On the p-integrable trajectories of the nonlinear control system described by the Urysohn-type integral equation

Proposition 1

Every admissible control function u ∗ ( ⋅ ) ∈ U p , r generates a unique trajectory of system (1).

Proof

For given u ∗ ( ⋅ ) ∈ U p , r , let us define a map x ( ⋅ ) → F ( x ( ⋅ ) ) ∣ ( ⋅ ) , x ( ⋅ ) ∈ L p ( Ω ; R n ) , setting

First, let us show that F ( x ( ⋅ ) ) ∣ ( ⋅ ) ∈ L p ( Ω ; R n ) for every x ( ⋅ ) ∈ L p ( Ω ; R n ) . Let us fix x ( ⋅ ) ∈ L p ( Ω ; R n ) . Since x ( ⋅ ) and u ∗ ( ⋅ ) are Lebesgue measurable functions, then conditions A, B and Theorem 1.4.22 of [20] imply that the map ξ → F ( x ( ⋅ ) ) ∣ ( ξ ) , ξ ∈ Ω , is the Lebesgue measurable function.

Let x ( ⋅ ) ∈ L p ( Ω ; R n ) be fixed and ∥ x ( ⋅ ) ∥ p = β ∗ < + ∞ . Conditions A and B, inclusion u ∗ ( ⋅ ) ∈ U p , r , Hölder’s inequality, and (2) yield

for a.a. ξ ∈ Ω , and hence

for a.a. ξ ∈ Ω . Integrating the obtained inequality on the set Ω and taking into consideration (3), (4), and (5), we finally have

From the last inequality, we conclude that F ( x ( ⋅ ) ) ∣ ( ⋅ ) ∈ L p ( Ω ; R n ) . Now, let us show that the map F ( x ( ⋅ ) ) ∣ ( ⋅ ) : L p ( Ω ; R n ) → L p ( Ω ; R n ) is contractive one. Let us choose x 1 ( ⋅ ) ∈ L p ( Ω ; R n ) and x 2 ( ⋅ ) ∈ L p ( Ω ; R n ) . From conditions A and B, Hölder’s inequality, and (2), it follows that

and consequently,

for a.a. ξ ∈ Ω . Integrating the last inequality on the set Ω and taking into consideration (3), we obtain

According to the condition C, we have 2 p − 1 p [ α 0 p + α 1 p ] 1 p < 1 , and thus the map F ( x ( ⋅ ) ) ∣ ( ⋅ ) : L p ( Ω ; R n ) → L p ( Ω ; R n ) is contractive. According to the Banach fixed point theorem, this map has unique fixed point x ∗ ( ⋅ ) ∈ L p ( Ω ; R n ) , which is a unique function satisfying the equation

for a.a. ξ ∈ Ω .□

Proposition 2

For every x ( ⋅ ) ∈ X p , r , the inequality ∥ x ( ⋅ ) ∥ p ≤ σ ∗ is satisfied, where σ ∗ > 0 is defined by equality (6).

Proof

Choose an arbitrary x ( ⋅ ) ∈ X p , r generated by the control function u ( ⋅ ) ∈ U p , r . By virtue of conditions A and B, Hölder’s inequality, and the inclusion u ( ⋅ ) ∈ U p , r and (2), we have

for a.a. ξ ∈ Ω , and hence

for a.a. ξ ∈ Ω . Integrating the last inequality on the set Ω , we obtain

where α i , ( i = 1 , 2 ) , g 1 , and g 2 , are defined by (3)–(5), respectively. From the obtained inequality, condition C, and (6), we conclude that

Proposition 3

Let x 1 ( ⋅ ) ∈ X p , r and x 2 ( ⋅ ) ∈ X p , r be the trajectories of system (1) generated by the admissible control functions u 1 ( ⋅ ) ∈ U p , r and u 2 ( ⋅ ) ∈ U p , r , respectively. Then, the inequality

is held where k ∗ > 0 is defined by (7).

Proof

From conditions A and B and Hölder’s inequality, we obtain that

for a.a. ξ ∈ Ω , where α 0 is defined by (2). Raising to the power p and integrating the last inequality on the set Ω , we have

where α i , ( i = 1 , 2 ) , is defined by (3). Finally, condition C, (7), and (8) imply that

Definition 1

Let ( E , d ( ⋅ , ⋅ ) ) be a metric space. The set Q ⊂ E is said to be path-connected if for each x ∈ Q and y ∈ Q , there exists a continuous function ψ ( ⋅ ) : [ 0 , 1 ] → Q such that ψ ( 0 ) = x and ψ ( 1 ) = y .

