The Linear Problem of Pursuing Two Coordinated Evaders in Time Scales

In this note, we propose a method to analyze systems in a time scale which is varied depending on the state such as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">dt/d\tau = s(x)</tex> (where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</tex> and τ are the actual time scale and that of new one, respectively, and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s(x)</tex> is the function which we call time scaling function). Analysis of the system in the new time scale τ enables us to investigate the intrinsic structure of the system. A linearization problem in the new time scale is formulated as wide-sense feedback equivalence and is solved. It is also shown that the time scaling function which makes the system linear is derived as the solution of differential equations.

This paper is mainly concerned with a class of optimal controlproblems of systems governed by the first order linear dynamicequations on time scales with quadratic cost functionals.Introducing the weak solutions of the first order linear dynamicequations and presenting the Arzela-Ascoli theorem on time scales,we prove the existence of solution to a class of linear quadraticoptimal control problems on time scales.

This paper discusses the spectrum properties of a linear fourth-order dynamic boundary value problem on time scales and obtains the existence result of positive solutions to a nonlinear fourth-order dynamic boundary value problem. The key condition which makes nonlinear problem have at least one positive solution is related to the first eigenvalue of the associated linear problem. The proof of the main result is based upon the Krein-Rutman theorem and the global bifurcation techniques on time scales.

Some delta-nabla type maximum principles for second-order dynamic equations on time scales are proved. By using these maximum principles, the uniqueness theorems of the solutions, the approximation theorems of the solutions, the existence theorem, and construction techniques of the lower and upper solutions for second-order linear and nonlinear initial value problems and boundary value problems on time scales are proved, the oscillation of second-order mixed delat-nabla differential equations is discussed and, some maximum principles for second order mixed forward and backward difference dynamic system are proved.

This paper offers conditions ensuring the existence of solutions of linear boundary value problems for systems of dynamic equations on time scales. Utilizing a method of Moore–Penrose pseudo‐inverse matrices leads to an analytical form of a criterion for the existence of solutions in a relevant space and, moreover, to the construction of a family of linearly independent solutions of such problems in a general case with the number of boundary conditions (defined by a linear vector functional) not coinciding with the number of unknowns of a system of dynamic equations. As an example of an application of the presented results, the problem of bifurcation of solutions of boundary value problems for systems of dynamic equations on time scales with a small parameter is considered.

A general theory for treating nonlinear stability problems in dissipative plasmas is developed, based on a two-time method extended to a system of partial differential equations. Assuming small dissipation, the first-order system is identical to that for the usual linear stability problem in magnetohydrodynamics; dissipative and nonlinear effects appear on the slow time scale in higher-order systems. For modes with finite frequencies on the fast time scale, the amplitudes are evolving on the slow time scale and are governed by ordinary differential equations with coefficients depending on the global effects of the eigenfunctions in the first-order system. For modes with low frequencies or small growth rates, nothing changes on the fast time scale; waves and nonlinear effects interact with dissipative effects on the slow time scale, resulting in a higher-order equation. Several possible scalings for dissipation are studied and stability related criteria on the slow time scale are obtained. Ohmic heating effects are also discussed.

A wide variety of inverse heat conduction problems have been studied in the last two decades for the estimation of boundary or initial conditions, thermophysical properties, geometrical parameters, or heat source intensities. In most transient heat conduction problems, the mathematical models were cast in dimensionless forms, by using a diffusion time scale. As the thermal diffusivities are usually small, the physical time scales turn to be rather long. In this way, most works show that the inverse analysis yields satisfactory results, without addressing the implications of the physical time scale. The physical time scale, in fact, influences significantly the quality of the inverse solution. We present here a unified treatment for one-dimensional, linear inverse heat conduction problems using the conjugate gradient method with an adjoint equation, and also show that there are physical limitations by the time scale on the inverse solutions.

