The dynamical systems method (DSM), for solving operator equations, especially nonlinear and ill-posed, is developed in this paper. Consider an operator equation F (u) = 0 in a Hilbert space H and assume that this equation is solvable. Let us call the problem of solving this equation illposed if the operator F ′(u) is not boundedly invertible, and well-posed otherwise. The DSM for solving linea...

In this paper, first the space of hyperbolic tangent functions is introduced and then the universal approximator property of this space is proved. In fact, by using this space, any nonlinear continuous function can be uniformly approximated with any degree of accuracy. Also, as an application, this space of functions is utilized to design feedback control for a nonlinear dynamical system.

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A data-driven identification of dynamical systems requiring only minimal prior knowledge is promising whenever no analytically derived model structure is available, e.g., from first principles in physics. However, meta-knowledge on the system’s behavior is often given and should be exploited: Stability as fundamental property is essential when the model is used for controller design or movement...

Firdaus E. Udwadia"'*, Prasanth B. Kogantib, Thanapat Wanichanon' and Du5an M. Stipanovi6'l d Depurtment of Aerospat'e antl Mechanical Engineering, Civil Engirteering, Muthematic's, and Infitrmation and Operutions Management,430K Olin Hall, Universitl,of Southern Calfornia, Los Angeles, CA 90089-1153, (/SA;hDepartment of Civil Engineering, Universitl, of South.ern CaliJbrnia, Los Angeles, CA 90...

It is demonstrated that very general nonlinear dynamical systems covering all cases arising in practice can be brought down to rate equations of chemical kinetics

This paper proposes a method to trace bifurcation sets for a piecewise-defined differential equation. In this system, the trajectory is continuous, but it is not differentiable at break points of the characteristics. We define the Poincaré mapping by suitable local sections and local mappings, and thereby it is possible to calculate bifurcation parameter values. As an illustrated example, we an...

In this paper, a conditional Gaussian framework for filtering complex turbulent systems is introduced. Despite the conditional Gaussianity, such systems are nevertheless highly nonlinear and are able to capture the non-Gaussian features of nature. The special structure of the filter allows closed analytical formulae for updating the posterior states and is thus computationally efficient. An inf...

A nonlinear equation is one in which a variable on the right side of the equation is raised to an exponent which is not equal to one (e.g., f (x) = x2). Nonlinear systems are generally thought of as systems whose motion is governed by mechanisms which behave as nonlinear equations. The sigmoidal-shaped oxygen-saturation curve of hemoglobin and the relation between vascular resistance and blood ...

The non-isothermal continuous stirred tank reactor (CSTR) is a classical yet complex case study of nonlinear dynamical systems. Power-shaping control is a recent approach for the control of nonlinear systems based on the physics of the dynamical system and it rests on the formulation of the dynamics in the Brayton-Moser form. The Brayton-Moser formulation of the non-isothermal continuous stirre...