F. Camilli, L. Grune and F. Wirth. A generalization of Zubov's method to perturbed systems. SIAM J. Control Optim., 2000. to appear. Home/Search Document Details and Download
Summary Related Articles Check |
This paper is cited in the following contexts:
Computing Control Lyapunov Functions via a Zubov Type Algorithm - Grüne, Wirth (2000) (1 citation) Self-citation (Gr Wirth) (Correct)
No context found.
F. Camilli, L. Grune and F. Wirth. A generalization of Zubov's method to perturbed systems. SIAM J. Control Optim., 2000. to appear.
A Generalization of Zubov's Method to Perturbed Systems - Grüne, Wirth (2 citations) Self-citation (Camilli Gr Wirth) (Correct)
No context found.
F. Camilli, L. Grune, and F. Wirth. A generalization of Zubov's method to perturbed systems. SIAM J. Control Optim. 40(2002), 496-515.
A Maximum Time Approach to the Computation of Robust.. - Falcone, Grüne, Wirth (1 citation) Self-citation (Grune Wirth) (Correct)
No context found.
F. Camilli, L. Grune, and F. Wirth. A Generalization of Zubov's method to perturbed systems. Preprint.
Characterizing Controllability Probabilities of Stochastic .. - Camilli, Grüne, Wirth (2004) Self-citation (Camilli Une Wirth) (Correct)
No context found.
F. Camilli, L. Gr une and F. Wirth, A generalization of Zubov's method to perturbed systems, SIAM J. Control Optim. 40 (2001), 496--515.
A generalization of Zubov's method to perturbed systems - Camilli, Grüne (2 citations) Self-citation (Camilli Gr Wirth) (Correct)
....assumption that D R is a locally asymptotically stable compact set for all admissible perturbation functions a we try to nd the set of points which are attracted to D under all these perturbations a. For the special case of D being just one xed point this set has been considered e.g. in [14, 15, 5, 8], for the case where D is a periodic orbit see e.g. 2] The present paper follows the approach of [5] where a generalization of Zubov s classical method [22] has been developed in the framework of viscosity solutions for the characterization of the domain of attraction of an exponentially ....
....a we try to nd the set of points which are attracted to D under all these perturbations a. For the special case of D being just one xed point this set has been considered e.g. in [14, 15, 5, 8] for the case where D is a periodic orbit see e.g. 2] The present paper follows the approach of [5], where a generalization of Zubov s classical method [22] has been developed in the framework of viscosity solutions for the characterization of the domain of attraction of an exponentially stable xed point of a perturbed system. We slightly extend the results from [5] by allowing arbitrary ....
[Article contains additional citation context not shown here]
F. Camilli, L. Grune, and F. Wirth. A generalization of Zubov's method to perturbed systems. SIAM J. Control Optim. 40(2002), 496-515.
A regularization of Zubov's equation for robust domains of .. - Camilli, Grüne, Wirth (2000) (1 citation) Self-citation (Camilli Gr Wirth) (Correct)
....values in some compact set A R m . Under the assumption that x 2 R n is a locally exponentially stable xed point for all admissible perturbation functions a( we try to nd the set of points which are attracted to x for all admissible a( This set has been considered e.g. in [14,15,4,7]. In particular, in [14] and [7] numerical procedures based on optimal control techniques for the computation of robust domains of attraction are presented. The techniques in these papers have in common that a numerical approximation of the optimal value function of a suitable optimal control ....
....the method in [7] needs just one such solution, but requires some knowledge about the local behavior around x in order to avoid discontinuities in the optimal value functions causing numerical problems. In this paper we use a similar optimal control technique, but start from recent results in [4] where the classical equation of Zubov [20] is generalized to perturbed systems. Under very mild conditions on the problem data this equation admits a continuous or even Lipschitz viscosity solution. The main problem in a numerical approximation is the inherent singularity of the equation at the ....
[Article contains additional citation context not shown here]
F. Camilli, L. Grune, and F. Wirth. A Generalization of Zubov's method to perturbed systems. Preprint 24/99, DFG-Schwerpunkt \Ergodentheorie, Analysis und eziente Simulation dynamischer Systeme", submitted.
Computing control Lyapunov functions via a Zubov type algorithm - Grüne, Wirth (2000) (1 citation) Self-citation (Wirth) (Correct)
....Dv(x) f(x) h(x) 1 v(x) p 1 kf(x)k 2 : 1) Namely, under suitable assumptions on h the domain of attraction is the set v 1 ( 0; 1) The v constructed via this equation is automatically a Lyapunov function and smooth. This result has recently be extended to perturbed systems in [2]. Numerical treatment of this equation is discussed in [3] As we already know that it is unreasonable to expect continuously di erentiable control Lyapunov functions, we will look for solutions of a generalization of (1) in the viscosity sense. We refer to [4] for an introduction to the theory ....
....function which is a viscosity solution of (9) on O and satis es P (0) 0 and P (x) 1 for x O and for jxj 1. Then P coincides with V from (3) and O = D. In particular, the function V from (3) is the unique positive continuous viscosity solution of equation (9) on D with V (0) 0. As in [2] it can be shown that we can restrict ourselves to a proper open subset O of the state space and still obtain our solution v, provided D O. We omit the discussion of this aspect, for reasons of space. Finally, we return to the point that the function v is a control Lyapunov function for the ....
F. Camilli, L. Grune and F. Wirth. A generalization of Zubov's method to perturbed systems. SIAM J. Control Optim., 2000. to appear.
A Maximum Time Approach to the Computation of Robust.. - Falcone, Grüne, Wirth (1 citation) Self-citation (Wirth) (Correct)
....to the fixed point regardless of the perturbation considered, under a local stability assumption which guarantees that it is reasonable to consider this set. This is what we call the robust domain of attraction. This subset of the domain of the unperturbed system x = f(x; a 0 ) is also studied in [4, 12, 13]. In particular an algorithm for the approximation of the robust domain of attraction based on ideas from optimal control is presented in [12] There the robust domain of attraction is approximated by the sublevel sets of a sequence of optimal value functions. In this paper we will present a ....
....the following characterization: Consider a compact neighbourhood N ae B(0; r) of the origin. Then we define the maximum time function by T (x) sup a2A t(x; a) It is immediate from Lemma 2.2 that x 2 D if and only if T (x) 1. Furthermore it is easily seen that T (x) 1 as x D, cp. e.g. [4]. Modifying the proof of [1, Chapter IV, Theorem 1.16] there for the minimum time function) we obtain the following result on continuity of T . Proposition 3.1 Assume N is C 2 , and sup a2A f(x; a)j(x) 0 for all x 2 N , where j(x) denotes the outward normal at x 2 N . Then T is ....
F. Camilli, L. Grune, and F. Wirth. A Generalization of Zubov's method to perturbed systems. Preprint.
Further Results on the Bellman Equation for Optimal Control.. - Malisoff (2002) (Correct)
No context found.
Camilli, F., L. Grune, and F. Wirth, \A generalization of Zubov's method to perturbed systems," SIAM J. Control Optim., 40(2001), pp. 496-515.
Further Results on Lyapunov Functions and Domains of Attraction.. - Malisoff (2002) (Correct)
No context found.
F. Camilli, L. Grune, and F. Wirth, A generalization of Zubov's method to perturbed systems, SIAM J. Control Optim., 40, (2001) 496-515.
Online articles have much greater impact More about CiteSeer.IST Add search form to your site Submit documents Feedback
CiteSeer.IST - Copyright Penn State and NEC