Control problems for parabolic equations on controlled domains

Given the one-dimensional heat equation v_t = v_xx on the controlled domain Q(y) = {(t, x); 0 < x < y(t), 0 < t < T} subject to some initial-boundary conditions, we study the problem of optimally selecting y(·) from some admissible class so as to maximize a given payoff of fixed duration. Q(y) is thus a controlled domain. We also study the problem in which the heat equation holds in Q(y, z) = {z(t) < x < y(t), 0 < t < T}; z minimizing, y maximizing, i.e., the differential game. The principle techniques involved are (i) transforming the controlled domain to an uncontrolled domain and then (ii) using the method of lines for parabolic equations to enable us to use known results for control systems governed by ordinary differential equations. Sufficient conditions for existence in an admissible class is given and the method of lines allows numerical techniques to be applied to determine the optimal control in our class.     

Globally smooth solutions of quasilinear hyperbolic systems in diagonal form

We prove the existence of a large class of globally smooth solutions of the Cauchy problem for the system of n equations u_t + Λ(x, t, u)u_x = 0, where Λ is a diagonal matrix. We show that, under certain monotonicity conditions on both Λ and the initial data u₀, the solution u will be locally Lipschitz continuous at positive times. Since u₀ is a function of locally bounded variation, our result thus provides both for the smoothing of discontinuities in u₀ as well as for the global preservation of smoothness. The global existence results from an a priori estimate of $\text{?u}\text{?x}$, which we obtain by a device which enables us to effectively uncouple the system of equations for $\text{?u}\text{?x}$. Finally, we prove a partial converse which demonstrates that our hypotheses are not overly restrictive.     

行和列和为常量的(0,1)-矩阵的计数

Let fs,t(m,n) be the number of (0,1) - matrices of size m x n such that each row has exactly s ones and each column has exactly t ones (sm = nt). How to determine fs,t(m,n)? As R. P. Stanley has observed (Enumerative CombinatoricsⅠ(1997), Example 1.1.3), the determination of fs,t(m, n) is an unsolved problem, except for very small s, t. In this paper the closed formulas for f2,2(n,n), f3,2(m,n), f4,2(m,n) are given. And recursion formulas and generating functions are discussed.     

Time-varying minimum cost flow problems

In this paper, we study a minimum cost flow problem on a time-varying network. Let N(V,A,l,b,c_r,c_w) be a network with an arc set A and a vertex set V. Each a∈A is associated with three integer parameters: a positive transit time b(a,t), an arbitrary transit cost c_r(a,t), and a positive capacity limit l(a,t). Each x∈V is associated with two integer parameters: a waiting cost c_w(x,t) and a vertex capacity l(x,t). All these parameters are functions of the discrete time t=0,1,2,… The objective is to find an optimal schedule to send a flow from the origin (the source vertex) to its destination (the sink vertex) with the minimum cost, subject to the constraint that the flow must arrive at the destination before a deadline T. Three versions of the problem are examined, which are classified depending on whether waiting at the intermediate vertices of the network is strictly prohibited, arbitrarily allowed, or bounded. Three algorithms with pseudopolynomial time complexity are proposed, which can find optimal solutions to the three versions of the problem, respectively.     

A general Trotter-Kato formula for a class of evolution operators

In this article we prove new results concerning the existence and various properties of an evolution system UA+B(t,s)_0?s?t?T generated by the sum −(A(t)+B(t)) of two linear, time-dependent and generally unbounded operators defined on time-dependent domains in a complex and separable Banach space B. In particular, writing L(B) for the algebra of all linear bounded operators on B, we can express UA+B(t,s)_0?s?t?T as the strong limit in L(B) of a product of the holomorphic contraction semigroups generated by −A(t) and −B(t), respectively, thereby proving a product formula of the Trotter-Kato type under very general conditions which allow the domain D(A(t)+B(t)) to evolve with time provided there exists a fixed set D⊂_?t∈[0,T]D(A(t)+B(t)) everywhere dense in B. We obtain a special case of our formula when B(t)=0, which, in effect, allows us to reconstruct UA(t,s)_0?s?t?T very simply in terms of the semigroup generated by −A(t). We then illustrate our results by considering various examples of nonautonomous parabolic initial-boundary value problems, including one related to the theory of time-dependent singular perturbations of self-adjoint operators. We finally mention what we think remains an open problem for the corresponding equations of Schrödinger type in quantum mechanics.     

