Title data
Baier, Robert ; Hafstein, Sigurdur Freyr:
Numerical computation of control Lyapunov functions in the sense of generalized gradients.
Department of Mathematics, University of Bayreuth
Bayreuth
,
2014
.  8 p.
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Abstract in another language
The existence of a control Lyapunov function with the weak infinitesimal decrease via the Dini or the proximal
subdifferential and the lower Hamiltonian characterizes asymptotic controllability of nonlinear control systems and differential inclusions. We study the class of nonlinear differential inclusions with a righthand side formed by the convex hull of active C² [$C^2$] functions which are defined on subregions of the domain. For a simplicial triangulation we parametrize a control Lyapunov function (clf) for nonlinear control systems by a continuous, piecewise affine (CPA) function via its values at the nodes and demand a suitable negative upper bound in the weak decrease condition on all vertices of all simplices.
Applying estimates of the proximal subdifferential via active gradients we can set up a mixed integer linear problem (MILP) with inequalities at the nodes of the triangulation which can be solved to obtain a CPA function. The computed function is a clf for the nonlinear control system.
We compare this novel approach with the one applied to
compute Lyapunov functions for strongly asymptotically stable differential inclusions and give a first numerical example.
Further data
Item Type:  Preprint, postprint, working paper, discussion paper 

Additional notes:  Contents:
I. Preliminaries, II. Control Lyapunov Functions, III. Approach with Mixed Integer Programming, IV. Numerical Example, V. Conclusions. © 2014 IEEE. Reuse of this content is subject to the IEEE Copyright. This content will be published in: Proceedings on the 21st International Symposium on Mathematical Theory of Networks and Systems (MTNS 2014), July 7–11, 2014, University of Groningen, Groningen, Netherlands, check the forthcoming abstract in IEEE Explore. 
Keywords:  control Lyapunov functions; asymptotic controllability; nonlinear control systems; continuous, piecewise affine functions; mixed integer linear programming 
Subject classification:  Mathematics Subject Classification Code: 93D30 (93B05 90C11) 
Institutions of the University:  Faculties Faculties > Faculty of Mathematics, Physics und Computer Science Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics V (Applied Mathematics) Profile Fields Profile Fields > Advanced Fields Profile Fields > Advanced Fields > Nonlinear Dynamics 
Result of work at the UBT:  Yes 
DDC Subjects:  500 Science > 510 Mathematics 
Date Deposited:  07 Mar 2015 22:00 
Last Modified:  09 Mar 2015 11:26 
URI:  https://eref.unibayreuth.de/id/eprint/7991 
Available Versions of this Item
 Numerical computation of control Lyapunov functions in the sense of generalized gradients. (deposited 07 Mar 2015 22:00) [Currently Displayed]