<1, t>0 subject to appropriate initial and natural boundary conditions prescribed, where k>0 and h>0 are mesh sizes in t- and x-directions, respectively. The proposed cubic spline methods require 3-spatial grid points and are applicable to problems in both rectangular and polar coordinates. The convergence analysis at advanced time level is briefly discussed. The proposed methods are then compared with the corresponding successive over relaxation (SOR) and Newton-SOR iterative methods both in terms of accuracy and performance.     

New explicit filters and smoothers for diffusions with nonlinear drift and measurements

The optimal least-squares filtering of a diffusion x(t) from its noisy measurements {y(τ); 0 τ t} is given by the conditional mean E[x(t)|y(τ); 0 τ t]. When x(t) satisfies the stochastic diffusion equation dx(t) = f(x(t)) dt + dw(t) and y(t) = ∫₀^tx(s) ds + b(t), where f(·) is a global solution of the Riccati equation /xf(x) + f(x)² = f(x)² = αx² + βx + γ, for some , and w(·), b(·) are independent Brownian motions, Benes gave an explicit formula for computing the conditional mean. This paper extends Benes results to measurements y(t) = ∫₀^tx(s) ds + ∫₀^t dx(s) + b(t) (and its multidimensional version) without imposing additional conditions on f(·). Analogous results are also derived for the optimal least-squares smoothed estimate E[x(s)|y(τ); 0 τ t], s < t. The methodology relies on Girsanov's measure transformations, gauge transformations, function space integrations, Lie algebras, and the Duncan-Mortensen-Zakai equation.     

A new critical theorem for adaptive nonlinear stabilization

It is fairly well known that there are fundamental differences between adaptive control of continuous-time and discrete-time nonlinear systems. In fact, even for the seemingly simple single-input single-output control system y_t+1=θ₁f(y_t)+u_t+w_t+1 with a scalar unknown parameter θ₁ and noise disturbance {w_t}, and with a known function f(⋅) having possible nonlinear growth rate characterized by |f(x)|=Θ(|x|^b) with b≥1, the necessary and sufficient condition for the system to be globally stabilizable by adaptive feedback is b<4. This was first found and proved by Guo (1997) for the Gaussian white noise case, and then proved by Li and Xie (2006) for the bounded noise case. Subsequently, a number of other types of “critical values” and “impossibility theorems” on the maximum capability of adaptive feedback were also found, mainly for systems with known control parameter as in the above model. In this paper, we will study the above basic model again but with additional unknown control parameter θ₂, i.e., u_t is replaced by θ₂u_t in the above model. Interestingly, it turns out that the system is globally stabilizable if and only ifb<3. This is a new critical theorem for adaptive nonlinear stabilization, which has meaningful implications for the control of more general uncertain nonlinear systems.     

Time rescaling and asymptotic behavior of some fourth-order degenerate diffusion equations

In this paper, we investigate the large-times behavior of weak solutions to the fourth-order degenerate parabolic equation u_t = −(|u|ⁿu_xxx)_x modeling the evolution of thin films. In particular, for all n > 0, we prove exponential decay of u(x, t) towards its mean value (1/|Ω|) ∫_Ω u dx in L¹-norm for long times and we give the explicit (n-dependent) rate of decay. The result is based on classical entropy estimates, and on detailed lower bounds for the entropy production.     

Three point discretization of order four and six for (du :dx) of the solution of non-linear singular two point boundary value problem

We present finite difference methods of order four and six for the numerical solution of (du/dx) for the non-linear differential equation u″ = f(x,u,u′), 0 < x > 1 subject to the boundary conditions u(0) = A, u(l) =B. The proposed methods require only three grid points and applicable to both singular and non-singular problems. Numerical examples are given to illustrate the methods and their convergence.     

Positive solutions of singular Dirichlet and periodic boundary value problems

Let A and T be positive numbers. The singular differential equation (r(x)x′)′ = μq(t)f(t, x) is considered. Here r > 0 on (0, A] may be singular at x = 0, and f(t, x) ≤ 0 may be singular at x = 0 and x = A. Effective sufficient conditions imposed on r, μ, q, and f are given for the existence of a solution x to the above equation satisfying either the Dirichlet conditions x(0) = x(T) = 0 or the periodic conditions x(0) = x(T), x′(0) = x′(T), and, in addition, 0 < x < A on (0, T).     