Statistical characteristics of continuous functions and statistically weakly invariant sets of controllable system

We continue the investigation of expansion of a concept of invariance for sets which consists in studying statistically invariant sets with respect to control systems and differential inclusions. We consider the statistical characteristics of continuous functions: Upper and lower relative frequency of containing for graph of a function in a given set. We obtain conditions under which statistical characteristics of two various asymptotical equivalent functions coincide; then by the value of one of them it is possible to calculate the value of another one. We adduce the equality for finding relative frequencies of hitting functions the given set in the case when the distance from the graph of one of functions to the given set is a periodic function. A consequence of these statements are conditions of statistically weak invariance of a set with respect to controlled system. For some almost periodic functions we obtain the formulas by which we can calculate the mean values and the statistical characteristics. We also consider the following problem. Let the number λ₀ ∈ [0, 1] be given. It is necessary to find the value c(λ₀) such that the upper solution z(t) of the Cauchy problem does not exceed c(λ₀) with the relative frequency being equal λ₀. Depending on statement of the problem, a value z(t) can be interpreted as the size of population, energy of a particle, concentration of substance, size of manufacture or the price of production.     

Nonlinear eigenvalue problems with positively convex operators

We consider the equation u = λAu (λ > 0), where A is a forced isotone positively convex operator in a partially ordered normed space with a complete positive cone K. Let Λ be the set of positive λ for which the equation has a solution u?K, and let Λ₀ be the set of positive λ for which a positive solution—necessarily the minimum one—can be obtained by an iteration u_n = λAu_n?1, u₀ = 0. We show that if K is normal, and if Λ is nonempty, then Λ₀ is nonempty, and each set Λ₀, Λ is an interval with $\text{inf}\text{(Λ}{}_0\text{) =}\text{inf}\text{(Λ) = 0}\text{and sup}\text{(Λ}{}_0\text{) =}\text{sup}\text{(Λ) (= λ1,}\text{say}\text{)}$; but we may have λ1 ? Λ₀ and λ1 ? Λ. Furthermore, if A is bounded on the intersection of K with a neighborhood of 0, then Λ₀ is nonempty. Let u⁰(λ) = lim_n→∞(λA)ⁿ(0) be the minimum positive fixed point corresponding to λ ? Λ₀. Then u⁰(λ) is a continuous isotone convex function of λ on Λ₀.     

On the reliability of stochastic systems

In this note we consider the problem of computing the probability R(t₀ = P(X(t) > Y(t) for 0 < t ? t₀), where X(t) and Y(t) are stochastic processes. This extends some of the existing results to the case of stochastic processes. Related estimation problems are also considered.     

Properties of the set of positive solutions to Dirichlet boundary value problems with time singularities

The paper investigates the structure and properties of the set S of all positive solutions to the singular Dirichlet boundary value problem u″(t) + au′(t)/t ? au(t)/t ² = f(t, u(t),u′(t)), u(0) = 0, u(T) = 0. Here a ∈ (?∞,?1) and f satisfies the local Carathéodory conditions on [0,T]×D, where D = [0,∞)×?. It is shown that S _c = {u ∈ S: u′(T) = ?c} is nonempty and compact for each c ≥ 0 and S = ∪ _c≥0 S _c . The uniqueness of the problem is discussed. Having a special case of the problem, we introduce an ordering in S showing that the difference of any two solutions in S _c ,c≥ 0, keeps its sign on [0,T]. An application to the equation v″(t) + kv′(t)/t = ψ(t)+g(t, v(t)), k ∈ (1,∞), is given.     

