An optimal Gauss–Markov approximation for a process with stochastic drift and applications

We consider a linear stochastic differential equation with stochastic drift. We study the problem of approximating the solution of such equation through an Ornstein–Uhlenbeck type process, by using direct methods of calculus of variations. We show that general power cost functionals satisfy the conditions for existence and uniqueness of the approximation. We provide some examples of general interest and we give bounds on the goodness of the corresponding approximations. Finally, we focus on a model of a neuron embedded in a simple network and we study the approximation of its activity, by exploiting the aforementioned results.     

Study of wave–wind interaction at a seawall using a numerical wave channel

This paper presents the study on wind and waves interactions at a seawall using a numerical wave channel. The numerical experiments were conducted for wave overtopping of a 1/4 sloping seawall using several conditions of incident waves and wind speeds. The numerical results were verified against laboratory data in a case for wave overtopping without wind effects. The interaction of waves and wind was analyzed in term of mean wave quantities, overtopping rate and variation of wind velocity at some selected locations. The results showed that the overtopping rate was strongly affected by wind and the wind field was also significantly modified by waves. There exists an effective range of wind speed in comparison with the local shallow wave speed at the breaking location, which gives significant effects to the wave overtopping rates. The maximum of wind adjustment coefficient f_w for wave overtopping rate was strongly related to the mean overtopping rate in the case for no wind. This study also showed that when the mean overtopping rate was greater than 5 × 10⁻⁴ m³/s/m, the maximum of wind adjustment coefficient f_w approached to a specific value of about 1.25.     

Error control for the approximation of Allen–Cahn and Cahn–Hilliard equations with a logarithmic potential

A fully computable upper bound for the finite element approximation error of Allen–Cahn and Cahn–Hilliard equations with logarithmic potentials is derived. Numerical experiments show that for the sharp interface limit this bound is robust past topological changes. Modifications of the abstract results to derive quasi-optimal error estimates in different norms for lowest order finite element methods are discussed and lead to weaker conditions on the residuals under which the conditional error estimates hold.     

Direct trajectory optimization and costate estimation of finite-horizon and infinite-horizon optimal control problems using a Radau pseudospectral method

A method is presented for direct trajectory optimization and costate estimation of finite-horizon and infinite-horizon optimal control problems using global collocation at Legendre-Gauss-Radau (LGR) points. A key feature of the method is that it provides an accurate way to map the KKT multipliers of the nonlinear programming problem to the costates of the optimal control problem. More precisely, it is shown that the dual multipliers for the discrete scheme correspond to a pseudospectral approximation of the adjoint equation using polynomials one degree smaller than that used for the state equation. The relationship between the coefficients of the pseudospectral scheme for the state equation and for the adjoint equation is established. Also, it is shown that the inverse of the pseudospectral LGR differentiation matrix is precisely the matrix associated with an implicit LGR integration scheme. Hence, the method presented in this paper can be thought of as either a global implicit integration method or a pseudospectral method. Numerical results show that the use of LGR collocation as described in this paper leads to the ability to determine accurate primal and dual solutions for both finite and infinite-horizon optimal control problems.     

The study of a predator–prey system with group defense and impulsive control strategy

A predator–prey system with group defense and impulsive control strategy is established. By using Floquet theorem and small amplitude perturbation skills, a locally asymptotically stable prey-eradication periodic solution is obtained when the impulsive period is less than some critical value. Otherwise, if the impulsive period is larger than the critical value, the system is permanent. By using bifurcation theory, we show the existence and stability of positive periodic solution when the pest-eradication lost its stability. Further, numerical examples show that the system considered has more complicated dynamics, such as: (1) quasi-periodic oscillating, (2) period-doubling bifurcation, (3) period-halving bifurcation, (4) non-unique dynamics (meaning that several attractors coexist), (5) attractor crisis, etc. Finally, the biological implications of the results and the impulsive control strategy are discussed.     

Semigroup wellposedness and asymptotic stability of a compressible Oseen–structure interaction via a pointwise resolvent criterion

In this study, we consider the Oseen structure of the linearization of a compressible fluid–structure interaction (FSI) system for which the interaction interface is under the effect of material derivative term. The flow linearization is taken with respect to an arbitrary, variable ambient vector field. This process produces extra “convective derivative” and “material derivative” terms, which render the coupled system highly nondissipative. We show first a new well-posedness result for the full incorporation of both Oseen terms, which provides a uniformly bounded semigroup via dissipativity and perturbation arguments. In addition, we analyze the long time dynamics in the sense of asymptotic (strong) stability in an invariant subspace (one-dimensional less) of the entire state space, where the continuous semigroup is uniformly bounded. For this, we appeal to the pointwise resolvent condition introduced in Chill and Tomilov [Stability of operator semigroups: ideas and results, perspectives in operator theory Banach center publications, 75 (2007), Institute of Mathematics Polish Academy of Sciences, Warszawa, 71–109], which avoids an immensely technical and challenging spectral analysis and provides a short and relatively easy-to-follow proof.     

