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.     

Control of singular problem via differentiation of a Min-Max

For Ω a smooth domain in Rⁿ with boundary Λ = Λ₀Λ₁, we are concerned with the wave equation y″ − Δy = S in Q_T =]0, T[ × Ω with = ∂/∂t, at source term satisfying S, S′ ε L¹(0, T L² (Ω)). A Dirichlet condition is imposed on Λ₀ and we consider an absorbing condition ∂y/∂n + uy′ = 0 in [0, T] × gL₁ where u is the control.parameter. We introduce the cost function. and using the Min-Max formulation of J we by-pasas the sensitivity analysis of u → y and obtain the gradient of J with a usual adjoint problem. We first present an abstract frame for this kind of problems. using the differentiability results of a Min-Max [1,2], which we very shortly deduce here, we show that the well posedness of the adjoint equation implies differentiability of the cost function governed by a linear well posed problem.     

Oscillation criteria of certain nonlinear partial difference equations

This paper is concerned with the nonlinear partial difference equation with continuous variables

,where a, σ_i, τ_i are positive numbers, h_i(x, y, u) ε C(R⁺ × R⁺ × R, R), uh_i(x, y, u) > 0 for u ≠ 0, h_i is nondecreasing in u, i = 1, …, m. Some oscillation criteria of this equation are obtained.     

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.     

Some new paranormed sequence spaces

Maddox defined the sequence spaces ℓ_∞(p), c(p) and c₀(p) in [Proc. Camb. Philos. Soc. 64 (1968) 335, Quart. J. Math. Oxford (2) 18 (1967) 345]. In the present paper, the sequence spaces a₀^r(u,p) and a_c^r(u,p) of non-absolute type have been introduced and proved that the spaces a₀^r(u,p) and a_c^r(u,p) are linearly isomorphic to the spaces c₀(p) and c(p), respectively. Besides this, the α-, β- and γ-duals of the spaces a₀^r(u,p) and a_c^r(u,p) have been computed and their basis have been constructed. Finally, a basic theorem has been given and later some matrix mappings from a₀^r(u,p) to the some sequence spaces of Maddox and to some new sequence spaces have been characterized.     

On the joint nonlinear filtering-smoothing of diffusion processes

The smoothing of diffusions dx_t = f(x_t) dt + σ(x_t) dw_t, measured by a noisy sensor dy_t = h(x_t) dt + dv_t, where w_t and v_t are independent Wiener processes, is considered in this paper. By focussing our attention on the joint p.d.f. of (x_τ x_t), 0 ≤ τ < t, conditioned on the observation path {y_s, 0 ≤ s ≤ t}, the smoothing problem is represented as a solution of an appropriate joint filtering problem of the process, together with its random initial conditions. The filtering problem thus obtained possesses a solution represented by a Zakai-type forward equation. This solution of the smoothing problem differs from the common approach where, by concentrating on the conditional p.d.f. of x_τ alone, a set of ‘forward and reverse’ equations needs to be solved.     

Feedback stabilization of real analytic systems in the plane

We investigate the local feedback stabilization of single input control affine analytic systems in the plane. New necessary and sufficient conditions for local stabilization with feedback laws of the form u = v(x₁,x₂), x2), (v/x₁(0, 0))² + (v/x₁ (0, 0))² 0≠ 0, v(0, 0) = 0, are obtained by using Lyapunov's stability theorems on two-dimensional analytic systems. If the sufficient conditions are satisfied, we also provide explicit feedback laws.     

Universal stabilization of a class of nonlinear systems with homogeneous vector fields

Under relative-degree-one and minimum-phase assumptions, it is well known that the class of finite-dimensional, linear, single-input (u), single-output (y) systems (A,b,c) is universally stabilized by the feedback strategy u = Λ(λ)y, λ = y², where Λ is a function of Nussbaum type (the terminology “universal stabilization” being used in the sense of rendering /s(0/s) a global attractor for each member of the underlying class whilst assuring boundedness of the function λ(·)). A natural generalization of this result to a class ^k of nonlinear control systems (a,b,c), with positively homogeneous (of degree k 1) drift vector field a, is described. Specifically, under the relative-degree-one (cb ≠ 0) and minimum-phase hypotheses (the latter being interpreted as that of asymptotic stability of the equilibrium of the “zero dynamics”), it is shown that the strategy u = Λ(λ)/vby/vb^k−1y, assures ^k-universal stabilization. More generally, the strategy u = Λ(λ)exp(/vby/vb)y, assures -universal stabilization, where = _{k 1} ^k.     

Paths in Möbius cubes and crossed cubes

The Möbius cube MQn and the crossed cube CQn are two important variants of the hypercube Qn. This paper shows that for any two different vertices u and v in G∈{MQn,CQn} with n?3, there exists a uv-path of every length from dG(u,v)+2 to n²−1 except for a shortest uv-path, where dG(u,v) is the distance between u and v in G. This result improves some known results.     

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.     