The Hele-Shaw problem with surface tension in a half-plane

In this paper, we consider the Hele-Shaw problem in a 2-dimensional fluid domain Ω(t) which is constrained to a half-plane. The boundary of Ω(t) consist of two components: Γ₀(t) which lies on the boundary of the half-plane, and Γ(t) which lies inside the half-plane. On Γ(t) we impose the classical boundary conditions with surface tension, and on Γ₀(t) we prescribe the normal derivative of the fluid pressure. At the point where Γ₀(t) and Γ(t) meet, there is an abrupt change in the boundary condition giving rise to a singularity in the fluid pressure. We prove that the problem has a unique solution with smooth free boundary Γ(t) for some small time interval.     

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.     

Maximal sets of points in finite projective space,no t-linearly dependent

Consider a finite (t + r ? 1)-dimensional projective space PG(t + r ? 1, s) based on the Galois field GF(s), where s is prime or power of a prime. A set of k distinct points in PG(t + r ? 1, s), no t-linearly dependent, is called a (k, t)-set and such a set is said to be maximal if it is not contained in any other (k¹, t)-set with $\text{k}{}^1\text{> k}$. The number of points in a maximal (k, t)-set with the largest k is denoted by m_t(t + r, s). Our purpose in the paper is to investigate the conditions under which two or more points can be adjoined to the basic set of E_i, i = 1, 2, …, t + r, where E_i is a point with one in i-th position and zeros elsewhere. The problem has several applications in the theory of fractionally replicated designs and information theory.     

Structure of cubic mapping graphs for the ring of Gaussian integers modulo n

Let ? _n [i] be the ring of Gaussian integers modulo n. We construct for ?n[i] a cubic mapping graph Γ(n) whose vertex set is all the elements of ?n[i] and for which there is a directed edge from a ∈ ?n[i] to b ∈ ?n[i] if b = a ³. This article investigates in detail the structure of Γ(n). We give suffcient and necessary conditions for the existence of cycles with length t. The number of t-cycles in Γ₁(n) is obtained and we also examine when a vertex lies on a t-cycle of Γ₂(n), where Γ₁(n) is induced by all the units of ?n[i] while Γ₂(n) is induced by all the zero-divisors of ?n[i]. In addition, formulas on the heights of components and vertices in Γ(n) are presented.     

Inverse problem with nonlocal observation of finding the coefficient multiplying <Emphasis Type="Italic">u</Emphasis> <Subscript> <Emphasis Type="Italic">t</Emphasis> </Subscript> in the parabolic equation

We study the inverse problem of the reconstruction of the coefficient ?(x, t) = ?⁰(x, t) + r(x) multiplying u_t in a nonstationary parabolic equation. Here ?⁰(x, t) ≥ ?⁰ > 0 is a given function, and r(x) ≥ 0 is an unknown function of the class L_∞(Ω). In addition to the initial and boundary conditions (the data of the direct problem), we pose the problem of nonlocal observation in the form ∫₀^Tu(x, t) dμ(t) = χ(x) with a known measure dμ(t) and a function χ(x). We separately consider the case dμ(t) = ω(t)dt of integral observation with a smooth function ω(t). We obtain sufficient conditions for the existence and uniqueness of the solution of the inverse problem, which have the form of ready-to-verify inequalities. We suggest an iterative procedure for finding the solution and prove its convergence. Examples of particular inverse problems for which the assumptions of our theorems hold are presented.     

On differential inclusions with prescribed attainable sets

In this article, the inverse problem of the differential inclusion theory is studied. For a given ε>0 and a continuous set valued map t→W(t), t∈[t₀,θ], where W(t)⊂Rⁿ is compact and convex for every t∈[t₀,θ], it is required to define differential inclusion so that the Hausdorff distance between the attainable set of the differential inclusion at the time moment t with initial set (t₀,W(t₀)) and W(t) would be less than ε for every t∈[t₀,θ].     

