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A SPECTRAL CONDITION FOR ASYMPTOTIC CONTROLLABILITY AND STABILIZATION AT SINGULAR POINTS

In this paper we present a spectral condition for the exponential stabilization of nonlinear control systems with constrained control range at singular points. The spectral approach in particular allows to formulate an equivalence result between exponentia

A SPECTRAL CONDITION FOR ASYMPTOTIC CONTROLLABILITY AND STABILIZATION AT SINGULAR POINTS Lars Grune Institut fur Mathematik, Universitat Augsburg, Universitatsstr. 14, 86135 Augsburg, Germany, E-Mail: Lars.Gruene@Math.Uni-Augsburg.de Abstract In this paper we present a spectral condition for the exponential stabilization of nonlinear control systems with constrained control range at singular points. The spectral approach in particular allows to formulate an equivalence result between exponential null controllability and exponential stabilization by means of a discrete feedback law. The key tool used is a discounted optimal control problem for the corresponding projected semilinear system, which also admits a numerical solution. we will frequently assume x= 0. Such singular situations do typically occur if the control enters in the parameters of an uncontrolled systems at a xed point, for instance when the restoring force of a nonlinear oscillator is controlled. Note that our general setup covers several models: The additional equation for y allows us to model systems where time varying parametric excitations governed by an additional (nonlinear) control or dynamical system enter the system to be stabilized. The case in which the control u does not enter explicitly in the function f and the case in which f does not depend on y occur as special situations in this setup, hence they are also covered. The main tool used for the stabilization is the linearization of (1.1) at the singular point which is given by

Keywords: nonlinear systems, singular points, stabilization, Lyapunov spectrum

AMS Classi cation: 93D15, 93D22 1 Introduction In this paper we will present a spectral condition for the exponential stabilization of nonlinear control systems with constrained control range at singular points, i.e. systems of the form x(t)= f (x(t); y(t); u(t)) _ (1.1) y_ (t)= g(y(t); u(t)) on Rd M where x 2 Rd and y 2 M, M some Riemannian Manifold and f and g are vector elds which are C 2 in x, Lipschitz in y and continuous in u. The control function u( ) may be chosen from the set U:= fu: R ! U j u( ) measurableg where U Rm is compact, i.e. we have a constrained set of control values. For each pair (x0; y0) of initial values the trajectories of (1.1) will be denoted by the pair (x(t; x0; y0; u( )); y(t; y0; u( ))) and we assume them to exist uniquely for all times. Our interest lies on the stabilization of the xcomponent at a singular point x, i.e. a point where f (x; y; u)= 0 for all (y; u) 2 M U . For simplicity 1

z_ (t)= A(y(t); u(t))z (t) y (t)= g(y(t); u(t)) _

(1.2)

d Here A(y; u):= dx f (x; y; u) 2 Rd d and f (x; y; u)= A(y; u)x+ f~(x; y; u) where the estimate kf~(x; y; u)k Cf kxk2 for some constant Cf holds in a neighborhood of x .

As above we denote the trajectories of (1.2) by (z (t; z0; y0; u( )); y(t; y0; u( ))) for the pair of initial values (z0; y0 ). The approach we follow is based on optimal control techniques. More precis

ely, we consider the Lyapunov exponents of the linearization and formulate a discounted optimal control problem in order to minimize these exponents, an idea that has rst been presented in 6]. Lyapunov exponents have recently turned out to be a suitable tool for the stability analysis of semilinear systems, see e.g. 3] and 4], and also for their stabilization 5]. However, due to the fact that for discounted optimal control problems optimal feedback laws are in general not available, we modify the feedback concept and introduce discrete feedback laws that are based on a discrete time sampled approximation of the given continuous time system. Using this approach it could be

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