Past Interval Research Articles

Stochastic mixed impulsive control and stability for stochastic functional differential systems with semi-Markov jump

This paper focuses on the problem of stochastic mixed impulsive control and stability for stochastic functional differential systems with semi-Markov jump. A new definition of average stochastic impulsive gain is formulated to estimate the stochastic mixed impulsive intensity. Functional systems states are referred to the historical system states in the past time interval rather than in the past time instant of the current time instant. The stability analysis is carried out based on the constructed appropriate Lyapunov functional, Dupire functional Itô’s formula, stochastic analysis theory and graph theory techniques. Moreover, we provide some exponential stability criteria for the investigated systems, which have close relation to not only the topological structures and semi-Markov jump of the functional systems but also average stochastic impulsive gain and stochastic disturbance intensity. Further we put it into practice of a new class of stochastic oscillators systems with semi-Markov jump and the corresponding numerical simulations results are on show to demonstrate the validity of the derived theoretical results.

The near future expected wave power in the coastal environment of the Iberian Peninsula

The main target of the proposed work is to assess the expected wave energy dynamics in the coastal environment of the Iberian Peninsula. This study is motivated by the fact that the Iberian nearshore is considered among the most promising European coastal environments for wave energy extraction with experimental wave projects already operating there since many years. The near future period considered for this analysis is the 20-year time interval 2026–2045 under the Representative Concentration Pathway scenario 4.5 (RCP4.5). In parallel, in order to have a coherent degree of comparison, the results are compared with those corresponding to the recent past 20-year time interval 2001–2020. A wave modelling system based on the third-generation spectral phase averaged model SWAN (acronym from Simulating WAves Nearshore) was implemented for the entire North Atlantic basin and focused on the Iberian nearshore. Furthermore, calibrations and extensive validations of this wave prediction system have been performed. Thus, system reliability was evaluated both against buoy and satellite data and the results were found as having good accuracy. The wave energy data corresponding to the near future (2026–2045) are compared with those from the other 20-year period from the recent past (2001–2020). In this way, a comprehensive picture of the expected wave power dynamics in the coastal environment of the Iberian Peninsula is provided. The results show that there are no significant differences between the two periods considered, especially in the transition seasons (spring and autumn), both in terms of maximum values and also as regards the general patterns of the wave power fields. However, in the near future, a decrease of the resources is expected near to the coast in the summer season, compared to the values simulated for the recent past (from 12 kW/m to 9 kW/m). A contrary tendency corresponds to the winter season when for the near future we find a maximum value higher by about 14 kW/m than in the recent past.

EXPONENTIAL STABILITY FOR DIFFERENCE EQUATIONS WITH MAXIMUM: THEORETICAL STUDY AND COMPUTER SIMULATIONS

Difference equations with a special type of delay is studied. Thistype of delay is characterized by the maximum value of the unknown function onthe past time interval with a constant length. It is studied the exponential stability.Some sufficient conditions are theoretically proved and computer simulated on severalexamples. The influence of the type of the coefficients is illustrated.

Trace-Length Independent Runtime Monitoring of Quantitative Policies

Metric linear-time logic (MTL) has been widely used to specify runtime policies. Traditionally this use of MTL is to capture the qualitative aspects of the monitored systems, but recent developments in its extensions with aggregate operators allow some quantitative policies to be specified. Our interest in MTL-based policy languages is driven by applications in runtime malware or intrusion detection in platforms like Android and autonomous vehicles, which requires the monitoring algorithm to be independent of the length of the system event traces so that its performance does not degrade as the traces grow. We propose a policy language based on a past-time variant of MTL, extended with an aggregate operator called the metric temporal counting quantifier to specify a policy based on the number of times some sub-policies are satisfied in the specified past time interval. We show that a broad class of policies, but not all policies, specified with our language can be monitored in a trace-length independent way, and provide a concrete algorithm to do so. We implement and test our algorithm in both an existing Android monitoring framework and an autonomous vehicle simulation platform, and show that our approach can effectively specify and monitor quantitative policies drawn from real-world studies.

Solvability for a class of integral inequalities with maxima on the theory of time scales and their applications

This paper is committed to introducing some Bihari type inequalities for scalar functions of one independent variable under an initial condition associated with an arbitrary time scale mathbb{T}. The integrals involve the maximum of an unknown function over a past time interval. We not only solve some new estimated bounds of a specific class of retarded and nonlinear dynamic inequalities but also derive and unify continuous inequalities along with the corresponding discrete analogs of some known results with ‘maxima’ on time scales. We illustrate some applications of the considered inequalities to represent the advantages of our work. The main results will be proved by utilizing some examination procedures and the basic technique of Keller’s chain rule on time scales.