On the existence and the uniqueness of the solution of a fluid–structure interaction scattering problem

The existence and uniqueness of the solution of a fluid–structure interaction problem is investigated. The proposed analysis distinguishes itself from previous studies by employing a weighted Sobolev space framework, the DtN operator properties, and the Fredholm theory. The proposed approach allows to extend the range of validity of the standard existence and uniqueness results to the case where the elastic scatterer is assumed to be only Lipschitz continuous, which is of more practical interest.     

On a general theory for quantum–gravity interaction and an experimental confirmation of heterotic strings

The note gives first a short systematic derivation of the basic topological structure of heterotic string theory using the electromagnetic fine structure constant. Subsequently, we argue that the physical reality of heterotic string theory is confirmed via an indirect comparison between certain theoretical predictions and the experimental results at what are presently accessible energy scales.     

Stability and bifurcation analysis of a mathematical model for tumor–immune interaction with piecewise constant arguments of delay

In this paper, we propose and analyze a Lotka–Volterra competition like model which consists of system of differential equations with piecewise constant arguments of delay to study of interaction between tumor cells and Cytotoxic T lymphocytes (CTLs). In order to get local and global behaviors of the system, we use Schur–Cohn criterion and constructed a Lyapunov function. Some algebraic conditions which satisfy local and global stability of the system are obtained. In addition, we investigate the possible bifurcation types for the system and observe that the system may undergo Neimark–Sacker bifurcation. Moreover, it is predicted a threshold value above which there is uncontrollable tumor growth, and below periodic solutions that leading to tumor dormant state occur.     

Coherent modeling of male and female mortality using Lee–Carter in a complex number framework

Forecasts of female and male mortality that are conducted independently run the risk of projecting implausible sex differentials and fail to exploit correlations that are known to exist between the sexes. We present a new model for the simultaneous modeling of female and male mortality. The model casts mortality as a complex-valued process where the real and imaginary parts correspond to female and male mortalities, respectively. Calculations proceed similarly to the usual Lee–Carter model, via the singular value decomposition, albeit in complex form. Initial applications suggest that the complex Lee–Carter gives fits that are broadly comparable to independent real fits, while offering the advantage of explicit modeling of the relationship between the sexes. Furthermore, model parameters are informative and easily-interpretable.     

Robust continuous-time and discrete-time flow control of a dam–river system. (II) Controller design

The paper presents robust design methods for the automatic control of a dam–river system, where the action variable is the upstream flow rate and the controlled variable the downstream flow rate. The system is modeled with a linear model derived analytically from simplified partial derivative equations describing open-channel flow dynamics. Two control methods (pole placement and Smith predictor) are compared in terms of performance and robustness. The pole placement is done on the sampled model, whereas the Smith predictor is based on the continuous model. Robustness is estimated with the use of margins and also with the use of a bound on multiplicative uncertainty taking into account the model errors, due to the nonlinear dynamics of the system. Simulations are carried out on a nonlinear model of the river and performance and robustness of both controllers are compared to the ones of a continuous-time PID controller.     

On the second-order approximate symmetry classification and optimal systems of subalgebras for a forced Korteweg–de Vries equation

We investigate a forced Korteweg–de Vries (fKdV) equation, $u_{\text{,}t}+{\mathit{cu}}_{\text{,}x}+\alpha {\mathit{uu}}_{\text{,}x}+\beta u_{\text{,}\mathit{xxx}}=\beta F(t)$, which arises in the modelling of tsunami generation by submarine landslides. Approximate symmetries are found for the fKdV equation using the method as proposed by Fushchich and Shtelen [6]. Symmetries are found to second order in the perturbation parameter using the MAPLE symmetry package ASP [11], an add-on to the symmetry package DESOLVII [18]. ASP also allows particular forms of the arbitrary function $F(t)$ to be found which extend the symmetry algebra and hence a full approximate symmetry classification of the fKdV equation is obtained. Optimal systems of one-dimensional subalgebras are also determined. Corresponding approximate invariant solutions to the fKdV equation are then constructed for particular $F(t)$ using DESOLVII routines.     

Behavior prediction of washing system in a paper industry using GA and fuzzy lambda–tau technique

Availability analysis has been an important issue in the design field of any Industrial system as the system structure has become more complicated. Also, the system availability is affected by many factors such as design, manufacturing, installation, etc., and so it may be extremely difficult to model, analyze and predict the failure behavior of the system. The purpose of this paper is to develop a new approach for computing various performance measures, namely reliability, availability, MTBF (mean time between failures), ENOF (expected number of failures), failure rate and repair time, for any industrial system. In the proposed approach, the failure rates and repair times of all constituent components are obtained using genetic algorithms and then various performance measures are computed using fuzzy lambda–tau methodology. Washing system, the major part of paper industry is the subject of study. The interactions among the working components are modeled using Petri nets. Failure and repair rates are represented using triangular fuzzy numbers as they allow expert opinion, linguistic variables, operating conditions, uncertainty and imprecision in reliability information to be incorporated into system model. Based on calculated reliability parameters, a structured framework has been developed that may help the maintenance engineers to analyze and predict the system behavior.     

