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

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1.1.6. Define the multimap
,

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

      Fig. 5

assuming F(x) = [e−x, 1] (Fig. 6):

      Fig. 6

2 → P(
2) assuming for x = (x1, x2) ∈
2:

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

      Fig. 7

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 Fr : X → P(Y) can be defined: Fr(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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