Multivalued Maps And Differential Inclusions: Elements Of Theory And Applications. Valeri Obukhovskii

Multivalued Maps And Differential Inclusions: Elements Of Theory And Applications

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1.1.6. Define the multimap

The graph of the multimap F is shown in the Fig. 5.

Fig. 5

Example 1.1.7. Define the multimap

assuming F(x) = [e^−x, 1] (Fig. 6):

Fig. 6

Example 1.1.8. Define the multimap F :

² → P(

²) assuming for x = (x₁, x₂) ∈

²:

The multimap F (but not the graph Γ_F!) is shown in the Fig. 7.

Fig. 7

Example 1.1.9 (Inverse functions). If X, Y are arbitrary sets and f : X → Y is a surjective map then the multimap F : Y → P(X), F(y) = {x|x ∈ X, f(x) = y} is the inverse to f.

Example 1.1.10 (Implicit functions). Let X, Y, Z be arbitrary sets, maps f : X × Y → Z and g : X → Z are such that for every x ∈ X there exists y ∈ Y such that f(x, y) = g(x). The implicit function defined by f and g, in a general case, is the multimap F : X → P(Y), F(x) = {y|y ∈ Y, f (x, y) = g(x)}.

Example 1.1.11. Let X, Y be arbitrary sets, f : X × Y →

a function. Let for a certain number r ∈

for every x ∈ X there exists y ∈ Y such that f(x, y) ≤ r. Then the following multimap F_r : X → P(Y) can be defined: F_r(x) = {y|y ∈ Y, f(x, y) ≤ r}.

Example 1.1.12. Generalized dynamical systems.

a) A multivalued translation operator.

Let a set X be the space of states of a certain dynamical system such that being at the initial moment in the state x ∈ X this system may move further along various trajectories. For example, such situation holds if the behavior of the system is governed by a differential equation which does not satisfy the uniqueness of a solution condition or contains a control parameter. A generalized dynamical system is defined if its reachable sets Q(x, t) ⊂ X are given, i.e., the sets of all states into which system can shift in the time t ≥ 0 from the state x ∈ X are indicated. The multimap Q : X ×

₊ → P(X) arising in such a manner is called the translation multioperator along the trajectories of the system. Notice that usually the translation multioperator satisfies the natural conditions:

1)Q(x, 0) = {x};

2)Q(x, t₁ + t₂) = Q(Q(x, t₁), t₂) for all x ∈ X;

b) Multivalued fields of directions.

Consider an important way of setting of a generalized dynamical system. Let

ⁿ be the state space of a system and for every state x ∈ Rⁿ the set F(x) ⊂ Rⁿ of velocities with which the system can leave x be given. The multimap F :

ⁿ → P(

ⁿ) which is defined in such a manner is called the multivalued field (multifield) of directions. A function x : Δ →

ⁿ, where Δ ⊂

is a certain interval is called an integral curve of the multifield F if at every (or almost every) point t ∈ Δ it has the derivative x′(t) and

for all (or almost all) t ∈ Δ. Such a relation is called a differential inclusion and the integral curve x is its solution.

A sulution x : Δ →

ⁿ is the trajectory of a given multifield of velocities. The collection Q(x, t) of points of such trajectories at the moment t emanating from a given point x ∈

ⁿ defines the translation multioperator Q along the trajectories of the multifield F.

Suppose, for example, that considered generalized dynamical system is a control system whose dynamics is governed by a differential equation

where f :

ⁿ ×

^m →

ⁿ is a map, u(t) ∈

^m a control parameter. The feedback in this system is described by a multimap U :

ⁿ → P(

^m) which defines for every given state x ∈

ⁿ a set of admissible controls U(x). Then the multifield

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