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Basic concepts


I - Feedback

The fundamental concept is that of feedback, allowing to pilot evolutions by using controls associated with each state of the system to satisfy some required properties.

Among these properties two have to be distinguished:

Viability Property:

It amounts to regulate "viable" evolutions in an environment defined by the viability constraints, that is they have to stay forever or as long as possible in this environment.

The viability kernel of an environment is the set of initial states from which starts at least one evolution viable in this environment forever.

Viability feedbacks govern evolutions either forever viable when the initial state is in the viability kernel either starting outside the viability kernel, as long as possible until (computed) time when all the evolutions instantly leave the environment.

Viable capturability Property:

It amounts to regulate "viable" evolutions in an environment defined by the viability constraints until they reach a target in the environment.

The viable capture basin of a target in an environment is the set of initial states from which starts at least one evolution viable in this environment until it reaches the target in finite time.

Capturability feedbacks govern evolutions viable until they reach the target when the initial state is the capture basin.

Viability algorithms compute feedbacks piloting viable evolutions:
- The longest time possible (feedback viability)
- Until they reach the target as quickly as possible (feedback capturability)

These viability and capturability properties arise naturally in some areas, but they are also pervasive in many other problems and when not directly perceptive, revealed through a specific mathematical analysis.

II - Evolutionary systems (dynamics under uncertainty)

(PNG)

This diagram represents an evolutionary input-output system consisting of two boxes, the first one, taking the control input and providing output state evolutions (figured below), which in turn feeds back on the control, compelling them to obey the constraints depending on the state.

(PNG)

The following diagram displays the mathematical translation of this example of evolutionary system.

(PNG)

III - Viability Kernels and Capture Basins

The first question is to know the set of the initial states from which starts at least one evolution viable in the blue environment.

(PNG)

This set is the viability kernel, in green on the following diagram.

(PNG)

The target (in red) is a subset of the environment (in blue). The capture basin of the target viable in the environment is the set (in pink) of initial states from which starts at least one evolution viable until it reaches the target in finite time.

(PNG)

Feedbacks associate to each control the only state used to maintain the evolution viable in the environment.

(PNG)

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