Steady-state,self-oscillating and chaotic behavior of a PID controlled nonlinear servomechanism by using Bogdanov–Takens and Andronov–Poincaré–Hopf bifurcations

This paper analyzes a controlled servomechanism with feedback and a cubic nonlinearity by means of the Bogdanov–Takens and Andronov–Poincaré–Hopf bifurcations, from which steady-state, self-oscillating and chaotic behaviors will be investigated using the center manifold theorem. The system controller is formed by a Proportional plus Integral plus Derivative action (PID) that allows to stabilize and drive to a prescribed set point a body connected to the shaft of a DC motor. The Bogdanov–Takens bifurcation is analyzed through the second Lyapunov stability method and the harmonic-balance method, whereas the first Lyapunov value is used for the Andronov–Poincaré–Hopf bifurcation. On the basis of the results deduced from the bifurcation analysis, we show a procedure to select the parameters of the PID controller so that an arbitrary steady-state position of the servomechanism can be reached even in presence of noise. We also show how chaotic behavior can be obtained by applying a harmonical external torque to the device in self-oscillating regime. The advantage of achieving chaotic behavior is that it can be used so that the system reaches a set point inside a strange attractor with a small control effort. The analytical calculations have been verified through detailed numerical simulations.     

Modeling and analysis of a modified Leslie–Gower type three species food chain model with an impulsive control strategy

This paper describes a modified Leslie–Gower type three species food chain model with harvesting. We have incorporated impulsive control strategy to the system. Theories of impulsive differential equations, small amplitude perturbation skills and comparison technique are used to study dynamical behavior of the system. Sufficient conditions are derived to ensure global stability of the lowest-level prey and mid-level predator eradication periodic solution. Sufficient conditions are also derived to examine the permanence of the system. Numerical simulations are carried out to verify the analytical results, and the system is analyzed through graphical illustrations. It is observed that the stability of the system exhibits several states, ranging from stable situation to cyclic oscillatory behavior, under different favorable conditions. These results are useful to study the dynamic complexity of ecological systems. The computation of the largest Lyapunov exponent demonstrates the chaotic dynamic nature of the system. The qualitative nature of strange attractor is examined. It is to be noted that the harvesting effort can cause a stable equilibrium to become unstable and even a switching of stabilities.     

Lie symmetry,nonlocal symmetry analysis,and interaction of solutions of a (2+1)-dimensional KdV–mKdV equation

Theoretical and Mathematical Physics - We use the method of Lie symmetry analysis to investigate the properties of a (2+1)-dimensional KdV–mKdV equation. Using the Ibragimov method, which...     

Mathematical model of liquid–liquid equilibrium for a ternary system using the GMDH-type neural network and genetic algorithm

A GMDH type-neural network was used to predict liquid phase equilibrium data for the (water + ethanol + trans-decalin) ternary system in the temperature range of 300.2–315.2 K. In order to accomplish modeling, the experimental data were divided into train and test sections. The data set was divided into two parts: 70% were used as data for “training” and 30% were used as a test set. The predicted values were compared with those of experimental values in order to evaluate the performance of the GMDH neural network method. The results obtained by using GMDH type neural network are in excellent agreement with the experimental results.     

Approximate solution of u′=–A2u using a relation between exp(–tA2) and exp(itA)

Let A be a positive definite operator in a Hilbert space and consider the initial value problem for u _t =–A² u. Using a representation of the semi group exp(–A² t) in terms of the group exp(iAt) we express u in terms of the solution of the standard heat equation w _t = w _yy , with initial values v solving the initial value problem for v _y = iAv. This representation is used to construct a method for approximating u in terms of approximations of v. In the case that A is a 2^nd order elliptic operator the method is combined with finite elements in the spatial variable and then reduces the solution of the 4th order equation for u to that of the 2^nd order equation for v, followed by the solution of the heat equation in one space variable.     

N-lump and interaction solutions of localized waves to the (2 + 1)-dimensional asymmetrical Nizhnik–Novikov–Veselov equation arise from a model for an incompressible fluid

The present article deals with M-soliton solution and N-soliton solution of the (2 + 1)-dimensional asymmetrical Nizhnik–Novikov–Veselov equation by virtue of Hirota bilinear operator method. The obtained solutions for solving the current equation represent some localized waves including soliton, breather, lump, and their interactions, which have been investigated by the approach of the long-wave limit. Mainly, by choosing the specific parameter constraints in the M-soliton and N-soliton solutions, all cases of the one breather or one lump can be captured from the two, three, four, and five solitons. In addition, the performances of the mentioned technique, namely, the Hirota bilinear technique, are substantially powerful and absolutely reliable to search for new explicit solutions of nonlinear models. Meanwhile, the obtained solutions are extended with numerical simulation to analyze graphically, which results in localized waves and their interaction from the two-, three-, four-, and five-soliton solutions profiles. They will be extensively used to report many attractive physical phenomena in the fields of acoustics, heat transfer, fluid dynamics, classical mechanics, and so on.