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Impulsive stabilization of high-order nonlinear retarded differential equations

Juan Liu, Xiaodi Li (2013)

Applications of Mathematics

In this paper, impulsive stabilization of high-order nonlinear retarded differential equations is investigated by using Lyapunov functions and some analysis methods. Our results show that several non-impulsive unstable systems can be stabilized by imposition of impulsive controls. Some recent results are extended and improved. An example is given to demonstrate the effectiveness of the proposed control and stabilization methods.

Inversion in indirect optimal control of multivariable systems

François Chaplais, Nicolas Petit (2008)

ESAIM: Control, Optimisation and Calculus of Variations

This paper presents the role of vector relative degree in the formulation of stationarity conditions of optimal control problems for affine control systems. After translating the dynamics into a normal form, we study the Hamiltonian structure. Stationarity conditions are rewritten with a limited number of variables. The approach is demonstrated on two and three inputs systems, then, we prove a formal result in the general case. A mechanical system example serves as illustration.

Local Controllability around Closed Orbits

Marek Grochowski (2014)

Bulletin of the Polish Academy of Sciences. Mathematics

We give a necessary and sufficient condition for local controllability around closed orbits for general smooth control systems. We also prove that any such system on a compact manifold has a closed orbit.

Lower semicontinuous differential inclusions. One-sided Lipschitz approach

Tzanko Donchev (1998)

Colloquium Mathematicae

Some properties of differential inclusions with lower semicontinuous right-hand side are considered. Our essential assumption is the one-sided Lipschitz condition introduced in [4]. Using the main idea of [10], we extend the well known relaxation theorem, stating that the solution set of the original problem is dense in the solution set of the relaxed one, under assumptions essentially weaker than those in the literature. Applications in optimal control are given.

Measure solutions for semilinear evolution equations with polynomial growth and their optimal control

N.U. Ahmed (1997)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

In this paper we introduce a new concept of generalized solutions generalizing the notion of relaxed solutions recently introduced by Fattorini. We present some results on the question of existence of generalized or measure valued solutions for semilinear evolution equations on Banach spaces with polynomial nonlinearities. The results are illustrated by two examples one of which arises in nonlinear quantum mechanics. The results are then applied to some control problems.

Minimax control of nonlinear evolution equations

Nikolaos S. Papageorgiou (1995)

Commentationes Mathematicae Universitatis Carolinae

In this paper we study the minimax control of systems governed by a nonlinear evolution inclusion of the subdifferential type. Using some continuity and lower semicontinuity results for the solution map and the cost functional respectively, we are able to establish the existence of an optimal control. The abstract results are then applied to obstacle problems, semilinear systems with weakly varying coefficients (e.gȯscillating coefficients) and differential variational inequalities.

Mixed type semicontinuous differential inclusions in Banach spaces

Tzanko Donchev (2001)

Annales Polonici Mathematici

We consider a class of differential inclusions in (nonseparable) Banach spaces satisfying mixed type semicontinuity hypotheses and prove the existence of solutions for a problem with state constraints. The cases of dissipative type conditions and with time lag are also studied. These results are then applied to control systems.

Necessary Optimality Conditions for a Lotka-Volterra Three Species System

N. C. Apreutesei (2010)

Mathematical Modelling of Natural Phenomena

An optimal control problem is studied for a Lotka-Volterra system of three differential equations. It models an ecosystem of three species which coexist. The species are supposed to be separated from each others. Mathematically, this is modeled with the aid of two control variables. Some necessary conditions of optimality are found in order to maximize the total number of individuals at the end of a given time interval.

Nonlinear observers for locally uniformly observable systems

Hassan Hammouri, M. Farza (2003)

ESAIM: Control, Optimisation and Calculus of Variations

This paper deals with the observability analysis and the observer synthesis of a class of nonlinear systems. In the single output case, it is known [4, 5, 6] that systems which are observable independently of the inputs, admit an observable canonical form. These systems are called uniformly observable systems. Moreover, a high gain observer for these systems can be designed on the basis of this canonical form. In this paper, we extend the above results to multi-output uniformly observable systems....

Nonlinear observers for locally uniformly observable systems

Hassan Hammouri, M. Farza (2010)

ESAIM: Control, Optimisation and Calculus of Variations

This paper deals with the observability analysis and the observer synthesis of a class of nonlinear systems. In the single output case, it is known [4-6] that systems which are observable independently of the inputs, admit an observable canonical form. These systems are called uniformly observable systems. Moreover, a high gain observer for these systems can be designed on the basis of this canonical form. In this paper, we extend the above results to multi-output uniformly observable systems....

Observer design for systems with unknown inputs

Stefen Hui, Stanisław Żak (2005)

International Journal of Applied Mathematics and Computer Science

Design procedures are proposed for two different classes of observers for systems with unknown inputs. In the first approach, the state of the observed system is decomposed into known and unknown components. The unknown component is a projection, not necessarily orthogonal, of the whole state along the subspace in which the available state component resides. Then, a dynamical system to estimate the unknown component is constructed. Combining the output of the dynamical system, which estimates the...

Observers for Canonic Models of Neural Oscillators

D. Fairhurst, I. Tyukin, H. Nijmeijer, C. van Leeuwen (2010)

Mathematical Modelling of Natural Phenomena

We consider the problem of state and parameter estimation for a class of nonlinear oscillators defined as a system of coupled nonlinear ordinary differential equations. Observable variables are limited to a few components of state vector and an input signal. This class of systems describes a set of canonic models governing the dynamics of evoked potential in neural membranes, including Hodgkin-Huxley, Hindmarsh-Rose, FitzHugh-Nagumo, and Morris-Lecar...

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