Determining observations for stability and bifurcation on a finite time interval in variational control systems with a parameter

We study the Cauchy problem for the retarded functional differential equations that model the dynamics of some living systems. We find certain conditions ensuring the existence, uniqueness, and nonnegativity of solutions on finite and infinite time intervals. We obtain upper bounds for solutions and prove the continuous dependence of solutions on the initial data on finite time intervals.

The problem of optimal control over a finite time interval for a mathematical model of a three-sector economic cluster is posed. The economic system is reduced by means of transformations to the optimal control problem for one class of nonlinear systems with coefficients depending on the state of the control object. Two optimal control problems for one class of nonlinear systems with and without control constraints are considered. The nonlinear objective functional in these problems depends on the control and state of the object. Then, using the results of solving optimal control problems on a finite interval, an algorithm for solving the problem for a nonlinear system of a three-sector economic cluster is developed. A nonlinear control based on the feedback principle using Lagrange multipliers of a special kind is found. The results obtained for nonlinear systems are used to construct the control parameters of a mathematical model of a three-sector economic cluster at a finite time interval with a given functional and various initial conditions. The results of the system state calculation are shown in the figures, the optimal controls satisfy the given constraints. The optimal distribution of labor and investment resources for a three-sector economic cluster is determined. They ensure that the system is brought into an equilibrium state and satisfy balance ratios. These results are useful for practice and are important because there are a number of optimal control problems when it is necessary to transfer a system from an initial state to a desired final state for a given time interval. Such problems often arise for an economic system when a certain level of development is required.

Sufficient conditions for stability on a finite time interval are derived for a class of evolution variational inequalities with the use of Lyapunov functions and frequency-domain conditions. These inequalities are considered for Hilbert spaces and Sobolev spaces of infinite order in the sense of Yu.A. Dubinskii. We show how to use the stability result on a finite interval to characterize bifurcations. An algorithm for finding observation functionals is presented that uses an isomorphism between the algebra of pseudodifferential operators with constant coefficients whose symbols are real-analytic functions in some domain and an algebra of analytic matrix-valued functions.

We consider the first-come first-serve (FCFS) GI/GI/n queueing model in which inter-arrival times, service times and finite time interval are independent and generally distributed. For such systems, obtained either stationary characteristics, or features, in a heavy traffic. Another approach is approximation of the systems. The method under consideration allows investigating the cyclic system, in which the units are cyclically distributed through channels regardless of the system state. The following correlations for distributions are obtained: numbers of arrivals at finite time interval and service time, numbers of services at finite time interval and processing time for an unlimited number of channels. The paper provides a numerical example, which confirms that replacement of the real exponential service time distribution leads to considerable distortions of the numerical system performance indicators.

The approach problem of a nonlinear controlled system in a finite time interval

ABSTRACTThis paper explores the stability of a class of multi-agent systems with proposed relay tracking strategy over a finite time interval. In this paper, agents are deployed in an area to monitor and track the intruded targets. According to proposed relay tracking strategy, the tracking agents and their communication topologies switch during the whole tracking process. This results in impulsive effects on the overall tracking errors. The relationship of finite time interval against desired overall tracking errors and control parameter is derived quantitatively for the multi-agent relay tracking system. Moreover, stability conditions for the system with disturbances and impulsive effects are obtained. Performance of proposed relay tracking strategy and correctness of derived results on stability over finite time intervals are demonstrated by a set of simulations.

We are concerned with time-dependent inverse source problems in elastodynamics. The source term is supposed to be the product of a spatial function and a temporal function with compact support. We present frequency-domain and time-domain approaches to show uniqueness in determining the spatial function from wave fields on a large sphere over a finite time interval. The stability estimate of the temporal function from the data of one receiver and the uniqueness result using partial boundary data are proved. Our arguments rely heavily on the use of the Fourier transform, which motivates inversion schemes that can be easily implemented. A Landweber iterative algorithm for recovering the spatial function and a non-iterative inversion scheme based on the uniqueness proof for recovering the temporal function are proposed. Numerical examples are demonstrated in both two and three dimensions.