Analysis and combination of four technique-individual EOP time series

Based on four intra-technique combined solutions in SINEX format, we tried at first to assess the accuracy of each single-technique Earth Orientation Parameters (EOPs) series over a past time interval of at least15 years by comparison with IERS EOP 08C04 combined solution as the reference in this paper. The EOPs studied here mainly consist of four elements, i.e. polar motion (XPO, YPO), Universal Time (UT1-UTC) and length-of-day (LOD). We combined these intra-technique EOP series, each of them associated with a given space geodetic technique by taking advantage of the relationship of the first three EOP components and three rotational parameters which carry the orientation of technique-related reference frame with respect to the estimated weekly inter-technique combined reference frame. Results indicated that the discrepancy between the pole coordinates (XPO, YPO) series, extracted from the intra-SLR combined loose-constraint solutions and the IERS EOP 08C04, seemed to be clearly characterized by systematic errors. Although both the XPO and YPO series determined by intra-VLBI combination had no significant characteristic of system error, they had relatively large difference values at some point with respect to the IERS EOP 08C04, which may be limited by the quantity of observation stations. Since the number of GPS stations is on the increase aimed at better global coverage, the accuracy of pole coordinates provided by IGS was superior to that derived from other space-geodetic techniques. As for DORIS XPO and YPO series from intra-DORIS combined minimal-constraint solutions, the discrepancy range of the former with respect to IERS EOP 08C04 was a little smaller than that of the latter. The objective of this study is twofold: on the one hand to analysis individual EOP series derived from the various space-geodetic techniques, on the other hand to present the new inter-technique combined EOP solution consistent with weekly inter-technique combined reference frame.

Modeling the dynamics of polymer chains in water solution. Application to sensor design

This paper is devoted to a mathematical model of a chaotic dynamics of a polymer chain in water. The model consists of a parabolic equation that is derived according to the self-consistent field approach. This model is employed for the numerical simulation of a biological sensor that detects the presence of a specific protein in the fluid. The sensor is absolutely simple and seems to be new. Besides that, the suggested equation is interesting from the mathematical point of view. It includes a non-local operator of integration not only over the past time interval as in the problems with memory but also over the future time interval. It is unusual for parabolic problems.

Quasilinearization for nonlinear boundary value problems for delay-type difference equations with maxima

The paper deals with an approximate method for solving a mixed boundary value problem for nonlinear difference equations containing a maximum of the unknown function over a past time interval. Every successive approximation to the unknown solution is the unique solution of an appropriately constructed initial value problem for a linear difference equation with maxima, and an algorithm for its explicit obtaining is suggested. Also, each approximation is a lower/upper solution of the given mixed problem. The rapid convergence of the successive approximations is proved. The suggested algorithm is realized as a computer program and it is applied to an example.MSC: 39A23, 39A99, 65Q10.

Monotone-iterative method for mixed boundary value problems for generalized difference equations with “maxima”

In this paper the authors investigate special type of difference equations which involve both delays and the maximum value of the unknown function over a past time interval. This type of equations is used to model a real process which present state depends significantly on its maximal value over a past time interval. An appropriate mixed boundary value problem for the given nonlinear difference equation is set up. An algorithm, namely, the monotone iterative technique is suggested to solve this problem approximately. An important feature of our algorithm is that each successive approximation of the unknown solution is equal to the unique solution of an appropriately constructed initial value problem for a linear difference equation with “maxima”, and a formula for its explicit form is given. Also, each approximation is a lower/upper solution of the given nonlinear boundary value problem. Several numerical examples are considered to illustrate the practical application of the suggested algorithm.

Monotone-Iterative Method for Solving Antiperiodic Nonlinear Boundary Value Problems for Generalized Delay Difference Equations with Maxima

A nonlinear generalized difference equation with both delays and the maximum value of the unknown function over a discrete past time interval are studied. A nonlinear boundary value problem of antiperiodic type for the given difference equation is set up. One of the main characteristics of the considered difference equation is the presence of the unknown function in both sides of the equation. It leads to impossibility for using the step method for explicit solving of the nonlinear difference equation. In this paper, an approximate method, namely, the monotone iterative technique, is applied to solve the problem. An important feature of the given algorithm is that each successive approximation of the unknown solution is equal to the unique solution of an appropriately constructed initial value problem for a linear difference equation with “maxima,” and an algorithm for its explicit solving is given. Also, each approximation is a lower/upper solution of the given nonlinear boundary value problem. The suggested scheme for approximate solving is computer realized, and it is applied to a particular example, which is a generalization of a model in population dynamics.