Three positive solutions to a discrete focal boundary value problem

We are concerned with the discrete focal boundary value problem Δ³x(t − k) = f(x(t)), x(a) = Δx(t₂) = Δ²x(b + 1) = 0. Under various assumptions on f and the integers a, t₂, and b we prove the existence of three positive solutions of this boundary value problem. To prove our results we use fixed point theorems concerning cones in a Banach space.     

Determining tension parameters in rational gamma-spline interpolation

Inspired by the classic γ-spline, we propose a method for constructing a G² rational γ-spline curve that interpolates a given set of distinct ordered data-points (planar or spatial). The only input of our method is just these data-points. We also present a procedure to solve the key problem of determining the tension parameters γ_i which are computed in terms of exponential functions that determine the eccentricities of the common conic osculants at the junction points while keeping in geometrical agreement with data-points. This allows the resulting curve to be modified in the close vicinity of each data-point.     

Estimation of states and parameters in continuous non-linear systems with discrete observations

We consider a class of continuous non-linear systems defined by the ordinary differential equation x = f(x, t) + g(x, t)u, where u is an unknown input representing noise or disturbances. The object is to estimate states and parameters in these systems by means of a fixed number of discrete observations y_i = h(x(t_i), t_i) + v_i, 1 ? i ? m, where the v_i represents unknown errors in the measurements y_i. No statistical assumptions are made concerning the nature of the unknown input u or the unknown measurement errors v_i. A weighted least squares criterion is defined as a measure of the optimal estimate. A result concerning the existence of solutions of the differential equation which minimize the criterion is presented. The necessary conditions for an optimal estimate, a set of Euler-Lagrange equations and multi-point discontinuous non-linear boundary conditions, are given. The multi-point problem is converted to an equivalent continuous two-point boundary value problem of larger dimension in the case in which the observations are assumed to be linear functions of the state. A pair of equivalent quasilinearization algorithms is defined for the two-point system and the multi-point system. Quadratic convergence for these algorithms is proved.     

Numerical solutions with a priori error bounds for coupled self-adjoint time dependent partial differential systems

This paper is concerned with the construction of accurate continuous numerical solutions for partial self-adjoint differential systems of the type (P(t) u_t)_t = Q(t)u_xx, u(0, t) = u(d, t) = 0, u(x, 0) = f(x), u_t(x, 0) = g(x), 0 ≤ x ≤ d, t >- 0, where P(t), Q(t) are positive definite oR^r×r-valued functions such that P′(t) and Q′(t) are simultaneously semidefinite (positive or negative) for all t ≥ 0. First, an exact theoretical series solution of the problem is obtained using a separation of variables technique. After appropriate truncation strategy and the numerical solution of certain matrix differential initial value problems the following question is addressed. Given T > 0 and an admissible error ϵ > 0 how to construct a continuous numerical solution whose error with respect to the exact series solution is smaller than ϵ, uniformly in D(T) = {(x, t); 0 ≤ x ≤ d, 0 ≤ t ≤ T}. Uniqueness of solutions is also studied.     

Decay of solutions of some nonlinear parabolic inequalities associated with time-dependent convexes

A simple result concerning integral inequalities enables us to give an alternative proof of Waltman's theorem: lim_{t → ∞} ∝^t₀a(s) ds = ∞ implies oscillation of the second order nonlinear equation y″(t) + a(t) f(y(t)) = 0; to prove an analog of Wintner's theorem that relates the nonoscillation of the second order nonlinear equations to the existence of solutions of some integral equations, assuming that lim_{t → ∞} ∝^t₀a(s) ds exists; and to give an alternative proof and to extend a result of Butler. An often used condition on the coefficient a(t) is given a more familiar equivalent form and an oscillation criterion involving this condition is established.