We provide a diagrammatic formulation of perturbative quantum field theory in a finite interval of time τ, on a compact space manifold Ω. We explain how to compute the evolution operator U(tf, ti) between the initial time ti and the final time tf = ti + τ, study unitarity and renormalizability, and show how to include purely virtual particles, by rendering some physical particles (and all the ghosts, if present) purely virtual. The details about the restriction to finite τ and compact Ω are moved away from the internal sectors of the diagrams (apart from the discretization of the three-momenta), and coded into external sources. Unitarity is studied by means of the spectral optical identities, and the diagrammatic version of the identity U†(tf, ti)U(tf, ti) = 1. The dimensional regularization is extended to finite τ and compact Ω, and used to prove, under general assumptions, that renormalizability holds whenever it holds at τ = ∞, Ω = ℝ3. Purely virtual particles are introduced by removing the on-shell contributions of some physical particles, and the ghosts, from the core diagrams, and trivializing their initial and final conditions. The resulting evolution operator Uph(tf, ti) is unitary, but does not satisfy the more general identity Uph(t3, t2)Uph(t2, t1) = Uph(t3, t1). As a consequence, Uph(tf, ti) cannot be derived from a Hamiltonian in a standard way, in the presence of purely virtual particles.

We consider the interaction of a rigid, frictionless, inelastic particle with a rigid boundary that has a corner. Typically, two possible final outcomes can occur: the particle escapes from the corner after experiencing a certain number of collisions with the corner, or the particle experiences an inelastic collapse in which an infinite number of collisions can occur in a finite time interval. For the former case, we determine the number of collisions that the particle will experience with the boundary before escaping the corner. For the latter case, we determine the conditions for which inelastic collapse can occur. For a corner composed of two straight walls, we derive simple analytic solutions and show that for a given coefficient of restitution, there is a critical corner angle above which inelastic collapse cannot occur. We show that as the corner angle tends to the critical corner angle from below, the process of inelastic collapse takes infinitely long. We also show a surprising phenomenon that if the corner has the form of a cusp, the particle can have an infinite number of collisions with the boundary in a finite time interval without losing all of its energy, and eventually escapes from the corner.

A nonclassical nonlinear third-order partial differential equation modeling nonstationary processes in a semiconductor medium is studied. Seven classes of exact solutions of this equation given as explicit, implicit, and quadrature formulas are constructed. It is shown that, among these solutions, there are some bounded globally with respect to time and some approaching infinity on finite time intervals. These results are of interest in view of the following property of initial boundary-value problems for the given equation. Here the method of energy estimates used in many papers to justify the boundedness or unboundedness of solutions does not provide sufficient conditions for the unboundedness of solutions on finite time intervals.

Equation for survival probability in a finite time interval in case of non-zero real interest force

Diagrammatical expansion of the quantum field theory with the finite time interval is discussed by evaluating the time evolution kernel. The coherent state representation is adopted which is the most convenient formalism for this purpose. It is applied to quantum electrodynamics (QED) to derive the Feynman rule for the kernel. The rule for the matrix element of any operator is also given. Our method can also be a systematic approach to the time dependent perturbation theory. As an example, a full order formula of the transition amplitude between the number states is given in terms of the connected Green's functions.

It has become customary to compute pressure transient model solutions in the Laplace domain. This is because it generally simplifies the solution, and allows easy incorporation of wellbore storage, commingled systems or dual-porosity formations. On the other hand, the convolution and parameter estimation are usually done in the time domain. Thus, when the solution is used as the forward model in an optimization scheme, each forward evaluation requires a Laplace inversion of the model (using the Stehfest algorithm) and a time domain convolution. Here we explore the alternative of transforming the data into the Laplace domain (a one-time operation) and then doing the fitting in the Laplace domain. The chief new result is a self-consistent prescription for dealing with the fact that the data are available only over a finite time interval. The technique is illustrated for the case of a conventional well test which reaches radial flow on the time scale of the test, a situation for which the method is particularly efficient.

Applying the steady state principle to a chemically realistic reaction scheme we obtained a kinetic differential equation system whose solutions are defined only on a finite time interval. This is very unusual but not a mathematical artifact; the solutions describe the self-accelerating character of the system. The solutions are not of “blow up” type: they are not connected to matter inflow and the concentrations remain finite in the system.

Areas of attraction for nonautonomous differential equations on finite time intervals