#### Table of Contents

# SVM Viability Kernel Approximation (3/3)

## Example: population system

### Description of the system

We consider a simple dynamical system of population growth on a limited space. The state *(x(t), y(t))* of the system represents the size of a population *x(t)*, which grows or diminishes with the evolution rate *y(t)*. The size of the population must remain in an interval *K=[a,b]*, with *a > 0*. The inertia bound *c* limits the derivative of the evolution rate at each time step. The system in discrete time defined by a time interval *dt* can be written as follows:

with

The viability constraint set is the set *K=[a,b] x [d,e]*. For the following example, we used the following parameters: *a=0.2*, *b=3*, *c=0.5*, *d=-2*, *e=2*.

### Progressive approximation of the viability kernel

The following pictures show the progressive approximation of the viability kernel. The points in blue are viable states, those in red non-viable states and in blue, the approximated viability kernel. The black lines represents the theoretical curves of the viability kernel.

### SVM heavy controller

The following two pictures present a trajectory obtained by using the SVM heavy controller. On the left, the number of steps of anticipation is 2 while on the right, it is 10. The light blue zone inside the approximation of the viability kernel correponds to the security distance.

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