Exponential stability in mean square of neutral stochastic differential functional equations

The aim of this paper is to investigate the exponential stability in mean square for a neutral stochastic differential functional equation of the form d[x(t) − G(x_t)] = [f(t,x(t)) + g(t, x_t)]dt + σ(t, x_t)dw(t), where x_t = {x(t + s): − τ s 0}, with τ > 0, is the past history of the solution. Several interesting examples are a given for illustration.     

Sorting a sequence of strong kings in a tournament

A king in a tournament is a player who beats any other player directly or indirectly. According to the existence of a king in every tournament, Wu and Sheng [Inform. Process. Lett. 79 (2001) 297-299] recently presented an algorithm for finding a sorted sequence of kings in a tournament of size n, i.e., a sequence of players u₁,u₂,…,u_n such that u_i→u_i+1 (u_i beats u_i+1) and u_i is a king in the sub-tournament induced by {u_i,u_i+1,…,u_n} for each i=1,2,…,n−1. With each pair u,v of players in a tournament, let b(u,v) denote the number of third players used for u to beat v indirectly. Then, a king u is called a strong king if the following condition is fulfilled: if v→u then b(u,v)>b(v,u). In the sequel, we will show that the algorithm proposed by Wu and Sheng indeed generates a sorted sequence of strong kings, which is more restricted than the previous one.     

Fault diagnosis in loop-connected systems

The paper considers fault diagnosis in a large system comprising a collection of small subsystems or units which can test one another for the existence of a faulty condition. If subsystem α is not faulty and tests subsystem β, a correct indication of the status of β is obtained; if α is faulty, the test outcome contains meaningless information. A particular form of interconnection is examined. For a system with n units u_o,u₁,…,u_{n ? 1}, for each i unit u_i tests u_{i + 1},u_{i + 2},…,u_{i + A} (modulo n arithmetic being understood), where A is a preselected integer. If t is the maximum number of faulty units, we show that when t ? A, all faults are immediately diagnosable if n ? 2t + 1; we also show that when t ? A, at least A faults can be diagnosed if and only if n ? s(t ? As) + t + A + 1, where s is the integer which maximizes the quadratic function f(x) = x(t ? Ax) of the integer variable x.     

Nonlinear eigenvalue problems for quasilinear systems

The paper deals with the existence of positive solutions for the quasilinear system (Φ(u'))' + λh(t)f(u) = 0,0 < t < 1 with the boundary condition u(0) = u(1) = 0. The vector-valued function Φ is defined by Φ(u) = (q(t)(p(t)u₁), …, q(t)(p(t)u_n)), where u = (u₁, …, u_n), andcovers the two important cases (u) = u and (u) = up > 1, h(t) = diag[h₁(t), …, h_n(t)] and f(u) = (f¹(u), …, fⁿ (u)). Assume that fⁱ and h_i are nonnegative continuous. For u = (u₁, …, u_n), let

, f₀ = maxf¹₀, …, fⁿ₀ and f_∞ = maxf¹_∞, …, fⁿ_∞. We prove that the boundary value problem has a positive solution, for certain finite intervals of λ, if one of f₀ and f_∞ is large enough and the other one is small enough. Our methods employ fixed-point theorem in a cone.     

The existence of positive solutions of singular fractional boundary value problems

We discuss the existence of positive solutions for the singular fractional boundary value problem D^αu+f(t,u,u^′,D^μu)=0, u(0)=0, u^′(0)=u^′(1)=0, where 2<α<3, 0<μ<1. Here D^α is the standard Riemann-Liouville fractional derivative of order α, f is a Carathéodory function and f(t,x,y,z) is singular at the value 0 of its arguments x,y,z.     

On the non-existence of stationary diffusions which satisfy the Bene condition

The one-dimensional diffusion x_t satisfying dx_t = f(x_t)dt + dw_t, where w_t is a standard Brownian motion and f(x) satisfies the Bene condition f′(x) + f²(x) = ax² + bx + c for all real x, is considered. It is shown that this diffusion does not admit a stationary probability measure except for the linear case f(x) = αx + β, α < 0.