# Maintenir la viabilité ou la résilience d'un système : les machines à vecteurs de support pour rompre la malédiction de la dimensionnalité ?

Abstract : Viability theory proposes concepts and tools to control a dynamical system such that it can remain inside a viability constraint set. The applications are frequent in ecology, economics or robotics, where the systems die or badly deteriorate when they leave some regions of the state space. Starting from the viability kernel or capture basin of a system, it enables providing control functions that maintain viability. Nevertheless, current algorithms approximating viability kernel or capture basin show several restrictions; particularly, they suffer the dimensionality curse, and their application is thus limited to problems in low dimension (in the state and control space). The aim of this thesis is to develop and evaluate new algorithms using a specific learning method: Support Vector Machines (SVMs). We propose a new viability kernel approximation algorithm, based on the Saint-Pierre algorithm, which uses a classification method to define the boundary of the kernel approximation. We establish the mathematical conditions that the classification procedure should fulfill, and we consider SVMs in this context. This classification method defines a kind of barrier function on the boundary of the approximation, which allows using optimisation techniques to compute a viable control, and thus work in higher dimensional control space. This function also enables to define more or less cautious controllers. We apply the algorithm on a fisheries management problem, examining which yield policies allow to ensure the sustainability of a marine ecosystem. This example shows the performance of the algorithm: the system is defined in 6 dimensions for the state space, and 17 dimensions for the control space. Starting from the viability kernel approximation algorithm using SVMs, we derive a capture basin approximation algorithm and resolution of hitting target problems. Approximating minimal time function comes down to approximate the viability kernel of an extended system. We present a procedure that approximates capture basin in the initial state space and thus avoid the cost of adding one supplementary dimension (computing and time cost). We describe two variants of this algorithm: the first one provides an outer approximation and the second an inner approximation. Comparing the two results gives an evaluation of the approximating error. Inner approximation enables to define a controller that guarantees to reach the target in minimal time. The procedure can be extended to the problem of minimizing a cost function, when it meets some general conditions. We illustrate this point on the computation of resilience values. We apply the algorithm on a problem of computing resilience values on a model of lake eutrophication. The proposed algorithms enable solving the exponential growth of the computing time with the control space dimension but still suffer the dimensionality curse for the state space: the training set size growth exponentially with the dimension of the space. We introduce active learning techniques to select the most informative'' states to define the SVM function, and then save memory, while keeping an accurate approximation. We illustrate the procedure on a bike control problem on a track, a 6 dimensional state space problem.
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https://tel.archives-ouvertes.fr/tel-00499465
Contributor : Laetitia Chapel <>
Submitted on : Friday, July 9, 2010 - 5:35:33 PM
Last modification on : Monday, February 25, 2019 - 3:14:05 PM
Long-term archiving on : Monday, October 11, 2010 - 10:04:52 AM

### Identifiers

• HAL Id : tel-00499465, version 1

### Citation

Laëtitia Chapel. Maintenir la viabilité ou la résilience d'un système : les machines à vecteurs de support pour rompre la malédiction de la dimensionnalité ?. Modélisation et simulation. Université Blaise Pascal - Clermont-Ferrand II, 2007. Français. ⟨tel-00499465⟩

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