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Best guaranteed bounds for simulations of nonlinear systems using linear Lyapunov-like functions

The models of biological or technical applications are generally represented by nonlinear systems. Moreover these systems contain uncertain or unknown parameters. These uncertainties are reasons for some numerical and analytical problems in finding guaranteed bounds for the solution of the state space representation. Unfortunately several industrial applications are demanding exactly these guaranteed bounds in order to fulfill regulations set by the state authorities. To get an idea of the solution of systems with uncertainties the numerical integration of the systems differential equations has to be done with random selected values of the unknown parameters. This computation is done several times, in some circumstances more than thousand times, to cover the interval of the uncertain parameters. This approach is well known as the Monte-Carlo method, but this stochastic approach cannot deliver guaranteed bounds. Thus, we developed a novel method to find guaranteed bounds, which is using linear Lyapunov-like functions to solve this problem. In this scope, we improved this method to get the best bounds, which the approach with linear Lyapunov-like functions can provide. The results can be used as an initial interval for further methods based on interval arithmetic. We achieve guaranteed and finite simulation bounds as a result of our approach. The idea is to find an auxiliary function, which helps to bind the state variables. An example from an industrial application and its simulation results conclude the paper