Quasilinearization of the Initial Value Problem for Difference Equations with “Maxima”

The object of investigation of the paper is a special type of difference equations containing the maximum value of the unknown function over a past time interval. These equations are adequate models of real processes which present state depends significantly on their maximal value over a past time interval. An algorithm based on the quasilinearization method is suggested to solve approximately the initial value problem for the given difference equation. Every successive approximation of the unknown solution is the unique solution of an appropriately constructed initial value problem for a linear difference equation with “maxima,” and a formula for its explicit form is given. Also, each approximation is a lower/upper solution of the given mixed problem. It is proved the quadratic convergence of the successive approximations. The suggested algorithm is realized as a computer program, and it is applied to an example, illustrating the advantages of the suggested scheme.

Monotone‐Iterative Method for the Initial Value Problem with Initial Time Difference for Differential Equations with “Maxima”

The object of investigation of the paper is a special type of functional differential equations containing the maximum value of the unknown function over a past time interval. An improved algorithm of the monotone‐iterative technique is suggested to nonlinear differential equations with “maxima.” The case when upper and lower solutions of the given problem are known at different initial time is studied. Additionally, all initial value problems for successive approximations have both initial time and initial functions different. It allows us to construct sequences of successive approximations as well as sequences of initial functions, which are convergent to the solution and to the initial function of the given initial value problem, respectively. The suggested algorithm is realized as a computer program, and it is applied to several examples, illustrating the advantages of the suggested scheme.

Practical Stability in terms of Two Measures for Impulsive Differential Equations with “Supremum”

The object of investigations is a system of impulsive differential equations with “supremum.” These equations are not widely studied yet, and at the same time they are adequate mathematical model of many real world processes in which the present state depends significantly on its maximal value on a past time interval. Practical stability for a nonlinear system of impulsive differential equations with “supremum” is defined and studied. It is applied Razumikhin method with piecewise continuous scalar Lyapunov functions and comparison results for scalar impulsive differential equations. To unify a variety of stability concepts and to offer a general framework for the investigation of the stability theory, the notion of stability in terms of two measures has been applied to both the given system and the comparison scalar equation. An example illustrates the usefulness of the obtained sufficient conditions.

Stability analysis of asset flow differential equations

I study the stability analysis of the solutions for the dynamical system of nonlinear asset flow differential equations (AFDEs) in three versions. I show that the previous two versions are not structurally stable mathematically because there are infinitely many critical points. It is important to reformulate a problem in order to eliminate any hypersensitivity in the mathematical model. I find that there is no critical point in the new version unless the chronic discount over the past finite time interval is zero.

A strategy for a payoff-switching differential game based on fuzzy reasoning

In this paper, a new concept of a payoff-switching differential game is introduced. In this new game, any one player at any time may have several choices of payoffs for the future. Moreover, the payoff-switching process, including the time of payoff switching and the outcome payoff, of any one player is unknown to the other. Indeed, the overall payoff, which is a sequence of several payoffs, is unknown until the game ends. An algorithm for determining a reasoning strategy based on fuzzy reasoning is proposed. In this algorithm, the fuzzy theory is used to estimate the behavior of one player during a past time interval. By deriving two fuzzy matrices GSM, game similarity matrix, and VGSM, variation of GSM, the behavior of the player can be quantified. Two weighting vectors are selected to weight the relative importance of the player's behavior at each past time instant. Finally a simple fuzzy inference rule is adopted to generate a linear reasoning strategy. The advantage of this algorithm is that it provides a flexible way for differential game specialists to convert their knowledge into a “reasonable” strategy. A practical example of guarding three territories is given to illustrate our main ideas.

Variational method and computation forH ∞ control

The variational approach is further applied to the measurement feedback H∞ control problems. Based on the induced norm description of the system Γ, the variational functionals Jc and Jp of state feedback H∞ control for the future time interval (t, tf], and of H∞ filter for the past time interval [0, t), respectively, are combined together to generate the variational functional of the measurement feedback for the whole time interval [0, tf]. The connection condition at the present time t is that the estimated state vector\(\bar x\)(t) must be continuously extended to be the initial condition of the future state vector estimation. Another connection condition for the dual vector λ(t) can be naturally derived from the variational principle. The equations thus derived show that the third condition for the optimal parameter γ α -2 is again a bound of the smallest Rayleigh quotient. Therefore, the precise integration method developed formerly to determine the optimal parameter γ α -2 of H∞ control and of H∞ filter respectively can be further applied to the determination of the optimal parameter.