Eigenvalue distribution of second-order dynamic equations on time scales considered as fractals

In this paper some global aspects of the intraseasonal oscillations on the time scale of 30 to 50 days are explored. Noting that the variability of zonal flow of the monsoon, the atmospheric angular momentum and the length of day are strongly correlated on this time scale, we have made an effort to examine the global variability using the length of day as a point of reference. The scenario of this cycle is presented starting from a super cloud cluster at the near equatorial latitudes. This seems to be accompanied with an acceleration of zonal flows, an increase of the atmospheric angular momentum and an increase in the length of day. The transfer of westerly angular momentum from the earth to the atmosphere occurs over regions of the surface easterlies to the east of the super cloud clusters resulting in an increase in the length of day. During this transition from a mean length of day to a maximum length of day, an active phase of the Indian summer monsoon is noted. The interesting aspect of the length of day transition occurs on its return cycle when the near equatorial cloud cover eases or moves away from the equator with a decrease in the monsoonal zonal flows and a reduction of this component of atmospheric angular momentum. The length of day does not simply go back to an equilibrium value, but the long term data from the laser ranger shows an overshooting beyond that to a minimum value. This transition is characterized in general by monsoon break-like conditions, counter monsoon flows in the low levels and by a transition from high index to low index conditions in the upper troposphere of the middle latitudes. Phenomenologically, some blocking situations have been noted over the higher middle latitudes during this transition. The reduction of the angular momentum is attributed to the transfer of the westerly angular momentum from the atmosphere to the earth via frictional and mountain torques. These torques exhibit a clear relationship to the changes in the atmospheric angular momentum on this time scale. The behavior of the middle latitude low frequency variability is also in part explained by the meridional wave energy flux. That problem is examined in this context with the full non-linear equations in the frequency domain. It is shown that unlike the linear problems where such fluxes are inhibited beyond the critical latitude, the nonlinear problem permits the temporal oscillations of zonal flows on this time scale. As a consequence, a significant tropical-middle latitude coupling is noted by this process.A simple mathematical model of the oscillation is also presented. It is a local theory in which ocean and atmosphere interact. Initially, the atmosphere is stably stratified with weak winds at the sea surface and stronger winds aloft; the ocean has a surface mixed layer of temperature Ts lying over deep cold water. Solar heating gradually increases Ts which leads to atmospheric convection with associated transport of horizontal momentum and increased winds at the sea surface. Increased winds lead to deepening of the mixed layer and a drop in Ts because of mixing of deep cold water with surface waters. Convection ceases, winds decay, and the cycle repeats only after solar heating has once more increased Ts. The period of this oscillation is shown to be on the order of 30 days.

The problem of exact linearization of single-input nonlinear systems by time scale transformation (TST) is considered in this paper. The aim of this paper is to derive method of computing time scale function based on relative degree structure of nonlinear systems. The method using two independent functions which have the largest relative degree of the system is proposed. Because of no requirement of the solution of the nonlinear PDE, the time scale function can be computed easily. An example of the Acrobot is also given. The Acrobot can not be linearized by feedback transformation and state feedback, but this paper shows that the Acrobot is linearizable exactly by applying time scale transformation.

We are concerned with the following nonlinear second-order three-point boundary value problem on time scales , , , , where with and . A new representation of Green's function for the corresponding linear boundary value problem is obtained and some existence criteria of at least one positive solution for the above nonlinear boundary value problem are established by using the iterative method.

Lower bounds for the eigenvalues of first-order nonlinear Hamiltonian systems on time scales

In a finite-dimensional Euclidean space, we consider the problem of pursuit of two evaders by a group of pursuers, described by a linear system with a simple matrix on a given time scale. It is assumed that the evaders use the same control. The pursuers employ quasistrategies based on information about the initial positions and control history of the evaders. The set of admissible controls for each participant is a ball of unit radius centered at the origin, and the terminal sets are the origin. The goal of the group of pursuers is to capture the two evaders. In the study, we use the method of resolving functions as a base one, which allows us to obtain sufficient conditions for the solvability of the approach problem in a certain guaranteed time. In terms of the initial positions and parameters of the game, a sufficient condition for capturing the evaders is obtained.

This paper addresses a version of the linear quadratic control problem for mean-field stochastic differential equations with deterministic coefficients on time scales, which includes the discrete time and continuous time as special cases. Two coupled Riccati equations on time scales are given and the optimal control can be expressed as a linear state feedback. Furthermore, we give a numerical example.

This paper addresses a version of the stochastic linear quadratic control problem on time scales S Δ LQ , which includes the discrete time and continuous time as special cases. Riccati equations on time scales are given, and the optimal control can be expressed as a linear state feedback. Furthermore, we present the uniqueness and existence of the solution to the Riccati equation on time scales. Furthermore, we give an example to illustrate the theoretical results.