Theorem 1

The set of trajectories X p , r of system (1) is a path-connected subset of the space L p ( Ω ; R n ) .

Proposition 4

Let g 1 ( ⋅ ) : [ a , b ] → R and g 2 ( ⋅ ) : [ a , b ] → R are continuous functions and g 1 ( ξ ) = g 2 ( ξ ) for a.a. ξ ∈ [ a , b ] . Then, g 1 ( ξ ) = g 2 ( ξ ) for every ξ ∈ [ a , b ] . In addition, if the functions g 1 ( ⋅ ) and g 2 ( ⋅ ) are differentiable for a.a. ξ ∈ [ 0 , 1 ] , then g 1 ′ ( ξ ) = g 2 ′ ( ξ ) for a.a. ξ ∈ [ a , b ] .

Example 1

Consider the control system described by the following system of integral equations:

where p > 1 , u ( ⋅ ) ∈ U p , 1 ∗ , U p , 1 ∗ = { u ( ⋅ ) ∈ L p ( [ 0 , 1 ] ; R ) : ∥ u ( ⋅ ) ∥ p ≤ 1 } , ξ ∈ [ 0 , 1 ] ,

The set of trajectories of system (9) generated by all admissible control functions u ( ⋅ ) ∈ U p , 1 ∗ is denoted by symbol X p , 1 ∗ .

First of all, let us show that X p , 1 ∗ is a bounded subset of the space L p ( [ 0 , 1 ] ; R ) . Let ( x 1 ( ⋅ ) , x 2 ( ⋅ ) ) ∈ X p , 1 ∗ be an arbitrarily chosen trajectory generated by admissible control function u ( ⋅ ) ∈ U p , 1 ∗ . Then, (9), (11), and the inclusion u ( ⋅ ) ∈ U p , 1 ∗ imply that

for a.a. ξ ∈ [ 0 , 1 ] , and consequently,

for a.a. ξ ∈ [ 0 , 1 ] . (12) and (13) yield that

Inequality (14) means that the set of trajectories X p , 1 ∗ is a bounded subset of the space L p ( [ 0 , 1 ] ; R ) . Now, let us show that the set of trajectories X p , 1 ∗ is not a closed subset of the space L p ( [ 0 , 1 ] ; R ) .

Choose a sequence of uniform partitions { Δ ( k ) } k = 1 ∞ of the closed interval [ 0 , 1 ] , where Δ ( k ) = 0 , 1 2 k , 2 2 k , … , 2 k − 1 2 k , 1 , k = 1 , 2 , … Let the functions u k ( ⋅ ) : [ 0 , 1 ] → R be defined as follows:

where i = 0 , 1 , … , k − 1 . It is not difficult to verify that u ( k ) ( ⋅ ) ∈ U p , 1 ∗ for every k = 1 , 2 , … . Let ( x 1 ( k ) ( ⋅ ) , x 2 ( k ) ( ⋅ ) ) ∈ X p , 1 ∗ be the trajectory of system (9) generated by admissible control function u ( k ) ( ⋅ ) ∈ U p , 1 ∗ . Then,

for a.a. ξ ∈ [ 0 , 1 ] . From (11), (15), and (16), it follows that

for a.a. ξ ∈ [ 0 , 1 ] . Setting x 1 ∗ ( ξ ) = ξ , x 2 ∗ ( ξ ) = 0 , and ξ ∈ [ 0 , 1 ] , from (17), we obtain

for a.a. ξ ∈ [ 0 , 1 ] , and hence

for every k = 1 , 2 , … . From (18), it follows that ( x 1 ( k ) ( ⋅ ) , x 2 ( k ) ( ⋅ ) ) → ( x 1 ∗ ( ⋅ ) , x 2 ∗ ( ⋅ ) ) as k → ∞ in the space L p ( [ 0 , 1 ] ; R ) , where ( x 1 ( k ) ( ⋅ ) , x 2 ( k ) ( ⋅ ) ) ∈ X p , 1 ∗ for every k = 1 , 2 , … Finally, let us show that ( x 1 ∗ ( ⋅ ) , x 2 ∗ ( ⋅ ) ) ∉ X p , 1 ∗ . Assume the contrary. Let ( x 1 ∗ ( ⋅ ) , x 2 ∗ ( ⋅ ) ) ∈ X p , 1 ∗ . Then, there exists u ∗ ( ⋅ ) ∈ U p , 1 ∗ such that

for a.a. ξ ∈ [ 0 , 1 ] . (11) and (19) yield that