Lanczos <Emphasis Type="Italic">τ</Emphasis>-method regularization algorithm and its algebraic-programming implementation

An algebraic algorithm is developed for computing an algebraic polynomial y _n of order n ∈ N in computer algebra systems. This polynomial is the optimal approximation of the solution y = y(x), x ∈ [a,b], to a system of linear differential equations with polynomial coefficients and initial conditions at a regular singular zero point of this equation in a space C_{[ a,b ]}^k C_{\left[ {a,b} \right]}^k .     

On Word Equations in One Variable

For a word equation E of length n in one variable x occurring # _x times in E a resolution algorithm of O(n+# _x log n) time complexity is presented here. This is the best result known and for the equations that feature #_x < \fracnlogn\#_{x}<\frac{n}{\log n} it yields time complexity of O(n) which is optimal. Additionally it is proven here that the set of solutions of any one-variable word equation is either of the form F or of the form F∪(uv)⁺ u where F is a set of O(log n) words and u, v are some words such that uv is a primitive word.     

On the Attainable Order of Collocation Methods for the Neutral Functional-Differential Equations with Proportional Delays

In this paper, we extend the recent results of H. Brunner in BIT (1997) for the DDE y′(t)= by(qt), y(0)=1 and the DVIE y(t)=1+∫₀ ^t by(qs)ds with proportional delay qt, 0<q≤1, to the neutral functional-differential equation (NFDE): and the delay Volterra integro-differential equation (DVIDE) : with proportional delays p _i t and q _i t, 0<p _i ,q _i ≤1 and complex numbers a,b _i and c _i . We analyze the attainable order of m-stage implicit (collocation-based) Runge-Kutta methods at the first mesh point t=h for the collocation solution v(t) of the NFDE and the `iterated collocation solution u _it (t)' of the DVIDE to the solution y(t), and investigate the existence of the collocation polynomials M _m (t) of v(th) or M^ _m (t) of u _it (th), t∈[0,1] such that the rational approximant v(h) or u _it (h) is the (m,m)-Padé approximant to y(h) and satisfies |v(h)−y(h)|=O(h ^{2 m +1}). If they exist, then we actually give the conditions of M _m (t) and M^ _m (t), respectively. Received September 17, 1998; revised September 30, 1999     

Filtering of some nonlinear diffusions satisfying the general Bene condition

Under some regularity assumptions and the following generalization of the well-known Bene condition [1]:

Full-size image

, where F(t,z) = g⁻²(t)∫f(t,z)dz, F_t, F_z, F_zz, are partial derivatives of F, we obtain explicit formulas for the unnormalized conditional density q_t(z, x) α Px_t ε dz| y_s, 0 st, where diffusion x_t on R¹ solves x₀ = x, dx_t = [β(t) + α(t)x_t + f(t, x_t] dt + g(t) dw₁, and observation y_t = ∫_o^th(s)x_s ds + ∫_o^t(s) dw_2t, with w = (w₁, w₂) a two-dimensional Wiener process.     

Adaptive suboptimal control of a linear system with bounded disturbance

An adaptive suboptimal control of a linear discrete system with unknown parameters is proposed. An additive disturbance v_t acting on the system is supposed to be uniformly bounded. The criterion is sup_vtI(y^∞₁, u^∞₁), where y_t is the output, u_t is the control. The adaptive control law gives almost the same guaranteed value of the criterion as the optimal linear feedback does for a system with known parameters.     

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.     

Improved upper bounds on the L(2,1) -labeling of the skew and converse skew product graphs

An L(2,1)-labeling of a graph G is a function f from the vertex set V(G) to the set of all nonnegative integers such that |f(x)−f(y)|≥2 if d(x,y)=1 and |f(x)−f(y)|≥1 if d(x,y)=2, where d(x,y) denotes the distance between x and y in G. The L(2,1)-labeling number λ(G) of G is the smallest number k such that G has an L(2,1)-labeling with max{f(v):vV(G)}=k. Griggs and Yeh conjecture that λ(G)≤Δ² for any simple graph with maximum degree Δ≥2. This paper considers the graph formed by the skew product and the converse skew product of two graphs with a new approach on the analysis of adjacency matrices of the graphs as in [W.C. Shiu, Z. Shao, K.K. Poon, D. Zhang, A new approach to the L(2,1)-labeling of some products of graphs, IEEE Trans. Circuits Syst. II: Express Briefs (to appear)] and improves the previous upper bounds significantly.     

A Criterion for Extractability of Mutual Information for a Triple of Strings

We say that the mutual information of a triple of binary strings a, b, c can be extracted if there exists a string d such that a, b, and c are independent given d, and d is simple conditional to each of the strings a, b, and c. It is proved that the mutual information between a, b, and c can be extracted if and only if the values of the conditional mutual informations I(a : b|c), I(a : c|b), and I(b : c|a) are negligible. The proof employs a non-Shannon-type information inequality (a generalization of the recently discovered Zhang–Yeung inequality).     

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)