Positive solutions to a nonlinear abel-type integral equation on the whole line

We consider the existence of positive solutions to the nonlinear integral equation

where g is a continuous, nondecreasing function such that g(0) = 0. We show that the equation always has nontrivial solutions and we give a necessary and sufficient condition for the existence of solutions u such that u(x) > − ∞. We also provide a condition which ensures that all the nontrivial solutions experience the blow-up behaviour.     

Monotone properties of certain classes of solutions of second-order difference equations

The authors consider the difference equations

δ(a_nδx_n)=q_nx_n+1

and

δ(a_nδx_n)=q_nf(x_n+1),

where a_n > 0, q_n > 0, and f: R å R is continuous with uf(u) > 0 for u ≠ 0. They obtain necessary and sufficient conditions for the asymptotic behavior of certain types of nonoscillatory solutions of (*) and sufficient conditions for the asymptotic behavior of certain types of nonoscillatory solutions of (**). Sufficient conditions for the existence of these types of nonoscillatory solutions are also presented. Some examples illustrating the results and suggestions for further research are included.     

Exact solution of mixed problems for variable coefficient one-dimensional diffusion equation

This paper deals with diffusion problems modeled by the equation a(t)u_xx = u_t, x > 0, t > 0, u(x, 0) = c(x) together with the boundary condition u(0, t) = b(t) or u_x(0, t) = b(t). By using Fourier transforms, existence conditions and exact solutions of the above mixed problems are given.     

Asymptotic formulae for a particular solution of linear nonhomogeneous discrete equations

The aim of this contribution is to develop the asymptotic formulae which describe the asymptotic behaviour of a particular solution of linear nonhomogeneous differential equation Δu(k) = A(k)u(k) + g(k) if k → ∞. It is shown that asymptotic formulae are generated by partial sums of certain formal series that satisfy the given equation. A comparison with the known explicit formula for solution of the initial problem is given and, moreover, some illustrative examples are discussed. The used method of investigation is based on results published in [1].     

A novel numerical method for a class of problems with the transition layer and Burgers’ equation

A numerical method for solving a class of quasi-linear singular two-point boundary value problems with a transition layer is presented in this paper. For the problem ? u _xx +a(u+f(x))u _x +b(x, u)=0, we develop a multiple scales method. First, this method solves the location of the transition layer, then it approximates the singular problem with reduced problems in the non-layer domain and pluses a layer corrected problem which nearly has an effect in the layer domain. Both problems are transformed into first-order problems which can be solved easily. For the problem ? u _xx +b(x, u)=0, we establish a similar method which approximate the problem with reduced problems and a two-point boundary value problem. Unsteady problems are also considered in our paper. We extend our method to solve Burgers’ equation problems by catching the transition layer with the formula of shock wave velocity and approximating it by a similar process.     

Two-sided bounds for the asymptotic behaviour of free nonlinear vibration systems with application of the differential calculus of norms

This paper deals with the free nonlinear dynamical system [xdot](t)=A x(t)+h(t, x(t)), t≥t ₀, x(t ₀)=x ₀, where A is an n×n-matrix and h a nonlinear vector function with h(t, u)=o(∥u∥). As the first novel point, a lower bound for the asymptotic behaviour on the solution x(t) is derived. Two methods are applied to determine the optimal two-sided bounds, where one of the methods is the differential calculus of norms. In this context, the second novel point enters; it consists of a new strategy to significantly reduce the computation time for the determination of the optimal constants in the two-sided bounds. The obtained results are especially of interest in engineering and cannot be obtained by the methods used so far.     

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.     

Random functions with given time correlation

First, on any sequence of real numbers (x_λ), $\text{λ ? [? Λ}{}_1\text{+ Λ] ?}\text{Z}$, the pseudo probability Pr(x, x′) of the event x_λ?[x, x′[ is defined to be the limit when Λ → ∞ of the ratio of the number of x_λ?[x, x′[ to the total number of x_λ. The a.d.f. (asymptotic distribution function) of the sequence is then defined by F(x) = Pr(? ∞, x); it possesses the properties of a d.f. (distribution function). Consequently, what is said below applies equally to a sequence of r.v. (random variables) or to a sequence of p.r.v. (pseudorandom variables) consisting of a sequence (_nx_λ), $\text{n ? [? N, + N] ?}\text{Z}$ of sequences _nx_λ, $\text{λ ? [? Λ, + Λ] ?}\text{Z}$.A weyl's polynomial ?_n(λ) is a polynomial such that one of its coefficieints other than ?(0) is irrational. Then any sequence, the fractional part of ?_n(λ), $\text{λ ? [? Λ, + Λ] ?}\text{Z}$, is asymptotically equidistributed on [0, 1].A property is given which permits the construction of a sequence (_nx_λ), $\text{n ? [? N, + N] ?}\text{Z}$ of pseudostochastically independent sequences _nx_λ, $\text{λ ? [? Λ, + Λ] ?}\text{Z}$.It is known that setting Y_n = F^{(? 1)}(X_n), it is possible to transform any sequence of r.v. X_n     

An accurate algorithm for computing the eigenvalues of a polygonal membrane

For a polygonal domain Ω, we consider the eigenvalue problem Δu + λu = 0 in Ω, u = 0 on the boundary of Ω. Ω is decomposed into subdomains Ω₁, Ω₂,...; on each $\text{Ω}{}_{\mathrm{i}}$, u is approximated by a linear combination of functions which satisfy the equation Δu + Δu = 0 and continuity conditions are imposed at the boundaries of the subdomains. We propose a non-conventional method based on the use of a Rayleigh quotient. We present numerical examples and a proof of the exponential convergence of the algorithm.     

