We present, with purely didactic aims, a simple and essentially self-contained proof of two necessary and sufficient conditions for existence of a measurable modification of a stochastic process with values in a separable complete metric space.Existence of a measurable modification of a stochastic process continuous in probability is an immediate consequence.

Boundary control to track the output reference of the heat equation is considered.A control input is implemented at one boundary while requiring the other boundary to track the output reference.By introducing the error system and backstepping transformation, the control law is designed.The undetermined coefficient method and two identities are used to obtain the analytical solution of the kernel function by complex mathematical calculation.This establishes an explicit control law and ensures that the error system can effectively achieve the desired closed-loop stability.Simulation results validate the proposed theoretical results.

This study focuses on parameter estimation for the generalized extreme value distribution (GEVD) using the regression method described by [27].A regression equation is derived from the cumulative distribution function and the scale parameter is estimated by applying the iterative re-weighted least squares in this regression equation.For estimating the shape parameter, a profile likelihood is constructed based on this regression equation.A comparison study of the regression method with other existing estimators derived from the method of moments, maximum likelihood, probability-weighted moments, l-moments, and maximum product spacing is performed for the GEVD.Also, the left truncated GEVD is considered and the behaviour of its hazard function is studied.The parameter estimates of the left truncated GEVD is also derived using the regression method.An extensive simulation study is conducted and the efficiencies of the estimation techniques are analysed.The bootstrap confidence intervals for the estimators are also constructed.Finally, a real data analysis is carried out to illustrate the applicability of the models and estimation techniques.

The stability and stabilization of systems with time-varying delays and external disturbances are the subject of this study.To circumvent the limitation of the Bessel-Legendre inequality, which cannot treat a time-varying delay system because the resulting limit contains reciprocal convexity, the generalized free-matrix-based integral inequality is used to generate less conservative stability criteria.Improved stabilization requirements are proposed in the form of linear matrix inequalities by developing a new augmented Lyapuno-Krasovskii function.To achieve resolved controller gains, a method for designing a H dynamic output feedback controller based on linear matrix inequalities is then provided.Finally, three examples are used to validate the advantages of the approach over existing methods.

We investigate Korteweg-de Vries-Burgers (KdVB) equation, where the dissipation and dispersion coefficients are unknown, but their lower bounds are known.First, we establish dynamic boundary controls with update laws to globally exponentially stabilize this uncertain system.Secondly, we demonstrate that the dynamic boundary control design is suboptimal to a meaningful functional after some minor modifications of the dynamic boundary controls.In addition, we also consider dynamic boundary controls for the case of unknown dissipation or dispersion coefficients, and obtain corresponding results.Finally, three examples are used to demonstrate the effectiveness of the proposed control design.