Backstepping boundary control of Burgers’ equation with actuator dynamics

In this paper, we propose a backstepping boundary control law for Burgers’ equation with actuator dynamics. While the control law without actuator dynamics depends only on the signals u(0,t) and u(1,t), the backstepping control also depends on u_x(0,t), u_x(1,t), u_xx(0,t) and u_xx(1,t), making the regularity of the control inputs the key technical issue of the paper. With elaborate Lyapunov analysis, we prove that all these signals are sufficiently regular and the closed-loop system, including the boundary dynamics, is globally H³ stable and well posed.     

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.     

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 …     

Solving the forced Duffing equation with integral boundary conditions in the reproducing kernel space

In this paper, we propose a new method to solve the forced Duffing equation with integral boundary conditions. Its exact solution is represented in the form of a series in the reproducing kernel space. The n-term approximation u _n (x) of the exact solution u(x) is proved to converge to the exact solution. Some numerical examples are displayed to demonstrate the accuracy of the present method.     

How to improve mamdani's approach to fuzzy control

Fuzzy control is a methodology that translates “if”-“then” rules, A_ji (x₁) &…& A_jn(x_n) → B_j(u), formulated in terms of a natural language, into an actual control strategy u(x). Implication of uncertain statements is much more difficult to understand than “and,” “or,” and “not.” So, the fuzzy control methodologies usually start with translating “if”-“then” rules into statements that contain only “and,” “not,” and “or.” the first such translation was proposed by Mamdani in his pioneer article on fuzzy control. According to this article, a fuzzy control is reasonable iff one of the rules is applicable, i.e., either the first rule is applicable (A₁₁(x₁) &…& A_1n(x_n) & B₁(u)), or the second one is applicable, etc. This approach turned out to be very successful, and it is still used in the majority of fuzzy control applications. However, as R. Yager noticed, in some cases, this approach is not ideal: Namely, if for some x, we know what u(x) should be, and add this crisp rule to our rules, then the resulting fuzzy control for this x may be different from the desired value u(x). to overcome this drawback, Yager proposed to assign priorities to the rules, so that crisp rules get the highest priority, and use these priorities while translating the rules into a control strategy u(x). In this article, we show that a natural modification of Mamdani's approach can solve this problem without adding any ad hoc priorities. © 1995 John Wiley & Sons, Inc.     

Parametric quintic spline solution of third-order boundary value problems

In this paper, we develop parametric quintic spline function to approximate the solution of third-order boundary value problems of the form u″′=f(x, u), a≤x≤b, subject to the boundary conditions u(a)=k ₁, u′(a)=k ₂ and u(b)=k ₃. The class of methods are second-, fourth- and sixth-order accurate. End equations of the splines are derived and truncation error is given. Three numerical examples are presented to illustrate the practical use of our methods as well as their accuracies when compared with some existing spline function methods. It is shown that the new methods give approximations, which are better than those produced by other methods.     

A class of interval-Newton-operators

A class of interval-Newton-operatorsN _a will be discussed. One of them,?, is optimal in the manner that?(X)?N _a (X). With the help of such an interval operator we can given an existence theorem for the solution x^* of the equationg(x)=0.     

Positivity properties in pencils of matrices†

If A and B are positive definite self-adjoint matrices, then all the matrices P_i, occurring in the expansion of an exterior power Λ^k(A+ λB) = P ₀ + P ₁λ + … + P _kλ^k are positive definite.     

Feedback representations of critical controls for well-posed linear systems

This is the first part in a three part study of the suboptimal full information H^∞ problem for a well-posed linear system with input space U, state space H, and output space Y. We define a cost function Q(x₀,u)=∫〈y(s),Jy(s)〉_Yds, where y∈L²_loc( R ⁺; Y) is the output of the system with initial state x₀∈H and control u∈L²_loc( R ⁺; U), and J is a self-adjoint operator on Y. The cost function Qis quadratic in x₀ and u, and we suppose (in the stable case) that the second derivative of Q(x₀, u) with respect to u is non-singular. This implies that, for each x₀∈H, there is a unique critical control u^crit such that the derivative of Q(x₀, u) with respect to u vanishes at u=u^crit. We show that u^crit can be written in feedback form whenever the input/output map of the system has a coprime factorization with a (J, S)-inner numerator; here S is a particular self-adjoint operator on U. A number of properties of this feedback representation are established, such as the equivalence of the (J, S)-losslessness of the factorization and the positivity of the Riccati operator on the reachable subspace. © 1998 John Wiley & Sons, Ltd.     

Stability of Some Difference Equations with Two Delays

In the framework of the general problem for clarifying the stability of the zero solution of the equation x(n) = a ₁ x(n – m) – a ₂ x(n – k) with delays k and m, some partial problems are solved. An appreciable dependence of the stability on the divisibility of one delay by the other is revealed.